Olympiad problems

1989 IMO Longlists, 52

Let $ f$ be a function from the real numbers to the real numbers such that $ f(1) \equal{} 1, f(a\plus{}b) \equal{} f(a)\plus{}f(b)$ for all $ a, b,$ and $ f(x)f \left( \frac{1}{x} \right) \equal{} 1$ for all $ x \neq 0.$ Prove that $ f(x) \equal{} x$ for all real numbers $ x.$

2007 France Team Selection Test, 2

Tags: search , symmetry , inequalities

Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

2008 Moldova Team Selection Test, 4

Tags: search , floor function , combinatorics

A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$ is good, for any positive integer $ a$. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...[/hide]

1998 Putnam, 5

Tags: search

Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.

2001 Pan African, 1

Tags: search , quadratic

Find all positive integers $n$ such that: \[ \dfrac{n^3+3}{n^2+7} \] is a positive integer.

2007 India Regional Mathematical Olympiad, 3

Tags: search , algebra

Find all pairs $ (a, b)$ of real numbers such that whenever $ \alpha$ is a root of $ x^{2} \plus{} ax \plus{} b \equal{} 0$, $ \alpha^{2} \minus{} 2$ is also a root of the equation. [b][Weightage 17/100][/b]

1962 Miklós Schweitzer, 1

Tags: advanced fields , algebra , polynomial , search

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

2011 Romania Team Selection Test, 2

Tags: euler , search , circumcircle , incenter , geometric transformation , homothety , geometry

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

2000 Greece National Olympiad, 2

Tags: search , number theory

Find all prime numbers $p$ such that $1 +p+p^2 +p^3 +p^4$ is a perfect square.

2005 Irish Math Olympiad, 5

Tags: search , inequalities

Let $ a,b,c$ be nonnegative real numbers. Prove that: $ \frac{1}{3}((a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2) \le a^2\plus{}b^2\plus{}c^2\minus{}3 \sqrt[3]{a^2 b^2 c^2 } \le (a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2.$

2007 Iran Team Selection Test, 2

Tags: ceiling function , pigeonhole principle , modular arithmetic , search , combinatorics

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

2004 Postal Coaching, 11

Tags: geometry , geometric transformation , rotation , homothety , rectangle , search , circumcircle

Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.

2009 Indonesia TST, 1

Tags: pigeonhole principle , search , number theory

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

2007 IMS, 7

Tags: linear algebra , search , number theory

$x_{1},x_{2},\dots,x_{n}$ are real number such that for each $i$, the set $\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}$ could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of $x_{i}$'s are zero. [hide="Hint"]It is a number theory problem.[/hide]

2008 China Team Selection Test, 3

Tags: search , arithmetic sequence , combinatorics , ramsey theory

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

2001 Taiwan National Olympiad, 3

Tags: search , combinatorics

Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint. what i have is [hide="this"]we may assume that the union of the $A_{i}$s is $S$. [/hide]

2014 NIMO Summer Contest, 12

Tags: search

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

1963 Miklós Schweitzer, 8

Tags: trigonometry , function , integration , search , real analysis

Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]

2005 South East Mathematical Olympiad, 2

Tags: search , analytic geometry , geometry

Circle $C$ (with center $O$) does not have common point with line $l$. Draw $OP$ perpendicular to $l$, $P \in l$. Let $Q$ be a point on $l$ ($Q$ is different from $P$), $QA$ and $QB$ are tangent to circle $C$, and intersect the circle at $A$ and $B$ respectively. $AB$ intersects $OP$ at $K$. $PM$, $PN$ are perpendicular to $QB$, $QA$, respectively, $M \in QB$, $N \in QA$. Prove that segment $KP$ is bisected by line $MN$.

1994 Baltic Way, 9

Tags: search , modular arithmetic , number theory

Find all pairs of positive integers $(a,b)$ such that $2^a+3^b$ is the square of an integer.

2002 AMC 12/AHSME, 17

Tags: function , trigonometry , search , trig identities , cosine double angle

Let $f(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}$. An equivalent form of $f(x)$ is $\textbf{(A) }1-\sqrt2\sin x\qquad\textbf{(B) }-1+\sqrt2\cos x\qquad\textbf{(C) }\cos\dfrac x2-\sin\dfrac x2$ $\textbf{(D) }\cos x-\sin x\qquad\textbf{(E) }\cos2x$

2007 Iran Team Selection Test, 2

Tags: algebra , polynomial , search , number theory

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

2009 Sharygin Geometry Olympiad, 22

Tags: search , quadratic , trigonometry , circumcircle , 3d geometry , prism , geometry

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

1994 AIME Problems, 11

Tags: search , function , modular arithmetic

Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?

2011 Macedonia National Olympiad, 4

Tags: function , search , algebra

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]