Olympiad problems

2015 May Olympiad, 1

The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?

2021 CMIMC, 2.3 1.1

Tags: combinatorics

Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box? [i]Proposed by Adam Bertelli[/i]

2020 AIME Problems, 15

Tags: geometry

Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$. Find $XY^2$.

2019 Finnish National High School Mathematics Comp, 1

Tags: algebra , equation

Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$

2020-IMOC, N6

Tags: number theory , greatest common divisor

$\textbf{N6.}$ Let $a,b$ be positive integers. If $a,b$ satisfy that \begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*} is also a positive integer, show that \begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*} is a Fibonacci number. [i]Proposed by usjl[/i]

2014 Czech and Slovak Olympiad III A, 6

Tags: inequalities

For arbitrary non-negative numbers $a$ and $b$ prove inequality $\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$, and find, where equality occurs. (Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)

2012 France Team Selection Test, 2

Tags: circumcircle , trigonometry , parallelogram , geometric transformation , reflection , ratio , geometry

Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.

2023 Turkey Junior National Olympiad, 1

Tags: combinatorics

Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is $6$ more than the number of balls from the white one or the number of balls from the white one is $16$ more than the number of balls from the red one. With that given information find all possible values of $n$

2020 Thailand TST, 1

Tags: binomial coefficient , combinatorics , trivial

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

2000 Swedish Mathematical Competition, 4

Tags: geometry , lattice , area , geometric inequalities , 3d geometry , combinatorial geometry

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

2009 Canadian Mathematical Olympiad Qualification Repechage, 10

Tags: function , combinatorics

Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.

1951 AMC 12/AHSME, 1

Tags: percent

The percent that $ M$ is greater than $ N$ is: $ \textbf{(A)}\ \frac {100(M \minus{} N)}{M} \qquad\textbf{(B)}\ \frac {100(M \minus{} N)}{N} \qquad\textbf{(C)}\ \frac {M \minus{} N}{N} \qquad\textbf{(D)}\ \frac {M \minus{} N}{M}$ $ \textbf{(E)}\ \frac {100(M \plus{} N)}{N}$

Denmark (Mohr) - geometry, 2009.4

Tags: geometry , square , equal segments

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

1969 IMO Longlists, 6

Tags: binomial coefficient , trigonometry , combinatorics

$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$

2014 AMC 8, 1

Tags:

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? $\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$

2019 Pan-African Shortlist, A3

Tags: algebra , functional equation , function

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y) $$ for all real numbers $x$ and $y$.

2022 Stanford Mathematics Tournament, 6

Tags:

Let \[f(x)=\cos(x^3-4x^2+5x-2).\] If we let $f^{(k)}$ denote the $k$th derivative of $f$, compute $f^{(10)}(1)$. For the sake of this problem, note that $10!=3628800$.

ICMC 3, 6

Tags:

Let $\varepsilon < \frac{1}{2}$ be a positive real number and let $U_{\varepsilon}$ denote the set of real numbers that differ from their nearest integer by at most $\varepsilon$. Prove that there exists a positive integer $m$ such that for any real number $x$, the sets $\left\{x, 2x, 3x, . . . , mx\right\}$ and $U_{\varepsilon}$ have at least one element in common. proposed by the ICMC Problem Committee

2009 AMC 12/AHSME, 16

Tags:

A circle with center $ C$ is tangent to the positive $ x$ and $ y$-axes and externally tangent to the circle centered at $ (3,0)$ with radius $ 1$. What is the sum of all possible radii of the circle with center $ C$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2012 Iran MO (3rd Round), 1

Tags: ratio , circumcircle , trigonometry , parallelogram , geometry

Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant. [i]Proposed by Mehdi E'tesami Fard[/i]

2022 239 Open Mathematical Olympiad, 8

Tags: trigonometry , algebra , number theory

Prove that there is positive integers $N$ such that the equation $$arctan(N)=\sum_{i=1}^{2020} a_i arctan(i),$$ does not hold for any integers $a_{i}.$

2010 May Olympiad, 5

Tags: coloring , combinatorics

In a $ 2\times 7$ board gridded in $1\times 1$ squares, the $24$ points that are vertices of the squares are considered. [img]https://cdn.artofproblemsolving.com/attachments/9/e/841f11ef9d6fc27cdbe7c91bab6d52d12180e8.gif[/img] Juan and Matías play on this board. Juan paints red the same number of points on each of the three horizontal lines. If Matthias can choose three red dots that are vertices of an acute triangle, Matthias wins the game. What is the maximum number of dots Juan can color in to make sure Matías doesn't win? (For the number found, give an example of coloring that prevents Matías from winning and justify why if the number is greater, Matías can always win.)

2006 Iran Team Selection Test, 3

Tags: geometry

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

1987 Tournament Of Towns, (159) 3

Tags: number theory , divisible , divide

Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .

1974 Canada National Olympiad, 3

Tags: algebra , polynomial

Let \[f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}\] be a polynomial with coefficients satisfying the conditions: \[0\le a_{i}\le a_{0},\quad i=1,2,\ldots,n.\] Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial \begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*} Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$.