Olympiad problems

1996 All-Russian Olympiad Regional Round, 11.2

Let us call the [i]median [/i] of a system of $2n$ points of a plane a straight line passing through exactly two of them, on both sides of which there are points of this system equally. What is the smallest number of [i]medians [/i] that a system of $2n$ points, no three of which lie on the same line?

1998 Portugal MO, 3

Tags: combinatorics

Could the set $\{1,2,3,...,3000\}$ contain a subset of $2000$ elements such that none of them is twice the size of another?

2017 Austria Beginners' Competition, 1

Tags: inequalities , algebra

The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that: $$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$ When do we have equality in the right inequality and when in the left inequality? [i]Proposed by Walther Janous [/i]

2020 Online Math Open Problems, 24

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Let $A$, $B$ be opposite vertices of a unit square with circumcircle $\Gamma$. Let $C$ be a variable point on $\Gamma$. If $C\not\in\{A, B\}$, then let $\omega$ be the incircle of triangle $ABC$, and let $I$ be the center of $\omega$. Let $C_1$ be the point at which $\omega$ meets $\overline{AB}$, and let $D$ be the reflection of $C_1$ over line $CI$. If $C \in\{A, B\}$, let $D = C$. As $C$ varies on $\Gamma$, $D$ traces out a curve $\mathfrak C$ enclosing a region of area $\mathcal A$. Compute $\lfloor 10^4 \mathcal A\rfloor$. [i]Proposed by Brandon Wang[/i]

1971 Canada National Olympiad, 9

Tags: trigonometry , quadratic , similar triangles , geometry

Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

V Soros Olympiad 1998 - 99 (Russia), 10.2

Tags: algebra

In $1748$, the great Russian mathematician Leonhard Euler published one of his most important works, Introduction to the Analysis of Infinites. In this work, in particular, Euler finds the values of two infinite sums $1 +\frac14 +\frac19+ \frac{1}{16}+...$ and $1 +\frac19+ \frac{1}{16}+...$ (the terms in the first sum are the inverses of the squares of the natural numbers, and in the second are the inverses of the squares of the odd numbers of the natural series). The value of the first sum, as Euler proved, equals $\frac{\pi^2}{6}$. Given this result, find the value of the second sum.

2009 AMC 12/AHSME, 6

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By inserting parentheses, it is possible to give the expression \[ 2\times3\plus{}4\times5 \]several values. How many different values can be obtained? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

2000 Turkey MO (2nd round), 2

Tags: polynomial , number theory , relatively prime , algebra

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

2025 Nepal National Olympiad, 3

Tags: geometry , incenter , projective geometry

Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point. [i](Kritesh Dhakal, Nepal)[/i]

2007 Puerto Rico Team Selection Test, 4

Tags: geometry

Just wondering: what exactly is Power of a Point?

2017 NIMO Problems, 2

Tags: geometry , trapezoid

Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid? [i]Proposed by David Altizio

1999 Kazakhstan National Olympiad, 6

Tags: number theory with sequences , sequence , divisible , remainder , number theory

In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

1999 Romania National Olympiad, 2a

Tags: inequalities

let $x_i,y_i 1 \le i \le n$ be positive numbers such that : $\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i$ Prove : $\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}$

2019 Portugal MO, 2

Tags: number theory , digit

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

2010 N.N. Mihăileanu Individual, 4

Tags: modular arithmetic , discrete mathematics , square grid , locusts , pigeonhole principle , combinatorics

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

2015 Baltic Way, 12

Tags: geometry

A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle $CKN$ is equilateral.

2002 Singapore Team Selection Test, 2

Tags: floor function , number theory , integer , divide , divisible

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

2021 JBMO Shortlist, C3

Tags: shortlist , combinatorics , process

We have a set of $343$ closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a triple of positive integers $(b; y; r) \in \{1; 2; ...; 7\}^3$ and flip all the jars whose number of blue, yellow and red marbles differ by not more than $1$ from $b, y, r$, respectively. After $n$ moves all the jars turned out to be with the caps down. Find the number of all possible values of $n$, if $n \le 2021$.

2021 Kyiv City MO Round 1, 10.5

Tags: number theory , sequence

The sequence $(a_n)$ is such that $a_{n+1} = (a_n)^n + n + 1$ for all positive integers $n$, where $a_1$ is some positive integer. Let $k$ be the greatest power of $3$ by which $a_{101}$ is divisible. Find all possible values of $k$. [i]Proposed by Kyrylo Holodnov[/i]

2010 Federal Competition For Advanced Students, P2, 1

Tags: algebra , inequalities

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

2010 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: table , combinatorics

In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times

2011 Postal Coaching, 3

Tags: geometry

Let $ABC$ be a scalene triangle. Let $l_A$ be the tangent to the nine-point circle at the foot of the perpendicular from $A$ to $BC$, and let $l_A'$ be the tangent to the nine-point circle from the mid-point of $BC$. The lines $l_A$ and $l_A'$ intersect at $A'$ . Deﬁne $B'$ and $C'$ similarly. Show that the lines $AA' , BB'$ and $CC'$ are concurrent.

1990 Baltic Way, 20

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A creative task: propose an original competition problem together with its solution.

2015 CCA Math Bonanza, T9

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The Fibonacci numbers are defined as the sequence $F_n$ with $F_0=1$, $F_1=1$ and $F_{n+2} = F_{n+1} +F_n$. How many ways can $10$ be written as an ordered sum of numbers found in the Fibonacci sequence? For example, $3$ can be written as $1 + 1 + 1$, $2 + 1$, $1 + 2$, and $3$, for a total of $4$ ways. [i]2015 CCA Math Bonanza Team Round #9[/i]

2000 Bundeswettbewerb Mathematik, 1b

Tags: number theory , sum , digit

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?