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Ukraine Contests
Random Geometry Problems from Ukrainian Contests
Ukrainian TYM Qualifying - geometry
Ukrainian TYM Qualifying - geometry
Part of
Random Geometry Problems from Ukrainian Contests
Subcontests
(83)
2019.17
1
Hide problems
2 player game, moving a chips along vertices of a regular n-gon
n
n
n
points are marked on the board points that are vertices of the regular
n
n
n
-gon. One of the points is a chip. Two players take turns moving it to the other marked point and at the same time draw a segment that connects them. If two points already connected by a segment, such a move is prohibited. A player who can't make a move, lose. Which of the players can guarantee victory?
2017.5
1
Hide problems
fixed area of triangle from Fibonacci lattice points
The Fibonacci sequence is given by equalities
F
1
=
F
2
=
1
,
F
k
+
2
=
F
k
+
F
k
+
1
,
k
∈
N
F_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N
F
1
=
F
2
=
1
,
F
k
+
2
=
F
k
+
F
k
+
1
,
k
∈
N
. a) Prove that for every
m
≥
0
m \ge 0
m
≥
0
, the area of the triangle
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
with vertices
A
1
(
F
m
+
1
,
F
m
+
2
)
A_1(F_{m+1},F_{m+2})
A
1
(
F
m
+
1
,
F
m
+
2
)
,
A
2
(
F
m
+
3
,
F
m
+
4
)
A_2 (F_{m+3},F_{m+4})
A
2
(
F
m
+
3
,
F
m
+
4
)
,
A
3
(
F
m
+
5
,
F
m
+
6
)
A_3 (F_{m+5},F_{m+6})
A
3
(
F
m
+
5
,
F
m
+
6
)
is equal to
0.5
0.5
0.5
. b) Prove that for every
m
≥
0
m \ge 0
m
≥
0
the quadrangle
A
1
A
2
A
4
A_1A_2A_4
A
1
A
2
A
4
with vertices
A
1
(
F
m
+
1
,
F
m
+
2
)
A_1(F_{m+1},F_{m+2})
A
1
(
F
m
+
1
,
F
m
+
2
)
,
A
2
(
F
m
+
3
,
F
m
+
4
)
A_2 (F_{m+3},F_{m+4})
A
2
(
F
m
+
3
,
F
m
+
4
)
,
A
3
(
F
m
+
5
,
F
m
+
6
)
A_3 (F_{m+5},F_{m+6})
A
3
(
F
m
+
5
,
F
m
+
6
)
,
A
4
(
F
m
+
7
,
F
m
+
8
)
A_4 (F_{m+7},F_{m+8})
A
4
(
F
m
+
7
,
F
m
+
8
)
is a trapezoid, whose area is equal to
2.5
2.5
2.5
. c) Prove that the area of the polygon
A
1
A
2
.
.
.
A
n
A_1A_2...A_n
A
1
A
2
...
A
n
,
n
≥
3
n \ge3
n
≥
3
with vertices does not depend on the choice of numbers
m
≥
0
m \ge 0
m
≥
0
, and find this area.
2015.18
1
Hide problems
several equal parts of a circle by 3 chords, not diameters
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
2014.1
1
Hide problems
angles =? (a-c)(a+c)^2+bc(a+c)=ab^2 and 48^o
In the triangle
A
B
C
ABC
A
BC
, one of the angles of which is equal to
4
8
o
48^o
4
8
o
, side lengths satisfy
(
a
−
c
)
(
a
+
c
)
2
+
b
c
(
a
+
c
)
=
a
b
2
(a-c)(a+c)^2+bc(a+c)=ab^2
(
a
−
c
)
(
a
+
c
)
2
+
b
c
(
a
+
c
)
=
a
b
2
. Express in degrees the measures of the other two angles of this triangle.
XII.17
1
Hide problems
3(A_1B_1C_1) <= S , max k_1, mion k_2 so that k_1 S<=(A_1B_1C_1) <= k_2 S
Given a triangle
A
B
C
ABC
A
BC
, inside which the point
M
M
M
is marked. On the sides
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
the following points
A
1
,
B
1
A_1,B_1
A
1
,
B
1
and
C
1
C_1
C
1
are chosen, respectively, that
M
A
1
∥
C
A
MA_1 \parallel CA
M
A
1
∥
C
A
,
M
B
1
∥
A
B
MB_1 \parallel AB
M
B
1
∥
A
B
,
M
C
1
∥
B
C
MC_1 \parallel BC
M
C
1
∥
BC
. Let S be the area of triangle
A
B
C
,
Q
M
ABC, Q_M
A
BC
,
Q
M
be the area of the triangle
A
1
B
1
C
1
A_1 B_1 C_1
A
1
B
1
C
1
. a) Prove that if the triangle
A
B
C
ABC
A
BC
is acute, and M is the point of intersection of its altitudes , then
3
Q
M
≤
S
3Q_M \le S
3
Q
M
≤
S
. Is there such a number
k
>
0
k> 0
k
>
0
that for any acute-angled triangle
A
B
C
ABC
A
BC
and the point
M
M
M
of intersection of its altitudes, such thatthe inequality
Q
M
>
k
S
Q_M> k S
Q
M
>
k
S
holds? b) For cases where the point
M
M
M
is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest
k
1
>
0
k_1> 0
k
1
>
0
and the smallest
k
2
>
0
k_2> 0
k
2
>
0
such that for an arbitrary triangle
A
B
C
ABC
A
BC
, holds the inequality
k
1
S
≤
Q
M
≤
k
2
S
k_1S \le Q_M\le k_2S
k
1
S
≤
Q
M
≤
k
2
S
(for the center of the circumscribed circle, only acute-angled triangles
A
B
C
ABC
A
BC
are considered).
XII.15
1
Hide problems
<ASB > <BSC cos \delta + <CSA cos \phi
Given a triangular pyramid
S
A
B
C
SABC
S
A
BC
, in which
∠
B
S
C
=
α
\angle BSC = \alpha
∠
BSC
=
α
,
∠
C
S
A
=
β
\angle CSA =\beta
∠
CS
A
=
β
,
∠
A
S
B
=
γ
\angle ASB = \gamma
∠
A
SB
=
γ
, and the dihedral angles at the edges
S
A
SA
S
A
and
S
B
SB
SB
have the value of
ϕ
\phi
ϕ
and
δ
\delta
δ
, respectively. Prove that
γ
>
α
⋅
cos
δ
+
β
⋅
cos
ϕ
.
\gamma > \alpha \cdot \cos \delta +\beta \cdot \cos \phi.
γ
>
α
⋅
cos
δ
+
β
⋅
cos
ϕ
.
$
XII.2
1
Hide problems
circumcenter construction, ruler only, given circumcircle, incenter
The figure shows a triangle, a circle circumscribed around it and the center of its inscribed circle. Using only one ruler (one-sided, without divisions), construct the center of the circumscribed circle.
XI.15
1
Hide problems
IW_1+ IW_2 + IW_3>= 2R + \sqrt{2Rr} , incenter, circumcircle
Let
I
I
I
be the point of intersection of the angle bisectors of the
△
A
B
C
\vartriangle ABC
△
A
BC
,
W
1
,
W
2
,
W
3
W_1,W_2,W_3
W
1
,
W
2
,
W
3
be point of intersection of lines
A
I
,
B
I
,
C
I
AI, BI, CI
A
I
,
B
I
,
C
I
with the circle circumscribed around the triangle,
r
r
r
and
R
R
R
be the radii of inscribed and circumscribed circles respectively. Prove the inequality
I
W
1
+
I
W
2
+
I
W
3
≥
2
R
+
2
R
r
.
IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}
I
W
1
+
I
W
2
+
I
W
3
≥
2
R
+
2
R
r
.
XI.13
1
Hide problems
locus of points from which cylindrical towers are visible at same angle
On the plane there are two cylindrical towers with radii of bases
r
r
r
and
R
R
R
. Find the set of all those points of the plane from which these towers are visible at the same angle. Consider the case of more towers.
XI.6
1
Hide problems
AQ, BN,CT equal and concurrent, centroids, AK^2-BC^2=BK^2-AC^ =CK^2-AB^2
Prove that there exists a point
K
K
K
in the plane of
△
A
B
C
\vartriangle ABC
△
A
BC
such that
A
K
2
−
B
C
2
=
B
K
2
−
A
C
2
=
C
K
2
−
A
B
2
.
AK^2 - BC^2 = BK^2 - AC^2 = CK^2 - AB^2.
A
K
2
−
B
C
2
=
B
K
2
−
A
C
2
=
C
K
2
−
A
B
2
.
Let
Q
,
N
,
T
Q, N, T
Q
,
N
,
T
be the points of intersection of the medians of the triangles
B
K
C
,
C
K
A
,
A
K
B
BKC, CKA, AKB
B
K
C
,
C
K
A
,
A
K
B
, respectively. Prove that the segments
A
Q
,
B
N
AQ, BN
A
Q
,
BN
and
C
T
CT
CT
are equal and have a common point.
XI.4
1
Hide problems
conctruction of point F on arc AB , PE/EQ = m/n
Chords
A
B
AB
A
B
and
C
D
CD
C
D
, which do not intersect, are drawn in a circle. On the chord
A
B
AB
A
B
or on its extension is taken the point
E
E
E
. Using a compass and construct the point
F
F
F
on the arc
A
B
AB
A
B
, such that
P
E
E
Q
=
m
n
\frac{PE}{EQ} = \frac{m}{n}
EQ
PE
=
n
m
, where
m
,
n
m,n
m
,
n
are given natural numbers,
P
P
P
is the point of intersection of the chord
A
B
AB
A
B
with the chord
F
C
FC
FC
,
Q
Q
Q
is the point of intersection of the chord
A
B
AB
A
B
with the chord
F
D
FD
F
D
. Consider cases where
E
∈
P
Q
E\in PQ
E
∈
PQ
and
E
∉
P
Q
E \notin PQ
E
∈
/
PQ
.
X.12
1
Hide problems
nP >= 4d in convex polygon where d=\sum |\overrightarrow {A_kM}|,
Inside the convex polygon
A
1
A
2
.
.
.
A
n
A_1A_2...A_n
A
1
A
2
...
A
n
, there is a point
M
M
M
such that
∑
k
=
1
n
A
k
M
→
=
0
→
\sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}
∑
k
=
1
n
A
k
M
=
0
. Prove that
n
P
≥
4
d
nP\ge 4d
n
P
≥
4
d
, where
P
P
P
is the perimeter of the polygon, and
d
=
∑
k
=
1
n
∣
A
k
M
→
∣
d=\sum_{k=1}^n |\overrightarrow {A_kM}|
d
=
∑
k
=
1
n
∣
A
k
M
∣
. Investigate the question of the achievement of equality in this inequality.
X.13
1
Hide problems
max circle inscribed in non-convex hexagon by folding a square
A paper square is bent along the line
ℓ
\ell
ℓ
, which passes through its center, so that a non-convex hexagon is formed. Investigate the question of the circle of largest radius that can be placed in such a hexagon.
IX.12
1
Hide problems
area of section of cube by plane equal to area of section of tetrahedron ...
Let
A
B
,
A
C
AB,AC
A
B
,
A
C
and
A
D
AD
A
D
be the edges of a cube,
A
B
=
α
AB=\alpha
A
B
=
α
. Point
E
E
E
was marked on the ray
A
C
AC
A
C
so that
A
E
=
λ
α
AE=\lambda \alpha
A
E
=
λ
α
, and point
F
F
F
was marked on the ray
A
D
AD
A
D
so that
A
F
=
μ
α
AF=\mu \alpha
A
F
=
μα
(
μ
>
0
,
λ
>
0
\mu> 0, \lambda >0
μ
>
0
,
λ
>
0
). Find (characterize) pairs of numbers
λ
\lambda
λ
and
μ
\mu
μ
such that the cross-sectional area of a cube by any plane parallel to the plane
B
C
D
BCD
BC
D
is equal to the cross-sectional area of the tetrahedron
A
B
E
F
ABEF
A
BEF
by the same plane.
VIII.2
1
Hide problems
AD/ sin \alpha=BD / sin \beta =CD/ sin \gamma in tetrahedron
Investigate the properties of the tetrahedron
A
B
C
D
ABCD
A
BC
D
for which there is equality
A
D
sin
α
=
B
D
sin
β
=
C
D
sin
γ
\frac{AD}{ \sin \alpha}=\frac{BD}{\sin \beta}=\frac{CD}{ \sin \gamma}
sin
α
A
D
=
sin
β
B
D
=
sin
γ
C
D
where
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
are the values of the dihedral angles at the edges
A
D
,
B
D
AD, BD
A
D
,
B
D
and
C
D
CD
C
D
, respectively.
VIII.3
1
Hide problems
max of p/R ( 1- r/3R )
Find the largest value of the expression
p
R
(
1
−
r
3
R
)
\frac{p}{R}\left( 1- \frac{r}{3R}\right)
R
p
(
1
−
3
R
r
)
, where
p
,
R
,
r
p,R, r
p
,
R
,
r
is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.
VII.12
1
Hide problems
\pi/ 3 <= (\alpha a +\beta b + \gamma c)/(a+b+c) < \pi}/2
Let
a
,
b
a, b
a
,
b
, and
c
c
c
be the lengths of the sides of an arbitrary triangle, and let
α
,
β
\alpha,\beta
α
,
β
, and
γ
\gamma
γ
be the radian measures of its corresponding angles. Prove that
π
3
≤
α
a
+
β
b
+
γ
c
a
+
b
+
c
<
π
2
.
\frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.
3
π
≤
a
+
b
+
c
α
a
+
β
b
+
γ
c
<
2
π
.
Suggest spatial analogues of this inequality.
VI.18
1
Hide problems
convex k-gon of largest area inscribed in convex n-gon
The convex polygon
A
1
A
2
.
.
.
A
n
A_1A_2...A_n
A
1
A
2
...
A
n
is given in the plane. Denote by
T
k
T_k
T
k
(
k
≤
n
)
(k \le n)
(
k
≤
n
)
the convex
k
k
k
-gon of the largest area, with vertices at the points
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2, ..., A_n
A
1
,
A
2
,
...
,
A
n
and by
T
k
(
A
+
1
)
T_k(A+1)
T
k
(
A
+
1
)
the convex k-gon of the largest area with vertices at the points
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2, ..., A_n
A
1
,
A
2
,
...
,
A
n
in which one of the vertices is in
A
1
A_1
A
1
. Set the relationship between the order of arrangement in the sequence
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2, ..., A_n
A
1
,
A
2
,
...
,
A
n
vertices: 1)
T
3
T_3
T
3
and
T
3
(
A
2
)
T_3 (A_2)
T
3
(
A
2
)
2)
T
k
T_k
T
k
and
T
k
(
A
1
)
T_k (A_1)
T
k
(
A
1
)
3)
T
k
T_k
T
k
and
T
k
+
1
T_{k+1}
T
k
+
1
VI.2
1
Hide problems
inscribed equilateral, midpoints, tangents, intersections
Let
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
be the midpoints of the sides of the
B
C
,
A
C
,
A
B
BC,AC, AB
BC
,
A
C
,
A
B
of an equilateral triangle
A
B
C
ABC
A
BC
. Around the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
is a circle
γ
\gamma
γ
, to which the tangents
B
2
C
2
B_2C_2
B
2
C
2
,
A
2
C
2
A_2C_2
A
2
C
2
,
A
2
B
2
A_2B_2
A
2
B
2
are drawn, respectively, parallel to the sides
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
. These tangents have no points in common with the interior of triangle
A
B
C
ABC
A
BC
. Find out the mutual location of the points of intersection of the lines
A
A
2
AA_2
A
A
2
and
B
B
2
BB_2
B
B
2
,
A
A
2
AA_2
A
A
2
and
C
C
2
CC_2
C
C
2
,
B
B
2
BB_2
B
B
2
and
C
C
2
CC_2
C
C
2
and the circumscribed circle
γ
\gamma
γ
. Try to consider the case of arbitrary points
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
located on the sides of the triangle
A
B
C
ABC
A
BC
.
VI.14
1
Hide problems
P<= ( r+\sqrt{r^2+4R^2})/2 for bicentric quadr.
A quadrilateral whose perimeter is equal to
P
P
P
is inscribed in a circle of radius
R
R
R
and is circumscribed around a circle of radius
r
r
r
. Check whether the inequality
P
≤
r
+
r
2
+
4
R
2
2
P\le \frac{r+\sqrt{r^2+4R^2}}{2}
P
≤
2
r
+
r
2
+
4
R
2
holds. Try to find the corresponding inequalities for the
n
n
n
-gon (
n
≥
5
n \ge 5
n
≥
5
) inscribed in a circle of radius
R
R
R
and circumscribed around a circle of radius
r
r
r
.
VI.9
1
Hide problems
chord of a polygon questions, area bisector, area>=1/3 of area of polygon
Consider an arbitrary (optional convex) polygon. It's chord is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than
1
3
\frac13
3
1
the area of the polygon?
VI.1
1
Hide problems
nonconvex quadrilaterals with fixed sum of distances
Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.
V.3
1
Hide problems
geometric miniatures from V All-Ukrainian TYM
Fix the triangle
A
B
C
ABC
A
BC
on the plane. 1. Denote by
S
L
,
S
M
S_L,S_M
S
L
,
S
M
and
S
K
S_K
S
K
the areas of triangles whose vertices are, respectively, the bases of bisectors, medians and points of tangency of the inscribed circle of a given triangle
A
B
C
ABC
A
BC
. Prove that
S
K
≤
S
L
≤
S
M
S_K\le S_L\le S_M
S
K
≤
S
L
≤
S
M
.2. For the point
X
X
X
, which is inside the triangle
A
B
C
ABC
A
BC
, consider the triangle
T
X
T_X
T
X
, the vertices of which are the points of intersection of the lines
A
X
,
B
X
,
C
X
AX, BX, CX
A
X
,
BX
,
CX
with the lines
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
, respectively. 2.1. Find the position of the point
X
X
X
for which the area of the triangle
T
x
T_x
T
x
is the largest possible. 2.2. Suggest an effective criterion for comparing the areas of triangles
T
x
T_x
T
x
for different positions of the point
X
X
X
. 2.3. Find the positions of the point
X
X
X
for which the perimeter of the triangle
T
x
T_x
T
x
is the smallest possible and the largest possible. 2.4. Propose an effective criterion for comparing the perimeters of triangles
T
x
T_x
T
x
for different positions of point
X
X
X
. 2.5. Suggest and solve similar problems with respect to the extreme values of other parameters (for example, the radius of the circumscribed circle, the length of the greatest height) of triangles
T
x
T_x
T
x
.3. For the point
Y
Y
Y
, which is inside the circle
ω
\omega
ω
, circumscribed around the triangle
A
B
C
ABC
A
BC
, consider the triangle
Δ
Y
\Delta_Y
Δ
Y
, the vertices of which are the points of intersection
A
Y
,
B
X
,
C
X
AY, BX, CX
A
Y
,
BX
,
CX
with the circle
ω
\omega
ω
. Suggest and solve similar problems for triangles
Δ
Y
\Delta_Y
Δ
Y
for different positions of point
Y
Y
Y
.4. Suggest and solve similar problems for convex polygons.5. For the point
Z
Z
Z
, which is inside the circle
ω
\omega
ω
, circumscribed around the triangle
A
B
C
ABC
A
BC
, consider the triangle
F
Z
F_Z
F
Z
, the vertices of which are orthogonal projections of the point
Z
Z
Z
on the lines
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
. Suggest and solve similar problems for triangles
F
Z
F_Z
F
Z
for different positions of the point
Z
Z
Z
.
V.8
1
Hide problems
3\sqrt3(cot AXC/2+cot AXB/2 )+ cot AXC/2+cot AXB/2>5 , BX+CX <3 AX
Let
X
X
X
be a point inside an equilateral triangle
A
B
C
ABC
A
BC
such that
B
X
+
C
X
<
3
A
X
BX+CX <3 AX
BX
+
CX
<
3
A
X
. Prove that
3
3
(
cot
∠
A
X
C
2
+
cot
∠
A
X
B
2
)
+
cot
∠
A
X
C
2
cot
∠
A
X
B
2
>
5
3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5
3
3
(
cot
2
∠
A
XC
+
cot
2
∠
A
XB
)
+
cot
2
∠
A
XC
cot
2
∠
A
XB
>
5
II.1
1
Hide problems
equal spheres in a cylinder (II All-Ukrainian Tourn.of Young Mathem. Final p1)
Inside a right cylinder with a radius of the base
R
R
R
are placed
k
k
k
(
k
≥
3
k\ge 3
k
≥
3
) of equal balls, each of which touches the side surface and the lower base of the cylinder and, in addition, exactly two other balls. After that, another equal ball is placed inside the cylinder so that it touches the upper base of the cylinder and all other balls. Find the volume
V
(
R
,
k
)
V (R, k)
V
(
R
,
k
)
of the cylinder.
III.11
1
Hide problems
broken line inside a regular polygon, spiral, equal angles, projections
A circle centered at point
O
O
O
is separated by points
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
on
n
n
n
equal parts (points are listed sequentially clockwise) and the rays
O
A
1
,
O
A
2
,
.
.
.
,
O
A
n
OA_1,OA_2,...,OA_n
O
A
1
,
O
A
2
,
...
,
O
A
n
are constructed. The angle
A
2
O
A
3
A_2OA_3
A
2
O
A
3
is divided by rays into two equal angles at vertex
O
O
O
, the angle
A
3
O
A
4
A_3OA_4
A
3
O
A
4
is divided into three equal angles, and so on, finally, the angle
A
n
O
A
1
A_nOA_1
A
n
O
A
1
divided into
n
n
n
equal angles at vertex
O
O
O
. A point belonging to the ray other than
O
A
1
OA_1
O
A
1
, is connected by a segment with its orthogonal projection
B
0
B_0
B
0
on the neighboring (clockwise) arrow) with ray
O
A
1
OA_1
O
A
1
, point
B
1
B_1
B
1
is connected by a segment with its orthogonal projection on the next (clockwise) ray, etc. As a result of such process it turns out the broken line
B
0
B
1
B
2
B
3
.
.
.
B_0B_1B_2B_3...
B
0
B
1
B
2
B
3
...
infinitely "twists". Consider the question of giving the thus obtained broken numerical value of "length"
L
(
n
)
L (n)
L
(
n
)
and explore the value of
L
(
n
)
L(n)
L
(
n
)
depending on
n
n
n
.
III.13
1
Hide problems
sequence of tangent circles inside regular polygon
Inside the regular
n
n
n
-gon
M
M
M
with side
a
a
a
there are
n
n
n
equal circles so that each touches two adjacent sides of the polygon
M
M
M
and two other circles. Inside the formed "star", which is bounded by arcs, these
n
n
n
equal circles are reconstructed so that each touches the two adjacent circles built in the previous step, and two more newly built circles. This process will take
k
k
k
steps. Find the area
S
n
(
k
)
S_n (k)
S
n
(
k
)
of the "star", which is formed in the center of the polygon
M
M
M
. Consider the spatial analogue of this problem.
II.2
1
Hide problems
are congruent triangles the faces of tetrahedron with same area?
Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?
II.16
1
Hide problems
3 symmetric points after finite no of steps lie outside a circle
Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?
II.18
1
Hide problems
tangent costruction, circle inside a circle, bisect segment
Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.
IV.8
1
Hide problems
diagonal in convex hexagon cuts off area <=1/6 area of hexagon
Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed
1
6
\frac16
6
1
of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than
1
6
\frac16
6
1
the area of the hexagon?
IV.11
1
Hide problems
S^2_{ABM}=CM^2/CD^2 S^2_{ABD}+(1- CM^2/CD^2)S_{ABC}, tetrahedron
In the tetrahedron
A
B
C
D
ABCD
A
BC
D
, the point
E
E
E
is the projection of the point
D
D
D
on the plane
(
A
B
C
)
(ABC)
(
A
BC
)
. Prove that the following statements are equivalent: a)
C
=
E
C = E
C
=
E
or
C
E
∥
A
B
CE \parallel AB
CE
∥
A
B
b) For each point M belonging to the segment
C
D
CD
C
D
, the following equation is satisfied
S
△
A
B
M
2
=
C
M
2
C
D
2
⋅
S
△
A
B
D
2
+
(
1
−
C
M
2
C
D
2
)
S
△
A
B
C
2
S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}
S
△
A
BM
2
=
C
D
2
C
M
2
⋅
S
△
A
B
D
2
+
(
1
−
C
D
2
C
M
2
)
S
△
A
BC
2
where
S
△
X
Y
Z
S_{\vartriangle XYZ}
S
△
X
Y
Z
means the area of triangle
X
Y
Z
XYZ
X
Y
Z
.
IV.10
1
Hide problems
2pr/R <=DE + EF + DF <= p, touchpoints with incircle
Given a triangle
A
B
C
ABC
A
BC
and points
D
,
E
,
F
D, E, F
D
,
E
,
F
, which are points of contact of the inscribed circle to the sides of the triangle.i) Prove that
2
p
r
R
≤
D
E
+
E
F
+
D
F
≤
p
\frac{2pr}{R} \le DE + EF + DF \le p
R
2
p
r
≤
D
E
+
EF
+
D
F
≤
p
(
p
p
p
is the semiperimeter,
r
r
r
and
R
R
R
are respectively the radius of the inscribed and circumscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
).ii). Find out when equality is achieved.
IV.7
1
Hide problems
PORS inscribed in ABCD with max area given sides, where PQRS has min perimeter
Let
A
B
C
D
ABCD
A
BC
D
be the quadrilateral whose area is the largest among the quadrilaterals with given sides
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, and let
P
O
R
S
PORS
PORS
be the quadrilateral inscribed in
A
B
C
D
ABCD
A
BC
D
with the smallest perimeter. Find this perimeter.
2010.16
1
Hide problems
regular tetrahedron if ABxACxADxBCxBDxCD= 512/27, A,B,C,D on sphere radius=1
Points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie on the sphere of radius
1
1
1
. It is known that
A
B
⋅
A
C
⋅
A
D
⋅
B
C
⋅
B
D
⋅
C
D
=
512
27
AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}
A
B
⋅
A
C
⋅
A
D
⋅
BC
⋅
B
D
⋅
C
D
=
27
512
. Prove that
A
B
C
D
ABCD
A
BC
D
is a regular tetrahedron.
2010.15
1
Hide problems
centroids coincide, right triangles on side of a triangle externally
On the sides of the triangle
A
B
C
ABC
A
BC
externally constructed right triangles
A
B
C
1
ABC_1
A
B
C
1
,
B
C
A
1
BCA_1
BC
A
1
,
C
A
B
1
CAB_1
C
A
B
1
. Prove that the points of intersection of the medians of the triangles
A
B
C
ABC
A
BC
and
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
coincide.
2010.12
1
Hide problems
tangent construction using only ruler at given point
On the plane is drawn a triangle
A
B
C
ABC
A
BC
and a circle
ω
\omega
ω
passing through the vertex
C
C
C
, the midpoints of the sides
A
C
AC
A
C
and
B
C
BC
BC
and the point of intersection of the medians of the triangle
A
B
C
ABC
A
BC
. The point
K
K
K
lies on the circle
ω
\omega
ω
such that
∠
A
K
B
=
9
0
o
\angle AKB=90^o
∠
A
K
B
=
9
0
o
. Using only with a ruler, draw a tangent to the circle
ω
\omega
ω
at point
K
K
K
.
2010.3
1
Hide problems
measuring distance of a river bank using thumbes and fixed points
The following method of approximate measurement is known for distances. Suppose, for example, that the observer is on the river bank at point
C
C
C
in order to measure its width. To do this, he fixes point
A
A
A
on the opposite bank so that the angle between the shoreline and the line
C
A
CA
C
A
is close to the line. Then the observer pulls forward the right hand with the raised thumb, closes left eye and aligns the raised finger with point
A
A
A
. Next, opens the left eye, closes right and estimates the distance between the point on the opposite bank to which the finger points, and point
A
A
A
. Multiply this distance by
10
10
10
and get the approximate value of the distance to point
A
A
A
, ie the width of the river. Justify this method of measuring distance.[hide=original wording]Відомий наступний спосіб наближеного вимірювання відстані. Нехай, наприклад, спостерігач знаходиться на березі річки у точці C і має на меті виміряти її ширину. Для цього він фіксує точку A на протилежному березі так, щоб кут між лінією берега і прямою CA був близьким до прямого. Потім спостерігач витягує вперед праву руку з піднятим вгору великим пальцем, заплющує ліве око і суміщає піднятий палець з точкою A. Далі, відкриває ліве око, заплющує праве і оцінює відстань між точкою на протилежному березі, на яку вказує палець, і точкою A. Цю відстань множить на 10 і отримує наближене значення відстані до точки A, тобто ширини річки. Обґрунтуйте цей спосіб вимірювання відстані.
2010.6
1
Hide problems
geometric and harmonic mean of distances of sides of triangle, max values
Find inside the triangle
A
B
C
ABC
A
BC
, points
G
G
G
and
H
H
H
for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment
G
H
GH
G
H
parallel to one of the sides of the triangle? Find the length of such a segment
G
H
GH
G
H
.
2011.2
1
Hide problems
computational, 8 equal circles inside a right triangle
Eight circles of radius
r
r
r
located in a right triangle
A
B
C
ABC
A
BC
(angle
C
C
C
is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle
A
B
C
ABC
A
BC
. https://cdn.artofproblemsolving.com/attachments/4/7/1b1cd7d6bc7f5004b8e94468d723ed16e9a039.png
2011.5
1
Hide problems
locus of circumcenters, circles tangent to fixed line intersecting fixed circle
The circle
ω
0
\omega_0
ω
0
touches the line at point A. Let
R
R
R
be a given positive number. We consider various circles
ω
\omega
ω
of radius
R
R
R
that touch a line
ℓ
\ell
ℓ
and have two different points in common with the circle
ω
0
\omega_0
ω
0
. Let
D
D
D
be the touchpoint of the circle
ω
0
\omega_0
ω
0
with the line
ℓ
\ell
ℓ
, and the points of intersection of the circles
ω
\omega
ω
and
ω
0
\omega_0
ω
0
are denoted by
B
B
B
and
C
C
C
(Assume that the distance from point
B
B
B
to the line
ℓ
\ell
ℓ
is greater than the distance from point
C
C
C
to this line). Find the locus of the centers of the circumscribed circles of all such triangles
A
B
D
ABD
A
B
D
.
2011.11
1
Hide problems
collinear wanted, tangents to circumcircle related
Let
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
be the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
, which intersect its angle bisector
A
L
AL
A
L
at two different points
P
P
P
and
Q
Q
Q
, respectively. Denote by
F
F
F
such a point that
P
F
∥
A
B
PF\parallel AB
PF
∥
A
B
and
Q
F
∥
A
C
QF\parallel AC
QF
∥
A
C
, and by
T
T
T
the intersection point of the tangents drawn at points
B
B
B
and
C
C
C
to the circumscribed circle of the triangle
A
B
C
ABC
A
BC
. Prove that the points
A
,
F
A, F
A
,
F
and
T
T
T
lie on the same line.
2011.14
1
Hide problems
concurrent and concyclic wanted, symmtetrics wrt lines, cyclic ABCD
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
, inscribed in a circle
ω
\omega
ω
such that
A
B
=
A
D
AB=AD
A
B
=
A
D
and
C
B
=
C
D
CB=CD
CB
=
C
D
. Take the point
P
∈
ω
P \in \omega
P
∈
ω
. Let the vertices of the quadrilateral
Q
1
Q
2
Q
3
Q
4
Q_1Q_2Q_3Q_4
Q
1
Q
2
Q
3
Q
4
be symmetric to the point P wrt the lines
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
, and
D
A
DA
D
A
, respectively. a) Prove that the points symmetric to the point
P
P
P
wrt lines
Q
1
Q
2
2
,
Q
2
Q
3
,
Q
3
Q
4
Q_1Q_22, Q_2Q_3, Q_3Q_4
Q
1
Q
2
2
,
Q
2
Q
3
,
Q
3
Q
4
and
Q
4
Q
1
Q_4Q_1
Q
4
Q
1
, lie on one line. b) Prove that when the point
P
P
P
moves in a circle
ω
\omega
ω
, then all such lines pass through one common point.
I.17
1
Hide problems
rotating a right trangle around a large leg to form a cone
A right triangle when rotating around a large leg forms a cone with a volume of
100
π
100\pi
100
π
. Calculate the length of the path that passes through each vertex of the triangle at rotation of
18
0
o
180^o
18
0
o
around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to
17
17
17
.
I.9
1
Hide problems
sum of distances >= 4S^2/ 9R^2
Prove that for any interior point of a triangle the sum of squares of distances from it to the sides of a triangle is not less than
4
S
2
9
R
2
\frac{4S^2}{9R^2}
9
R
2
4
S
2
. [hide=about S,R]they are not defined, but I suppose they mean it's area and circumradii respectively
I.8
1
Hide problems
area chasing in a triangle, 2 given ratios
One of the sides of the triangle is divided by the ratio
p
:
q
p: q
p
:
q
, and the other by
m
:
n
:
k
m: n: k
m
:
n
:
k
. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of this triangle to the area of the quadrilateral formed by three such lines and one of the sides of the triangle.
I.10
1
Hide problems
ratio of areas of max circumscribed to min inscribed triangle in a given circle
Given a circle of radius
R
R
R
. Find the ratio of the largest area of the circumscribed quadrilateral to the smallest area of the inscribed one.
I.13
1
Hide problems
reflections a man can see in dihedral mirror angle , a candle
A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?
I.7
1
Hide problems
ratio of areas, n-gon inscirbed in n-gon
Given a natural number
n
>
3
n> 3
n
>
3
. On the plane are considered convex
n
n
n
- gons
F
1
F_1
F
1
and
F
2
F_2
F
2
such that on each side of
F
1
F_1
F
1
lies one vertex of
F
2
F_2
F
2
and no two vertices
F
1
F_1
F
1
and
F
2
F_2
F
2
coincide. For each
n
n
n
, determine the limits of the ratio of the areas of the polygons
F
1
F_1
F
1
and
F
2
F_2
F
2
. For each
n
n
n
, determine the range of the areas of the polygons
F
1
F_1
F
1
and
F
2
F_2
F
2
, if
F
1
F_1
F
1
is a regular
n
n
n
-gon. Determine the set of values of this value for other partial cases of the polygon
F
1
F_1
F
1
.
I.5
1
Hide problems
all flat angles in pyramid at any vertex are either acute, or right, or obtuse
The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.
2012.11
1
Hide problems
3 concurrent incircles, inside a square
Let
E
E
E
be an arbitrary point on the side
B
C
BC
BC
of the square
A
B
C
D
ABCD
A
BC
D
. Prove that the inscribed circles of triangles
A
B
E
ABE
A
BE
,
C
D
E
CDE
C
D
E
,
A
D
E
ADE
A
D
E
have a common tangent.
2012.2
1
Hide problems
triangle construction, AB + AC = 2BC, feet of angle bisectors
The triangle
A
B
C
ABC
A
BC
is drawn on the board such that
A
B
+
A
C
=
2
B
C
AB + AC = 2BC
A
B
+
A
C
=
2
BC
. The bisectors
A
L
1
,
B
L
2
,
C
L
3
AL_1, BL_2, CL_3
A
L
1
,
B
L
2
,
C
L
3
were drawn in this triangle, after which everything except the points
L
1
,
L
2
,
L
3
L_1, L_2, L_3
L
1
,
L
2
,
L
3
was erased. Use a compass and a ruler to reconstruct triangle
A
B
C
ABC
A
BC
.
2013.15
1
Hide problems
concurrent wanted, 4 right angles givne
Inside the acute-angled triangle
A
B
C
ABC
A
BC
, mark the point
O
O
O
so that
∠
A
O
B
=
9
0
o
\angle AOB=90^o
∠
A
OB
=
9
0
o
, a point
M
M
M
on the side
B
C
BC
BC
such that
∠
C
O
M
=
9
0
o
\angle COM=90^o
∠
COM
=
9
0
o
, and a point
N
N
N
on the segment
B
O
BO
BO
such that
∠
O
M
N
=
9
0
o
\angle OMN = 90^o
∠
OMN
=
9
0
o
. Let
P
P
P
be the point of intersection of the lines
A
M
AM
A
M
and
C
N
CN
CN
, and let
Q
Q
Q
be a point on the side
A
B
AB
A
B
that such
∠
P
O
Q
=
9
0
o
\angle POQ = 90^o
∠
POQ
=
9
0
o
. Prove that the lines
A
N
,
C
O
AN, CO
A
N
,
CO
and
M
Q
MQ
MQ
intersect at one point.
2013.17
1
Hide problems
concurrency in tetrahedron iff
Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .
2013.9
1
Hide problems
many points concyclic wanted, AB^2 + AC^2 = 2BC^2
Given a triangle
P
Q
R
PQR
PQR
, the inscribed circle
ω
\omega
ω
which touches the sides
Q
R
,
R
P
QR, RP
QR
,
RP
and
P
Q
PQ
PQ
at points
A
,
B
A, B
A
,
B
and
C
C
C
, respectively, and
A
B
2
+
A
C
2
=
2
B
C
2
AB^2 + AC^2 = 2BC^2
A
B
2
+
A
C
2
=
2
B
C
2
. Prove that the point of intersection of the segments
P
A
,
Q
B
PA, QB
P
A
,
QB
and
R
C
RC
RC
, the center of the circle
ω
\omega
ω
, the point of intersection of the medians of the triangle
A
B
C
ABC
A
BC
, the point
A
A
A
and the midpoints of the segments
A
C
AC
A
C
and
A
B
AB
A
B
lie on one circle.
2013.6
1
Hide problems
locus wanted, altitudes and circumcircle realted
Given a circle
ω
\omega
ω
, on which marks the points
A
,
B
,
C
A,B,C
A
,
B
,
C
. Let
B
F
BF
BF
and
C
E
CE
CE
be the altitudes of the triangle
A
B
C
ABC
A
BC
,
M
M
M
be the midpoint of the side
A
C
AC
A
C
. Find a the locus of the intersection points of the lines
B
F
BF
BF
and E
M
M
M
for all positions of point
A
A
A
, as
A
A
A
moves on
ω
\omega
ω
.
2014.23
1
Hide problems
tangent circles wanted, incircle and 3 circumcircles related
The inscribed circle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
with center
I
I
I
touches the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
at points
C
1
,
A
1
,
B
1
C_1, A_1, B_1
C
1
,
A
1
,
B
1
. The circle circumsrcibed around
△
A
B
1
C
1
\vartriangle AB_1C_1
△
A
B
1
C
1
intersects the circumscribed circle of
A
B
C
ABC
A
BC
for second time at the point
K
K
K
. Let
M
M
M
be the midpoint
B
C
BC
BC
,
L
L
L
be the midpoint of
B
1
C
1
B_1C_1
B
1
C
1
. The circle circumsrcibed around
△
K
A
1
M
\vartriangle KA_1M
△
K
A
1
M
cuts intersects
ω
\omega
ω
for second time at the point
T
T
T
. Prove that the circumscribed circles of triangles
K
L
T
KLT
K
L
T
and
L
I
M
LIM
L
I
M
are tangent.
2014.22
1
Hide problems
orthocenter of ABC is incenter of XYZ, symmetric wrt lines, altitudes, midpoints
In
△
A
B
C
\vartriangle ABC
△
A
BC
on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
mark feet of altitudes
H
1
,
H
2
,
H
3
H_1, H_2, H_3
H
1
,
H
2
,
H
3
and the midpoint of sides
M
1
,
M
3
,
M
3
M_1, M_3, M_3
M
1
,
M
3
,
M
3
. Let
H
H
H
be orthocenter
△
A
B
C
\vartriangle ABC
△
A
BC
. Suppose that
X
2
,
X
3
X_2, X_3
X
2
,
X
3
are points symmetric to
H
1
H_1
H
1
wrt
B
H
2
BH_2
B
H
2
and
C
H
3
CH_3
C
H
3
. Lines
M
3
X
2
M_3X_2
M
3
X
2
and
M
2
X
3
M_2X_3
M
2
X
3
intersect at point
X
X
X
. Similarly,
Y
3
,
Y
1
Y_3,Y_1
Y
3
,
Y
1
are points symmetric to
H
2
H_2
H
2
wrt
C
3
H
C_3H
C
3
H
and
A
H
1
AH_1
A
H
1
.Lines
M
1
Y
3
M_1Y_3
M
1
Y
3
and
M
3
Y
1
M_3Y_1
M
3
Y
1
intersect at point
Y
.
Y.
Y
.
Finally,
Z
1
,
Z
2
Z_1,Z_2
Z
1
,
Z
2
are points symmetric to
H
3
H_3
H
3
wrt
A
H
1
AH_1
A
H
1
and
B
H
2
BH_2
B
H
2
. Lines
M
1
Z
2
M_1Z_2
M
1
Z
2
and
M
2
Z
1
M_2Z_1
M
2
Z
1
intersect at the point
Z
Z
Z
Prove that
H
H
H
is the incenter
△
X
Y
Z
\vartriangle XYZ
△
X
Y
Z
.
2014.9
1
Hide problems
construct point Q on ABC at least 2 of CQ, BQ, AQ bisect incircle
Construct a point
Q
Q
Q
in triangle
A
B
C
ABC
A
BC
such that at least two of the segments
C
Q
,
B
Q
,
A
Q
CQ, BQ, AQ
CQ
,
BQ
,
A
Q
, divide the inscribed circle in half. For which triangles is this possible?
2014.8
1
Hide problems
equal angles wanted and given, right angle for point on median
In the triangle
A
B
C
ABC
A
BC
on the ray
B
A
BA
B
A
mark the point
K
K
K
so that
∠
B
C
A
=
∠
K
C
A
\angle BCA= \angle KCA
∠
BC
A
=
∠
K
C
A
, and on the median
B
M
BM
BM
mark the point
T
T
T
so that
∠
C
T
K
=
9
0
o
\angle CTK=90^o
∠
CT
K
=
9
0
o
. Prove that
∠
M
T
C
=
∠
M
C
B
\angle MTC=\angle MCB
∠
MTC
=
∠
MCB
.
2015.24
1
Hide problems
points X', Y' ,Z' coincide wanted, D'X'= AX, E'Y' = BY, F'Z' = CZ, incircle
The inscribed circle
ω
\omega
ω
of the triangle
A
B
C
ABC
A
BC
touches its sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
at the points
D
,
E
D, E
D
,
E
, and
F
F
F
, respectively. Let the points
X
,
Y
X, Y
X
,
Y
, and
Z
Z
Z
of the circle
ω
\omega
ω
be diametrically opposite to the points
D
,
E
D, E
D
,
E
, and
F
F
F
, respectively. Line
A
X
,
B
Y
AX, BY
A
X
,
B
Y
and
C
Z
CZ
CZ
intersect the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
at the points
D
′
,
E
′
D', E'
D
′
,
E
′
and
F
′
F'
F
′
, respectively. On the segments
A
D
′
,
B
E
′
AD', BE'
A
D
′
,
B
E
′
and
C
F
′
CF'
C
F
′
noted the points
X
′
,
Y
′
X', Y'
X
′
,
Y
′
and
Z
′
Z'
Z
′
, respectively, so that
D
′
X
′
=
A
X
D'X'= AX
D
′
X
′
=
A
X
,
E
′
Y
′
=
B
Y
E'Y' = BY
E
′
Y
′
=
B
Y
,
F
′
Z
′
=
C
Z
F'Z' = CZ
F
′
Z
′
=
CZ
. Prove that the points
X
′
,
Y
′
X', Y'
X
′
,
Y
′
and
Z
′
Z'
Z
′
coincide.
2015.23
1
Hide problems
locus of points X_P lies to some two lines, orthocenter related
An acute-angled triangle
A
B
C
ABC
A
BC
is given, through the vertices
B
B
B
and
C
C
C
of which a circle
Ω
\Omega
Ω
,
A
∉
Ω
A \notin \Omega
A
∈
/
Ω
, is drawn. We consider all points
P
∈
Ω
P \in \Omega
P
∈
Ω
, that do not lie on none of the lines
A
B
AB
A
B
and
A
C
AC
A
C
and for which the common tangents of the circumscribed circles of triangles
A
P
B
APB
A
PB
and
A
P
C
APC
A
PC
are not parallel. Let
X
P
X_P
X
P
be the point of intersection of such two common tangents. a) Prove that the locus of points
X
P
X_P
X
P
lies to some two lines. b) Prove that if the circle
Ω
\Omega
Ω
passes through the orthocenter of the triangle
A
B
C
ABC
A
BC
, then one of these lines is the line
B
C
BC
BC
.
2015.22
1
Hide problems
criterion for a convex polygon to be cyclic if \alpha_i/ sin \beta_i = constant
Let
A
1
A
2
.
.
.
A
2
n
+
1
A_1A_2... A_{2n + 1}
A
1
A
2
...
A
2
n
+
1
be a convex polygon,
a
1
=
A
1
A
2
a_1 = A_1A_2
a
1
=
A
1
A
2
,
a
2
=
A
2
A
3
a_2 = A_2A_3
a
2
=
A
2
A
3
,
.
.
.
...
...
,
a
2
n
=
A
2
n
A
2
n
+
1
a_{2n} = A_{2n}A_{2n + 1}
a
2
n
=
A
2
n
A
2
n
+
1
,
a
2
n
+
1
=
A
2
n
+
1
A
1
a_{2n + 1} = A_{2n + 1}A_1
a
2
n
+
1
=
A
2
n
+
1
A
1
. Denote by:
α
i
=
∠
A
i
\alpha_i = \angle A_i
α
i
=
∠
A
i
,
1
≤
i
≤
2
n
+
1
1 \le i \le 2n + 1
1
≤
i
≤
2
n
+
1
,
α
k
+
2
n
+
1
=
α
k
\alpha_{k + 2n + 1} = \alpha_k
α
k
+
2
n
+
1
=
α
k
,
k
≥
1
k \ge 1
k
≥
1
,
β
i
=
α
i
+
2
+
α
i
+
4
+
.
.
.
+
α
i
+
2
n
\beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}
β
i
=
α
i
+
2
+
α
i
+
4
+
...
+
α
i
+
2
n
,
1
≤
i
≤
2
n
+
1
1 \le i \le 2n + 1
1
≤
i
≤
2
n
+
1
. Prove what if
α
1
sin
β
1
=
α
2
sin
β
2
=
.
.
.
=
α
2
n
+
1
sin
β
2
n
+
1
\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}
sin
β
1
α
1
=
sin
β
2
α
2
=
...
=
sin
β
2
n
+
1
α
2
n
+
1
then a circle can be circumscribed around this polygon. Does the inverse statement hold a place?
2015.21
1
Hide problems
right triangle construction, given 3 points
Let
C
H
CH
C
H
be the altitude of the triangle
A
B
C
ABC
A
BC
drawn on the board, in which
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
,
C
A
≠
C
B
CA \ne CB
C
A
=
CB
. The mathematics teacher drew the perpendicular bisectors of segments
C
A
CA
C
A
and
C
B
CB
CB
, which cut the line CH at points
K
K
K
and
M
M
M
, respectively, and then erased the drawing, leaving only the points
C
,
K
C, K
C
,
K
and
M
M
M
on the board. Restore triangle
A
B
C
ABC
A
BC
, using only a compass and a ruler.
2015.20
1
Hide problems
max ratio of largest side to inradii in triangle
What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?
2016.15
1
Hide problems
<BFC = 60^o wanted in triangle with <A=120^o
A non isosceles triangle
A
B
C
ABC
A
BC
is given, in which
∠
A
=
12
0
o
\angle A = 120^o
∠
A
=
12
0
o
. Let
A
L
AL
A
L
be its angle bisector,
A
K
AK
A
K
be it's median, drawn from vertex
A
A
A
, point
O
O
O
be the center of the circumcircle of this triangle,
F
F
F
be the point of intersection of the lines
O
L
OL
O
L
and
A
K
AK
A
K
. Prove that
∠
B
F
C
=
6
0
o
\angle BFC = 60^o
∠
BFC
=
6
0
o
.
2016.14
1
Hide problems
triangle construction given centroid, incenter, touchpoint of incircle with side
Using only a compass and a ruler, reconstruct triangle
A
B
C
ABC
A
BC
given the following three points: point
M
M
M
the intersection of its medians, point
I
I
I
is the center of its inscribed circle and the point
Q
a
Q_a
Q
a
is touch point of the inscribed circle to side
B
C
BC
BC
.
2016.1
1
Hide problems
ortocenters of BCK,ACN coincide iff BN/AK =/tan^2 A, right ABC
The points
K
K
K
and
N
N
N
lie on the hypotenuse
A
B
AB
A
B
of a right triangle
A
B
C
ABC
A
BC
. Prove that orthocenters the triangles
B
C
K
BCK
BC
K
and
A
C
N
ACN
A
CN
coincide if and only if
B
N
A
K
=
tan
2
A
.
\frac{BN}{AK}=\tan^2 A.
A
K
BN
=
tan
2
A
.
2017.4
1
Hide problems
integer right triangles with M inside such that MA, MB, MC integers also
Specify at least one right triangle
A
B
C
ABC
A
BC
with integer sides, inside which you can specify a point
M
M
M
such that the lengths of the segments
M
A
,
M
B
,
M
C
MA, MB, MC
M
A
,
MB
,
MC
are integers. Are there many such triangles, none of which are are similar?
2017.3
1
Hide problems
KQ=QH and fixed circle wanted, altitudes, 2 common tangents to circles
The altitude
A
H
,
B
T
AH, BT
A
H
,
BT
, and
C
R
CR
CR
are drawn in the non isosceles triangle
A
B
C
ABC
A
BC
. On the side
B
C
BC
BC
mark the point
P
P
P
; points
X
X
X
and
Y
Y
Y
are projections of
P
P
P
on
A
B
AB
A
B
and
A
C
AC
A
C
. Two common external tangents to the circumscribed circles of triangles
X
B
H
XBH
XB
H
and
H
C
Y
HCY
H
C
Y
intersect at point
Q
Q
Q
. The lines
R
T
RT
RT
and
B
C
BC
BC
intersect at point
K
K
K
. a). Prove that the point
Q
Q
Q
lies on a fixed line independent of choice
P
P
P
. b). Prove that
K
Q
=
Q
H
KQ = QH
K
Q
=
Q
H
.
2017.2
1
Hide problems
AX, BY, CZ concurrent iff AP, BQ, CR concurrent , 3 circles related
Points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
were marked on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Let
a
a
a
be tangent at point
A
A
A
to the circumcircle of triangle
A
Q
R
AQR
A
QR
,
b
b
b
be tangent at point
B
B
B
to the circumcircle of the triangle BPR,
c
c
c
be tangent at point
C
C
C
to the circumscribed circle triangle
C
P
Q
CPQ
CPQ
. Let
X
X
X
be the point of intersection of the lines
b
b
b
and
c
,
Y
c, Y
c
,
Y
be the point the intersection of lines
c
c
c
and
a
,
Z
a, Z
a
,
Z
is the point of intersection of lines
a
a
a
and
b
b
b
. Prove that the lines
A
X
,
B
Y
,
C
Z
AX, BY, CZ
A
X
,
B
Y
,
CZ
intersect at one point if and only if the lines
A
P
,
B
Q
,
C
R
AP, BQ, CR
A
P
,
BQ
,
CR
intersect at one point.
2017.1
1
Hide problems
isosceles trapezoid construction given 3 semgents, equal inradii
In an isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
D
AD
A
D
and
B
C
BC
BC
, diagonals intersect at point
P
P
P
, and lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at point
Q
Q
Q
.
O
1
O_1
O
1
and
O
2
O_2
O
2
are the centers of the circles circumscribed around the triangles
A
B
P
ABP
A
BP
and
C
D
P
CDP
C
D
P
,
r
r
r
is the radius of these circles. Construct the trapezoid ABCD given the segments
O
1
O
2
O_1O_2
O
1
O
2
,
P
Q
PQ
PQ
and radius
r
r
r
.
2018.18
1
Hide problems
point lies on midline, circles having AB,BH,CH,AC as diameters
In the acute triangle
A
B
C
ABC
A
BC
, the altitude
A
H
AH
A
H
is drawn. Using segments
A
B
,
B
H
,
C
H
AB,BH,CH
A
B
,
B
H
,
C
H
and
A
C
AC
A
C
as diameters circles
ω
1
,
ω
2
,
ω
3
\omega_1, \omega_2, \omega_3
ω
1
,
ω
2
,
ω
3
and
ω
4
\omega_4
ω
4
are constructed respectively. Besides the point
H
H
H
, the circles
ω
1
\omega_1
ω
1
and
ω
3
\omega_3
ω
3
intersect at the point
P
,
P,
P
,
and the circles
ω
2
\omega_2
ω
2
and
ω
4
\omega_4
ω
4
interext at point
Q
Q
Q
. The lines
B
Q
BQ
BQ
and
C
P
CP
CP
intersect at point
N
N
N
. Prove that this point lies on the midline of triangle
A
B
C
ABC
A
BC
, which is parallel to
B
C
BC
BC
.
2018.17
1
Hide problems
triangle construction, given b,c and AI where I incenter
Using a compass and a ruler, construct a triangle
A
B
C
ABC
A
BC
given the sides
b
,
c
b, c
b
,
c
and the segment
A
I
AI
A
I
, where
I
I
I
is the center of the inscribed circle of this triangle.
2018.16
1
Hide problems
triangle construction, given 2 rays and 1 point, incircle, excircle related
Let
K
,
T
K, T
K
,
T
be the points of tangency of inscribed and exscribed circles to the side
B
C
BC
BC
triangle
A
B
C
ABC
A
BC
,
M
M
M
is the midpoint of the side
B
C
BC
BC
. Using a compass and a ruler, construct triangle ABC given rays
A
K
AK
A
K
and
A
T
AT
A
T
(points
K
,
T
K, T
K
,
T
are not marked on them) and point
M
M
M
.
2019.8
1
Hide problems
3 isosceles triangles, same base, angles of integer degrees, BE=AC
Hannusya, Petrus and Mykolka drew independently one isosceles triangle
A
B
C
ABC
A
BC
, all angles of which are measured as a integer number of degrees. It turned out that the bases
A
C
AC
A
C
of these triangles are equals and for each of them on the ray
B
C
BC
BC
there is a point
E
E
E
such that
B
E
=
A
C
BE=AC
BE
=
A
C
, and the angle
A
E
C
AEC
A
EC
is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?
2019.11
1
Hide problems
incenter lies on a line, 3 excircles related
Let
ω
a
,
ω
b
,
ω
c
\omega_a, \omega_b, \omega_c
ω
a
,
ω
b
,
ω
c
be the exscribed circles tangent to the sides
a
,
b
,
c
a, b, c
a
,
b
,
c
of a triangle
A
B
C
ABC
A
BC
, respectively,
I
a
,
I
b
,
I
c
I_a, I_b, I_c
I
a
,
I
b
,
I
c
be the centers of these circles, respectively,
T
a
,
T
b
,
T
c
T_a, T_b, T_c
T
a
,
T
b
,
T
c
be the points of contact of these circles to the line
B
C
BC
BC
, respectively. The lines
T
b
I
c
T_bI_c
T
b
I
c
and
T
c
I
b
T_cI_b
T
c
I
b
intersect at the point
Q
Q
Q
. Prove that the center of the circle inscribed in triangle
A
B
C
ABC
A
BC
lies on the line
T
a
Q
T_aQ
T
a
Q
.
2019.10
1
Hide problems
orthocenter wanted, projections of point on altitude on sides
At the altitude
A
H
1
AH_1
A
H
1
of an acute non-isosceles triangle
A
B
C
ABC
A
BC
chose a point
X
X
X
, from which draw the perpendiculars
X
N
XN
XN
and
X
M
XM
XM
on the sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively. It turned out that
H
1
A
H_1A
H
1
A
is the angle bisector
M
H
1
N
MH_1N
M
H
1
N
. Prove that
X
X
X
is the point of intersection of the altitudes of the triangle
A
B
C
ABC
A
BC
.
2019.9
1
Hide problems
isosceles triangle construction given 2 incircles
On the base
B
C
BC
BC
of the isosceles triangle
A
B
C
ABC
A
BC
chose a point
D
D
D
and in each of the triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers the apex
A
A
A
is located . Use a compass and ruler to restore the triangle
A
B
C
ABC
A
BC
, if we know that : a)
A
D
AD
A
D
is angle bisector, b)
A
D
AD
A
D
is median.
2020.13
1
Hide problems
concyclic wanted with center on the line connecting excenters
In the triangle
A
B
C
ABC
A
BC
on the side
B
C
BC
BC
, the points
D
D
D
and
E
E
E
are chosen so that the angle
B
A
D
BAD
B
A
D
is equal to the angle
E
A
C
EAC
E
A
C
. Let
I
I
I
and
J
J
J
be the centers of the inscribed circles of triangles
A
B
D
ABD
A
B
D
and
A
E
C
AEC
A
EC
respectively,
F
F
F
be the point of intersection of
B
I
BI
B
I
and
E
J
EJ
E
J
,
G
G
G
be the point of intersection of
D
I
DI
D
I
and
C
J
CJ
C
J
. Prove that the points
I
,
J
,
F
,
G
I, J, F, G
I
,
J
,
F
,
G
lie on one circle, the center of which belongs to the line
I
b
I
c
I_bI_c
I
b
I
c
, where
I
b
I_b
I
b
and
I
c
I_c
I
c
are the centers of the exscribed circles of the triangle
A
B
C
ABC
A
BC
, which touch respectively sides
A
C
AC
A
C
and
A
B
AB
A
B
.
2020.12
1
Hide problems
concyclic and equal circles wanted, 3 squares related
On the side
C
D
CD
C
D
of the square
A
B
C
D
ABCD
A
BC
D
, the point
F
F
F
is chosen and the equal squares
D
G
F
E
DGFE
D
GFE
and
A
K
E
H
AKEH
A
K
E
H
are constructed (
E
E
E
and
H
H
H
lie inside the square). Let
M
M
M
be the midpoint of
D
F
DF
D
F
,
J
J
J
is the incenter of the triangle
C
F
H
CFH
CF
H
. Prove that: a) the points
D
,
K
,
H
,
J
,
F
D, K, H, J, F
D
,
K
,
H
,
J
,
F
lie on the same circle; b) the circles inscribed in triangles
C
F
H
CFH
CF
H
and
G
M
F
GMF
GMF
have the same radii.
2020.11
1
Hide problems
XY_|_PO iff AP is angle bisector
In the acute-angled triangle
A
B
C
ABC
A
BC
, the segment
A
P
AP
A
P
was drawn and the center was marked
O
O
O
of the circumscribed circle. The circumcircle of triangle
A
B
P
ABP
A
BP
intersects the line
A
C
AC
A
C
for the second time at point
X
X
X
, the circumcircle of the triangle
A
C
P
ACP
A
CP
intersects the line
A
B
AB
A
B
for the second time at the point
Y
Y
Y
. Prove that the lines
X
Y
XY
X
Y
and
P
O
PO
PO
are perpendicular if and only if
P
P
P
is the foor of the bisector of the triangle
A
B
C
ABC
A
BC
.
2020.10
1
Hide problems
X_1X_2/X_3X_4=Y_1Y_2/Y_3Y_4 if X_1,X_2,Y_1,Y_2 concyclic, excenter related
In triangle
A
B
C
ABC
A
BC
, point
I
I
I
is the center, point
I
a
I_a
I
a
is the center of the excircle tangent to the side
B
C
BC
BC
. From the vertex
A
A
A
inside the angle
B
A
C
BAC
B
A
C
draw rays
A
X
AX
A
X
and
A
Y
AY
A
Y
. The ray
A
X
AX
A
X
intersects the lines
B
I
BI
B
I
,
C
I
CI
C
I
,
B
I
a
BI_a
B
I
a
,
C
I
a
CI_a
C
I
a
at points
X
1
X_1
X
1
,
.
.
.
...
...
,
X
4
X_4
X
4
, respectively, and the ray
A
Y
AY
A
Y
intersects the same lines at points
Y
1
Y_1
Y
1
,
.
.
.
...
...
,
Y
4
Y_4
Y
4
respectively. It turned out that the points
X
1
,
X
2
,
Y
1
,
Y
2
X_1,X_2,Y_1,Y_2
X
1
,
X
2
,
Y
1
,
Y
2
lie on the same circle. Prove the equality
X
1
X
2
X
3
X
4
=
Y
1
Y
2
Y
3
Y
4
.
\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.
X
3
X
4
X
1
X
2
=
Y
3
Y
4
Y
1
Y
2
.