Olympiad problems

1984 IMO Shortlist, 8

Tags: geometry , combinatorial geometry , Coloring , Ramsey Theory , IMO , IMO 1984

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

1984 IMO, 3

Tags: number theory , equation , Divisibility , power of 2 , IMO , IMO 1984 , Hi

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1984 IMO Shortlist, 4

Tags: algebra , geometry , convex polygon , perimeter , IMO , IMO 1984 , geometric inequality

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

1984 IMO Longlists, 44

Tags: algebra , system of equations , Pythagorean Theorem , geometry , IMO Shortlist , IMO 1984

Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$ Prove that the system of equations \[\sqrt{y-a}+\sqrt{z-a}=1\] \[\sqrt{z-b}+\sqrt{x-b}=1\] \[\sqrt{x-c}+\sqrt{y-c}=1\] has exactly one solution $(x,y,z)$ in real numbers. It was proposed by Poland. Have fun! :lol:

1984 IMO Shortlist, 5

Tags: Inequality , three variable inequality , calculus , maximization , IMO , IMO 1984 , Hi

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1984 IMO Longlists, 20

Tags: Inequality , three variable inequality , calculus , maximization , IMO , IMO 1984 , Hi

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1984 IMO Longlists, 43

Tags: number theory , equation , Divisibility , power of 2 , IMO , IMO 1984 , Hi

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1984 IMO, 2

Tags: algebra , geometry , convex polygon , perimeter , IMO , IMO 1984 , geometric inequality

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

1984 IMO Longlists, 40

Tags: algebra , polynomial , modular arithmetic , number theory , Divisibility , IMO , IMO 1984

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

1984 IMO, 1

Tags: Inequality , three variable inequality , calculus , maximization , IMO , IMO 1984 , Hi

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1984 IMO Longlists, 33

Tags: algebra , geometry , convex polygon , perimeter , IMO , IMO 1984 , geometric inequality

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

1984 IMO, 1

Tags: geometry , convex quadrilateral , circle , parallel , diameter , IMO , IMO 1984

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

1984 IMO, 3

Tags: geometry , combinatorial geometry , Coloring , Ramsey Theory , IMO , IMO 1984

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

1984 IMO Longlists, 50

Tags: geometry , convex quadrilateral , circle , parallel , diameter , IMO , IMO 1984

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

1984 IMO Shortlist, 16

Tags: number theory , equation , Divisibility , power of 2 , IMO , IMO 1984 , Hi

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1984 IMO Shortlist, 14

Tags: geometry , convex quadrilateral , circle , parallel , diameter , IMO , IMO 1984

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

1984 IMO, 2

Tags: algebra , polynomial , modular arithmetic , number theory , Divisibility , IMO , IMO 1984

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

1984 IMO Shortlist, 12

Tags: algebra , polynomial , modular arithmetic , number theory , Divisibility , IMO , IMO 1984

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

1984 IMO Longlists, 47

Tags: geometry , combinatorial geometry , Coloring , Ramsey Theory , IMO , IMO 1984

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.