Olympiad problems

2009 Peru IMO TST, 1

Tags: algebra

Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$ and $x\ne y\ne z \ne x.$

2002 Estonia National Olympiad, 1

Tags: algebra , min

Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.

1967 Poland - Second Round, 4

Tags: diophantine , diophantine equation , number theory

Solve the equation in natural numbers $$ xy+yz+zx = xyz + 2. $$

2023 239 Open Mathematical Olympiad, 7

Tags: geometry

The diagonals of convex quadrilateral $ABCD$ intersect at point $E$. Triangles $ABE$ and $CED$ have a common excircle $\Omega$, tangent to segments $AE$ and $DE$ at points $B_1$ and $C_1$, respectively. Denote by $I$ and $J$ the centers of the incircles of these triangles, respectively. Segments $IC_1$ and $JB_1$ intersect at point $S$. It is known that $S$ lies on $\Omega$. Prove that the circumcircle of triangle $AED$ is tangent to $\Omega$. [i]Proposed by David Brodsky[/i]

2022 Malaysia IMONST 2, 3

Tags: number theory

Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.

1971 Miklós Schweitzer, 1

Tags: superior algebra , topology , search

Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$. [i]J. Erdos[/i]

1996 USAMO, 2

Tags: ratio , number theory , inequalities

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

2020 LMT Fall, 5

Tags:

For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8?$ [i]Proposed by Alex Li[/i]

2002 Regional Competition For Advanced Students, 4

Tags: algebra , inequalities , product

Let $a_0, a_1, ..., a_{2002}$ be real numbers. a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies: it is smaller or equal to $1/4$. b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ? c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ : the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$.

2018 Puerto Rico Team Selection Test, 5

Tags: geometry , square

In the square shown in the figure, find the value of $x$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/4659d5afa5b409d9264924735297d1188b0be3.png[/img]

2007 Czech and Slovak Olympiad III A, 4

Tags: arithmetic sequence , combinatorics

The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$, then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$, are in $M$. Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$?

1994 India National Olympiad, 6

Tags: function , continued fraction , algebra

Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.

1971 AMC 12/AHSME, 21

Tags: logarithm

If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to $\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$

2022-IMOC, A2

Tags: inequalities

Given positive integer $n>2,$ consider real numbers $a_1,a_2,\dots, a_n$ satisfying $a^{2}_1+a^2_2+\dots a^2_n=1.$ Find the maximal value of $$|a_1-a_2|+|a_2-a_3| +\dots +|a_n-a_1|.$$ [i]Proposed by ltf0501[/i]

LMT Speed Rounds, 2011.12

Tags:

In a round robin tournament of $7$ people, each person plays every other person exactly once in a game of table tennis. For each game played, the winner is given $2$ points, the loser is given $0$ points, and in the event of a tie, each player gets $1$ point. At the end of the tournament, what is the average score of the $7$ people?

2019 CMI B.Sc. Entrance Exam, 6

Tags: lebinitz rule , calculus , definite integral , logarithm

$(a)$ Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*} $(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\ \\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\ \\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$

2013 European Mathematical Cup, 4

Tags: inequality , three variable inequality

Let $a,b,c$ be positive reals satisfying : \[ \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} \] Then prove that : \[ \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}) \] [i]Proposed by Dimitar Trenevski[/i]

2014 Oral Moscow Geometry Olympiad, 2

Tags: parallelogram , equal segments , geometry

Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

2008 Argentina National Olympiad, 5

Tags: geometry , 3d geometry , number theory

Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.

2020 Iran Team Selection Test, 6

Tags: combinatorics , triangulation

$n$ positive numbers are given. Is it always possible to find a convex polygon with $n+3$ edges and a triangulation of it so that the length of the diameters used in the triangulation are the given $n$ numbers? [i]Proposed by Morteza Saghafian[/i]

2005 MOP Homework, 7

Tags: trigonometry , cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral who interior angle at $B$ is $60$ degrees. Show that if $BC=CD$, then $CD+DA=AB$. Does the converse hold?

2013 Poland - Second Round, 4

Tags: algebra , equation

Solve equation $(x^4 + 3y^2)\sqrt{|x + 2| + |y|}=4|xy^2|$ in real numbers $x$, $y$.

2018 Costa Rica - Final Round, 1

Tags: geometry , combinatorial geometry , combinatorics

There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).

2013 Saint Petersburg Mathematical Olympiad, 4

Tags: combinatorics

There are $100$ glasses, with $101,102,...,200$ cents.Two players play next game. In every move they can take some cents from one glass, but after move should be different number of cents in every glass. Who will win with right strategy?

2017 Princeton University Math Competition, 8

Tags: combinatorics

Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with $17$ orange and $5$ black marshmallows. If $\frac{p}{q}$ is the expected number of marshmallows remaining the instant that there is only one color left, and $p$ and $q$ are relatively prime positive integers, find $p + q$.