This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 19

1967 IMO, 1
Tags: geometry , parallelogram , trigonometry , geometric inequality , Trigonometric inequality , IMO , IMO 1967
The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1967 IMO, 4
Tags: geometry , Triangle , similar triangles , area of a triangle , IMO , IMO 1967
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

1967 IMO, 3
Tags: number theory , Divisibility , binomial coefficients , IMO , IMO 1967
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

1967 IMO Shortlist, 1
Tags: combinatorics , recurrence relation , system of equations , IMO , IMO 1967
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

1967 IMO Longlists, 24
Tags: combinatorics , recurrence relation , system of equations , IMO , IMO 1967
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

1967 IMO Longlists, 29
Tags: geometry , Triangle , similar triangles , area of a triangle , IMO , IMO 1967
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

1967 IMO, 2
Tags: geometry , 3D geometry , tetrahedron , Volume , geometric inequality , IMO , IMO 1967
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

2010 Morocco TST, 1
Tags: combinatorics , recurrence relation , system of equations , IMO , IMO 1967
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

1967 IMO Longlists, 40
Tags: geometry , 3D geometry , tetrahedron , Volume , geometric inequality , IMO , IMO 1967
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

1967 IMO Shortlist, 1
Tags: geometry , parallelogram , trigonometry , geometric inequality , Trigonometric inequality , IMO , IMO 1967
The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1967 IMO Shortlist, 1
Tags: algebra , Sequence , algebraic identities , IMO , IMO 1967
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

1967 IMO Longlists, 8
Tags: geometry , parallelogram , trigonometry , geometric inequality , Trigonometric inequality , IMO , IMO 1967
The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1967 IMO, 6
Tags: combinatorics , recurrence relation , system of equations , IMO , IMO 1967
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

1967 IMO Shortlist, 1
Tags: geometry , 3D geometry , tetrahedron , Volume , geometric inequality , IMO , IMO 1967
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

1967 IMO Shortlist, 1
Tags: geometry , Triangle , similar triangles , area of a triangle , IMO , IMO 1967
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

1967 IMO Shortlist, 1
Tags: number theory , Divisibility , binomial coefficients , IMO , IMO 1967
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

1967 IMO, 5
Tags: algebra , Sequence , algebraic identities , IMO , IMO 1967
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

1967 IMO Longlists, 17
Tags: number theory , Divisibility , binomial coefficients , IMO , IMO 1967
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

1967 IMO Longlists, 57
Tags: algebra , Sequence , algebraic identities , IMO , IMO 1967
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$