Olympiad problems

2010 Thailand Mathematical Olympiad, 6

Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$

2007 AMC 8, 21

Tags: probability

Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? $\textbf{(A)}\ \frac{2}{7} \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{4}{7} \qquad \textbf{(E)}\ \frac{5}{8}$

MOAA Accuracy Rounds, 2021.5

Tags: accuracy

If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$. \[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\] [i]Proposed by Jeffrey Shi[/i]

2012 Tuymaada Olympiad, 2

Tags: rectangle , symmetry , angle bisector , geometry

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$. [i]Proposed by S. Berlov[/i]

1976 IMO Longlists, 11

Tags: polynomial , recurrence relation , root , chebyshev polynomial , algebra

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

2013 Kosovo National Mathematical Olympiad, 5

Tags: geometry

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area

2004 Baltic Way, 7

Tags: ratio , number theory

Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.

2005 Iran Team Selection Test, 1

Tags: function , number theory

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

2017 Math Prize for Girls Problems, 3

Tags:

If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?

1959 Poland - Second Round, 5

Tags: geometry , combinatorial geometry

In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.

2007 Vietnam National Olympiad, 1

Tags: combinatorics

Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.

2013 Stanford Mathematics Tournament, 3

Tags: probability , expected value

Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?

2016 CMIMC, 7

Tags: geometry

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.

PEN A Problems, 9

Tags: number theory , relatively prime , divisibility theory

Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

2007 China Team Selection Test, 1

Tags: inequalities

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

2023 Belarusian National Olympiad, 11.3

Tags: number theory

Prove that for any fixed integer $a$ equation $$(m!+a)^2=n!+a^2$$ has finitely many solutions in positive integers $m,n$

1998 Romania Team Selection Test, 1

Tags: floor function , combinatorics

Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that: i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$ ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$ [i]Radu Todor[/i]

2007 Cono Sur Olympiad, 2

Tags: number theory

Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.

1955 AMC 12/AHSME, 39

Tags: parabola , conic

If $ y\equal{}x^2\plus{}px\plus{}q$, then if the least possible value of $ y$ is zero $ q$ is equal to: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{p^2}{4} \qquad \textbf{(C)}\ \frac{p}{2} \qquad \textbf{(D)}\ \minus{}\frac{p}{2} \qquad \textbf{(E)}\ \frac{p^2}{4}\minus{}q$

1997 National High School Mathematics League, 4

Tags: ellipse , conic

In rectangular coordinate system, if $m(x^2+y^2+2y+1)=(x-2y+3)^2$ refers to an ellipse, then the range value of $m$ is $\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)$

2000 Harvard-MIT Mathematics Tournament, 8

Tags:

A man has three daughters. The product of their ages is $168$, and he remembers that the sum of their ages is the number of trees in his yard. He counts the trees but cannot determine any of their ages. What are all possible ages of his oldest daughter?

2023 ISL, G3

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2013 Turkey Team Selection Test, 2

Tags: geometry , rectangle , induction , combinatorics

We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles.

VMEO III 2006, 11.3

Tags: algebra , inequalities

Let $x, y, z$ be non-negative real numbers whose sum is $ 1$. Prove that: $$\sqrt[3]{x - y + z^3} + \sqrt[3]{y - z + x^3} + \sqrt[3]{z - x + y^3} \le 1$$

Oliforum Contest III 2012, 5

Tags: geometry , cyclic quadrilateral , orthogonal

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.