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Problems
Contests
International Contests
Tuymaada Olympiad
1997 Tuymaada Olympiad
1997 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
1
1
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4 numbers, product of 3 is perfect square, prove all 4 are perfect squares
The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.
8
1
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find a right triangle that can be cut into 365 equal triangles
Find a right triangle that can be cut into
365
365
365
equal triangles.
7
1
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boys and girls visit the rink
It is known that every student of the class for Sunday once visited the rink, and every boy met there with every girl. Prove that there was a point in time when all the boys, or all the girls of the class were simultaneously on the rink.
6
1
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14 consecutive positive integers, each with one divisor <=11 other than 1
Are there
14
14
14
consecutive positive integers, each of which has a divisor other than
1
1
1
and not exceeding
11
11
11
?
5
1
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(1+1/q)(1+1/q^2)...(1+1/q^n)< (q-1)/(q-2)
Prove the inequality
(
1
+
1
q
)
(
1
+
1
q
2
)
.
.
.
(
1
+
1
q
n
)
<
q
−
1
q
−
2
\left(1+\frac{1}{q}\right)\left(1+\frac{1}{q^2}\right)...\left(1+\frac{1}{q^n}\right)<\frac{q-1}{q-2}
(
1
+
q
1
)
(
1
+
q
2
1
)
...
(
1
+
q
n
1
)
<
q
−
2
q
−
1
for
n
∈
N
,
q
>
2
n\in N, q>2
n
∈
N
,
q
>
2
2
1
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system in N: 3x^2+6y^2+5z^2=1997, 3x+6y+5z=161
Solve in natural numbers the system of equations
3
x
2
+
6
y
2
+
5
z
2
=
1997
3x^2+6y^2+5z^2=1997
3
x
2
+
6
y
2
+
5
z
2
=
1997
and
3
x
+
6
y
+
5
z
=
161
3x+6y+5z=161
3
x
+
6
y
+
5
z
=
161
.
3
1
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coloring numbers in 6 colors, sum of any 5 of different colors to be in 6th one
Is it possible to paint all natural numbers in
6
6
6
colors, for each one color to be used and the sum of any five numbers of different color to be painted in the sixth color?
4
1
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given angle \pi /7 and a ruler, construct angle \pi /14
Using only angle with angle
π
7
\frac{\pi}{7}
7
π
and a ruler, constuct angle
π
14
\frac{\pi}{14}
14
π