Olympiad problems

2018 Latvia Baltic Way TST, P15

Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations $$\begin{cases} x+y+z=n\\ xyz = 2t^3. \end{cases}$$

1997 AIME Problems, 7

Tags: analytic geometry , graphing lines , slope , calculus , conic

A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

2020 Bulgaria EGMO TST, 1

Tags: inequalities

The positive integers $a$, $p$, $q$ and $r$ are greater than $1$ and are such that $p$ divides $aqr+1$, $q$ divides $apr+1$ and $r$ divides $apq+1$. Prove that: a) There are infinitely many such quadruples $(a,p,q,r)$. b) For each such quadruple we have $a\geq \frac{pqr-1}{pq+qr+rp}$.

2002 Italy TST, 3

Tags: function , algebra

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

2023 pOMA, 1

Tags: permutation , combinatorics

Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.

2008 South africa National Olympiad, 2

Tags: geometry

Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.

2020 AMC 10, 2

Tags: arithmetic mean

The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$? $\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$

2018 CHMMC (Fall), 4

Tags: algebra , polynomial

Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.

2018 ASDAN Math Tournament, 3

Tags:

In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$. What is the probability of drawing a number from the bag that is divisible by $11$?

2013 Mediterranean Mathematics Olympiad, 2

Tags: induction , combinatorics

Determine the least integer $k$ for which the following story could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.

2019 Dutch IMO TST, 1

Tags: trinomial , quadratic trinomial , algebra , polynomial

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

2015 JBMO TST - Turkey, 2

Tags: geometry

Let $ABCD$ be a convex quadrilateral and let $\omega$ be a circle tangent to the lines $AB$ and $BC$ at points $A$ and $C$, respectively. $\omega$ intersects the line segments $AD$ and $CD$ again at $E$ and $F$, respectively, which are both different from $D$. Let $G$ be the point of intersection of the lines $AF$ and $CE$. Given $\angle ACB=\angle GDC+\angle ACE$, prove that the line $AD$ is tangent to th circumcircle of the triangle $AGB$.

1992 Turkey Team Selection Test, 3

Tags: inequalities

$x_1, x_2,\cdots,x_{n+1}$ are posive real numbers satisfying the equation $\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$ Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.

1961 AMC 12/AHSME, 27

Tags:

Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: ${{ \textbf{(A)}\ \text{infinite} \qquad\textbf{(B)}\ \text{finite, but greater than 2} \qquad\textbf{(C)}\ \text{Two} \qquad\textbf{(D)}\ \text{One} }\qquad\textbf{(E)}\ \text{Zero} } $

2024 IFYM, Sozopol, 2

Tags: geometry

Let $ n \geq 3 $ be an integer. For every two adjacent vertices $ A $ and $ B $ of a convex $ n $-gon, we find a vertex $ C $ such that the angle $ \angle ACB $ is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written $ n $ numbers.

1993 China Team Selection Test, 3

Tags: geometry , circumcircle , incenter , trigonometry , geometric transformation , reflection , ring theory

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2019 China Team Selection Test, 3

Tags: algebra

Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.

2021 Kyiv Mathematical Festival, 2

Tags: inequalities , algebra

Let $a,b,c\ge0$ and $a+b+c=3.$ Prove that $(3a-bc)(3b-ac)(3c-ab)\le8.$ (O. Rudenko)

2016 China Northern MO, 6

Tags: geometry

Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying: $$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$ Prove that $FH\perp EG$.

2017 Philippine MO, 2

Tags: inequalities , function

Find all positive real numbers $(a,b,c) \leq 1$ which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]

2011 Bogdan Stan, 1

Tags: inequalities , algebra , logarithm

If $ a,b,c $ are all in the interval $ (0,1) $ or all in the interval $ \left( 1,\infty \right), $ then $$ 1\le\sum_{\text{cyc}} \frac{\log_a^7 b\cdot \log_b^3c}{\log_c a +2\log_a b} . $$ [i]Gheorghe Duță[/i]

Putnam 1938, A1

Tags:

A solid in Euclidean $3$-space extends from $z = \frac{-h}{2}$ to $z = \frac{+h}{2}$ and the area of the section $z = k$ is a polynomial in $k$ of degree at most $3$. Show that the volume of the solid is $\frac{h(B + 4M + T)}{6},$ where $B$ is the area of the bottom $(z = \frac{-h}{2})$, $M$ is the area of the middle section $(z = 0),$ and $T$ is the area of the top $(z = \frac{h}{2})$. Derive the formulae for the volumes of a cone and a sphere.

2005 Germany Team Selection Test, 1

Tags: number theory

[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

2010 Balkan MO Shortlist, A4

Tags: algebra

Let $n>2$ be a positive integer. Consider all numbers $S$ of the form \begin{align*} S= a_1 a_2 + a_2 a_3 + \ldots + a_{k-1} a_k \end{align*} with $k>1$ and $a_i$ begin positive integers such that $a_1+a_2+ \ldots + a_k=n$. Determine all the numbers that can be represented in the given form.

2020 ASDAN Math Tournament, 7

Tags: team test

Alex scans the list of integers between $1$ and $2020$ inclusive using the following algorithm. First, he reads off perfect squares between $1$ and $2020$ in ascending order and removes these numbers from the list. Next, he reads off numbers now at perfect square indices in ascending order, which are $2$, $6$, $12$, $...$, and removes these numbers from the list. He repeats this algorithm until he reads off $2020$, which is the nth number he has read o so far. Compute $n$.