Olympiad problems

1980 IMO Shortlist, 11

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

2015 Kazakhstan National Olympiad, 6

Tags: geometry

The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?

1965 All Russian Mathematical Olympiad, 058

Tags: geometry , incenter , circumcenter

A circle is circumscribed around the triangle $ABC$. Chords, from the midpoint of the arc $AC$ to the midpoints of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.

V Soros Olympiad 1998 - 99 (Russia), 10.2

Tags: algebra , trigonometry

Solve the equation $$ |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.$$

2017 Switzerland - Final Round, 8

Tags: midpoint , isosceles , equal segments , geometry

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

2022 Bulgarian Spring Math Competition, Problem 8.3

Tags: algebra , inequality , inequalities

Given the inequalities: $a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ $b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$ For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.7

Tags: inequalities , algebra

Without using a calculator, prove that $$2^{1995} >5^{854},$$

2001 Singapore MO Open, 3

Tags: combinatorics , difference

Suppose that there are $2001$ golf balls which are numbered from $1$ to $2001$ respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither $5$ nor $8$. How many such golf balls the box can contain at most? Justify your answer.

1961 Czech and Slovak Olympiad III A, 1

Tags: sequence , consecutive , positive integer

Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$ Find the 1000th term of the sequence.

2008 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , divisibility , extremal combinatorics , additive combinatorics

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

1951 AMC 12/AHSME, 8

Tags:

The price of an article is cut $ 10\%$. To restore it to its former value, the new price must be increased by: $ \textbf{(A)}\ 10\% \qquad\textbf{(B)}\ 9\% \qquad\textbf{(C)}\ 11\frac {1}{9}\% \qquad\textbf{(D)}\ 11\% \qquad\textbf{(E)}\ \text{none of these answers}$

1999 Miklós Schweitzer, 7

Tags: real analysis , function

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.

2021 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Find all real numbers $a,b,c$ and $d$ such that: $a^2+b^2+c^2+d^2=a+b+c+d-ab=3.$

2014 Kurschak Competition, 3

Tags: geometry , geometric transformation , reflection , perimeter

Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$. Show that there exists a finite sequence of reflections in the sides of $K$, such that $K$ contains the image of $X$ after these reflections.

2022 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

2008 Stars Of Mathematics, 3

Tags: geometry , incenter , geometric transformation , homothety , parallelogram , ellipse , conic

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. [i]Dan Schwarz[/i]

2014 AIME Problems, 8

Tags: trigonometry , asymptote , geometry , geometric transformation , reflection , pythagorean theorem

Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the form $\sqrt{m} - n,$ where $m$ and $n$ are positive integers. Find $m+n$.

2012 AIME Problems, 2

Tags: ratio

Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$

2008 AMC 12/AHSME, 21

Tags:

A permutation $ (a_1,a_2,a_3,a_4,a_5)$ of $ (1,2,3,4,5)$ is heavy-tailed if $ a_1 \plus{} a_2 < a_4 \plus{} a_5$. What is the number of heavy-tailed permutations? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 52$

1954 AMC 12/AHSME, 34

Tags:

The fraction $ \frac{1}{3}$: $ \textbf{(A)}\ \text{equals 0.33333333} \qquad \textbf{(B)}\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\ \textbf{(C)}\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^9} \\ \textbf{(D)}\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\ \textbf{(E)}\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^9}$

2022 Bundeswettbewerb Mathematik, 4

Tags: enclosing circles , colored points , geometry , combinatorics

Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1. Prove that there exists a circle with diameter $\sqrt{2}$ such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.

2016 Online Math Open Problems, 1

Tags:

Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1,\dots,30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1+A_2+\dots+A_{30})$. [i]Proposed by Yang Liu[/i]

2025 India STEMS Category B, 5

Tags: rhombus , homothety , euler circle , arc midpoint

Let $ABC$ be an acute scalene triangle. Let $D, E$ be points on segments $AB, AC$ respectively, such that $BD=CE$. Prove that the nine-point centers of $ADE$, $ACD$, $ABC$, $AEB$ form a rhombus. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

1988 Canada National Olympiad, 5

Tags:

If $S$ is a sequence of positive integers let $p(S)$ be the product of the members of $S$. Let $m(S)$ be the arithmetic mean of $p(T)$ for all non-empty subsets $T$ of $S$. Suppose that $S'$ is formed from $S$ by appending an additional positive integer. If $m(S) = 13$ and $m(S') = 49$, find $S'$.

2024 Harvard-MIT Mathematics Tournament, 6

Tags: guts

In triangle $ABC,$ points $M$ and $N$ are the midpoints of $AB$ and $AC,$ respectively, and points $P$ and $Q$ trisect $BC.$ Given that $A, M, P, N,$ and $Q$ lie on a circle and $BC=1,$ compute the area of triangle $ABC.$