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Contests
National and Regional Contests
Israel Contests
Israel National Olympiad
2024 Israel National Olympiad (Gillis)
2024 Israel National Olympiad (Gillis)
Part of
Israel National Olympiad
Subcontests
(7)
P7
1
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Minimal mine-avoiding rook path
A rook stands in one cell of an infinite square grid. A different cell was colored blue and mines were placed in
n
n
n
additional cells: the rook cannot stand on or pass through them. It is known that the rook can reach the blue cell in finitely many moves. Can it do so (for every
n
n
n
and such a choice of mines, starting point, and blue cell) in at most (a)
1.99
n
+
100
1.99n+100
1.99
n
+
100
moves? (b)
2
n
−
2
3
n
+
100
2n-2\sqrt{3n}+100
2
n
−
2
3
n
+
100
moves?Remark. In each move, the rook goes in a vertical or horizontal line.
P6
1
Hide problems
AA' and CC' are parallel in cyclic quad
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. Let
ω
A
\omega_A
ω
A
,
ω
B
\omega_B
ω
B
,
ω
C
\omega_C
ω
C
,
ω
D
\omega_D
ω
D
be the incircles of triangles
D
A
B
DAB
D
A
B
,
A
B
C
ABC
A
BC
,
B
C
D
BCD
BC
D
,
C
D
A
CDA
C
D
A
respectively. The common external common tangent of
ω
A
\omega_A
ω
A
,
ω
B
\omega_B
ω
B
, different from line
A
B
AB
A
B
, meets the external common tangent of
ω
A
\omega_A
ω
A
,
ω
D
\omega_D
ω
D
, different from
A
D
AD
A
D
, at point
A
′
A'
A
′
. Similarly, the external common tangent of
ω
B
\omega_B
ω
B
,
ω
C
\omega_C
ω
C
different from
B
C
BC
BC
meets the external common tangent of
ω
C
\omega_C
ω
C
,
ω
D
\omega_D
ω
D
different from
C
D
CD
C
D
at
C
′
C'
C
′
. Prove that
A
A
′
∥
C
C
′
AA'\parallel CC'
A
A
′
∥
C
C
′
.
P5
1
Hide problems
Minimal prime is bounded
For positive integral
k
>
1
k>1
k
>
1
, we let
p
(
k
)
p(k)
p
(
k
)
be its smallest prime divisor. Given an integer
a
1
>
2
a_1>2
a
1
>
2
, we define an infinite sequence
a
n
a_n
a
n
by
a
n
+
1
=
a
n
n
−
1
a_{n+1}=a_n^n-1
a
n
+
1
=
a
n
n
−
1
for each
n
≥
1
n\geq 1
n
≥
1
. For which values of
a
1
a_1
a
1
is the sequence
p
(
a
n
)
p(a_n)
p
(
a
n
)
bounded?
P4
1
Hide problems
Lines parallel to UV, UW, VW concurrent
Acute triangle
A
B
C
ABC
A
BC
is inscribed in a circle with center
O
O
O
. The reflections of
O
O
O
across the three altitudes of the triangle are called
U
U
U
,
V
V
V
,
W
W
W
:
U
U
U
over the altitude from
A
A
A
,
V
V
V
over the altitude from
B
B
B
, and
W
W
W
over the altitude from
C
C
C
. Let
ℓ
A
\ell_A
ℓ
A
be a line through
A
A
A
parallel to
V
W
VW
VW
, and define
ℓ
B
\ell_B
ℓ
B
,
ℓ
C
\ell_C
ℓ
C
similarly. Prove that the three lines
ℓ
A
\ell_A
ℓ
A
,
ℓ
B
\ell_B
ℓ
B
,
ℓ
C
\ell_C
ℓ
C
are concurrent.
P3
1
Hide problems
Minimal number of diagonal pairs
A triangle is composed of circular cells arranged in
5784
5784
5784
rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called diagonal if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division? An example division into pairs is depicted in the image.
P2
1
Hide problems
Sum of fractional parts
A positive integer
x
x
x
satisfies the following:
{
x
3
}
+
{
x
5
}
+
{
x
7
}
+
{
x
11
}
=
248
165
\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}
{
3
x
}
+
{
5
x
}
+
{
7
x
}
+
{
11
x
}
=
165
248
Find all possible values of
{
2
x
3
}
+
{
2
x
5
}
+
{
2
x
7
}
+
{
2
x
11
}
\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}
{
3
2
x
}
+
{
5
2
x
}
+
{
7
2
x
}
+
{
11
2
x
}
where
{
y
}
\{y\}
{
y
}
denotes the fractional part of
y
y
y
.
P1
1
Hide problems
Symmetric system of specifications
Solve the following system (over the real numbers):
{
5
x
+
5
y
+
5
x
y
−
2
x
y
2
−
2
x
2
y
=
20
3
x
+
3
y
+
3
x
y
+
x
y
2
+
x
2
y
=
23
\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\ 3x+3y+3xy+xy^2+x^2y=23&\end{cases}
{
5
x
+
5
y
+
5
x
y
−
2
x
y
2
−
2
x
2
y
=
20
3
x
+
3
y
+
3
x
y
+
x
y
2
+
x
2
y
=
23