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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1984 Spain Mathematical Olympiad
1984 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
3
1
Hide problems
if p,q>0 and p+q = 1 show (p+1/p)^2+ (q+q)^2 >=25/2, also 2 inequalities
If
p
p
p
and
q
q
q
are positive numbers with
p
+
q
=
1
p+q = 1
p
+
q
=
1
, knowing that any real numbers
x
,
y
x,y
x
,
y
satisfy
(
x
−
y
)
2
≥
0
(x-y)^2 \ge 0
(
x
−
y
)
2
≥
0
, show that
x
+
y
2
≥
x
y
\frac{x+y}{2} \ge \sqrt{xy}
2
x
+
y
≥
x
y
,
x
2
+
y
2
2
≥
(
x
+
y
2
)
2
\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2
2
x
2
+
y
2
≥
(
2
x
+
y
)
2
,
(
p
+
1
p
)
2
+
(
q
+
1
q
)
2
≥
25
2
\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}
(
p
+
p
1
)
2
+
(
q
+
q
1
)
2
≥
2
25
8
1
Hide problems
find the remainder upon division by x^2-1 of the 4x4 determinant ...
Find the remainder upon division by
x
2
−
1
x^2-1
x
2
−
1
of the determinant
∣
x
3
+
3
x
2
1
0
x
2
+
5
x
3
0
2
x
4
+
x
2
+
1
2
1
3
x
5
+
1
1
2
3
∣
\begin{vmatrix} x^3+3x & 2 & 1 & 0 \\ x^2+5x & 3 & 0 & 2 \\x^4 +x^2+1 & 2 & 1 & 3 \\x^5 +1 & 1 & 2 & 3 \\ \end{vmatrix}
x
3
+
3
x
x
2
+
5
x
x
4
+
x
2
+
1
x
5
+
1
2
3
2
1
1
0
1
2
0
2
3
3
7
1
Hide problems
erasing first digits in natural numbers written in the decimal system
Consider the natural numbers written in the decimal system. (a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have? (b) Prove that there is no number that decreases
12
12
12
times when its first digit is erased. (c) Find the necessary and sufficient condition on
k
k
k
for the existence of a natural number which is divided by
k
k
k
when its first digit is erased.
6
1
Hide problems
locus in analytic geometry
Consider the circle
γ
\gamma
γ
with center at point
(
0
,
3
)
(0,3)
(
0
,
3
)
and radius
3
3
3
, and a line
r
r
r
parallel to the axis
O
x
Ox
O
x
at a distance
3
3
3
from the origin. A variable line through the origin meets
γ
\gamma
γ
at point
M
M
M
and
r
r
r
at point
P
P
P
. Find the locus of the intersection point of the lines through
M
M
M
and
P
P
P
parallel to
O
x
Ox
O
x
and
O
y
Oy
O
y
respectively.
5
1
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locus of midpoints of arcs of equal length in equal circles, 1 fixed point each
Let
A
A
A
and
A
′
A'
A
′
be fixed points on two equal circles in the plane and let
A
B
AB
A
B
and
A
′
B
′
A' B'
A
′
B
′
be arcs of these circles of the same length
x
x
x
. Find the locus of the midpoint of segment
B
B
′
BB'
B
B
′
when
x
x
x
varies: (a) if the arcs have the same direction, (b) if the arcs have opposite directions.
4
1
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limit of product of cos\frac{x}{2^k} from k=1 to n
Evaluate
lim
n
→
∞
c
o
s
x
2
c
o
s
x
2
2
c
o
s
x
2
3
.
.
.
c
o
s
x
2
n
\lim_{n\to \infty} cos\frac{x}{2}cos\frac{x}{2^2} cos\frac{x}{2^3}...cos\frac{x}{2^n}
lim
n
→
∞
cos
2
x
cos
2
2
x
cos
2
3
x
...
cos
2
n
x
2
1
Hide problems
No of 5-digit numbers whose square ends in the same 5 digits in same order
Find the number of five-digit numbers whose square ends in the same five digits in the same order.
1
1
Hide problems
determine, by ruler and compass, the reach of the gun against two tanks ...
At a position
O
O
O
of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments
A
B
AB
A
B
and
C
D
CD
C
D
attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length
ℓ
\ell
ℓ
.