Olympiad problems

1972 IMO Shortlist, 7

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

1972 IMO Shortlist, 12

Tags: combinatorics , pigeonhole principle , IMO , IMO 1972

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

1972 IMO, 1

Tags: algebra , inequality system , IMO , IMO 1972

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1972 IMO, 1

Tags: combinatorics , pigeonhole principle , IMO , IMO 1972

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

1972 IMO Shortlist, 8

Tags: factorial , number theory , binomial coefficients , recurrence relation , IMO , IMO 1972

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

1972 IMO Longlists, 7

Tags: function , algebra , functional equation , Functional inequality , IMO , IMO 1972

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

1972 IMO Shortlist, 9

Tags: algebra , inequality system , IMO , IMO 1972

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1972 IMO Longlists, 44

Tags: combinatorics , pigeonhole principle , IMO , IMO 1972

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

1972 IMO Shortlist, 10

Tags: geometry , trapezoid , dissection , cyclic quadrilateral , IMO , IMO 1972

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

1972 IMO, 3

Tags: geometry , 3D geometry , tetrahedron , IMO , IMO 1972

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

1972 IMO Shortlist, 1

Tags: function , algebra , functional equation , Functional inequality , IMO , IMO 1972

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

1972 IMO Longlists, 27

Tags: geometry , trapezoid , dissection , cyclic quadrilateral , IMO , IMO 1972

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

1972 IMO, 2

Tags: function , algebra , functional equation , Functional inequality , IMO , IMO 1972

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

1972 IMO, 3

Tags: factorial , number theory , binomial coefficients , recurrence relation , IMO , IMO 1972

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

1972 IMO Longlists, 26

Tags: algebra , inequality system , IMO , IMO 1972

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1972 IMO Longlists, 15

Tags: factorial , number theory , binomial coefficients , recurrence relation , IMO , IMO 1972

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

1972 IMO, 2

Tags: geometry , trapezoid , dissection , cyclic quadrilateral , IMO , IMO 1972

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.