Olympiad problems

2019 Middle European Mathematical Olympiad, 4

Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example, to form multidigit numbers (such as $222$) or powers (such as $2^2$). Valid examples: $$\left((2\times 2+2)\times 2-\frac{2}{2}\right)\times 2=22 \;\;, \;\; (2\times2\times 2-2)\times \left(2\times 2 +\frac{2+2+2}{2}\right)=42$$ [i]Proposed by Stephan Wagner, Austria[/i]

1997 VJIMC, Problem 2

Tags: complex analysis , calculus

Let $f:\mathbb C\to\mathbb C$ be a holomorphic function with the property that $|f(z)|=1$ for all $z\in\mathbb C$ such that $|z|=1$. Prove that there exists a $\theta\in\mathbb R$ and a $k\in\{0,1,2,\ldots\}$ such that $$f(z)=e^{i\theta}z^k$$for all $z\in\mathbb C$.

1980 Spain Mathematical Olympiad, 2

Tags: probability , combinatorics

A ballot box contains the votes for the election of two candidates $A$ and $B$. It is known that candidate $A$ has $6$ votes and candidate $B$ has $9$. Find the probability that, when carrying out the scrutiny, candidate $B$ always goes first.

2015 Kyiv Math Festival, P1

Tags: solve equation , algebra

Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

2016 NIMO Summer Contest, 12

Tags: number theory

Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse. [i]Proposed by David Altizio[/i]

2004 National Olympiad First Round, 33

Tags: circumcircle , trapezoid , perpendicular bisector , trigonometry , angle bisector , geometry

Let $ABCD$ be a trapezoid such that $|AB|=9$, $|CD|=5$ and $BC\parallel AD$. Let the internal angle bisector of angle $D$ meet the internal angle bisectors of angles $A$ and $C$ at $M$ and $N$, respectively. Let the internal angle bisector of angle $B$ meet the internal angle bisectors of angles $A$ and $C$ at $L$ and $K$, respectively. If $K$ is on $[AD]$ and $\dfrac{|LM|}{|KN|} = \dfrac 37$, what is $\dfrac{|MN|}{|KL|}$? $ \textbf{(A)}\ \dfrac{62}{63} \qquad\textbf{(B)}\ \dfrac{27}{35} \qquad\textbf{(C)}\ \dfrac{2}{3} \qquad\textbf{(D)}\ \dfrac{5}{21} \qquad\textbf{(E)}\ \dfrac{24}{63} $

JOM 2023, 4

Tags: algebra

Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

2020 BMT Fall, 24

Tags: number theory

Let $N$ be the number of non-empty subsets $T$ of $S = \{1, 2, 3, 4, . . . , 2020\}$ satisfying $\max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

2017 Simon Marais Mathematical Competition, A1

Tags: combinatorics , game

The ﬁve sides and ﬁve diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The ﬁrst player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the ﬁrst to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the ﬁrst player, the second player, or neither player can force a win.

2023 USA EGMO Team Selection Test, 3

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

2001 Brazil National Olympiad, 6

Tags: combinatorics

A one-player game is played as follows: There is a bowl at each integer on the $Ox$-axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either (A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or (B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left. Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$).

1964 AMC 12/AHSME, 28

Tags: arithmetic sequence

The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first interm is an integer, and $n>1$, then the number of possible values for $n$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

2002 Polish MO Finals, 2

Tags: geometry , rectangle , circumcircle

On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$, point $C$ and center of circumcircle are collinear.

IV Soros Olympiad 1997 - 98 (Russia), 9.6

Tags: rhombus , geometry

A rhombus is circumscribed around a square with side $1997$. Find its diagonals if it is known that they are equal to different integers.

2023/2024 Tournament of Towns, 2

Tags: number theory

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

2023 Yasinsky Geometry Olympiad, 4

Tags: circles , equal segments , geometry

Let $C$ be one of the two points of intersection of circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$, respectively. The line $O_1O_2$ intersects the circles at points $A$ and $B$ as shown in the figure. Let $K$ be the second point of intersection of line $AC$ with circle $\omega_2$, $L$ be the second point of intersection of line $BC$ with circle $\omega_1$. Lines $AL$ and $BK$ intersect at point $D$. Prove that $AD=BD$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/6/4/2cdccb43743fcfcb155e846a0e05ec79ba90e4.png[/img]

2017 Purple Comet Problems, 19

Tags: number theory

Find the greatest integer $n < 1000$ for which $4n^3 - 3n$ is the product of two consecutive odd integers.

2010 Contests, 3

Tags: geometry , geometric transformation , rotation

Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions: $i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$. $ii)$ There are no two lines of $S$ which are parallel.

2024 Korea Winter Program Practice Test, Q8

Tags: geometry , circumcircle

Let $\omega$ be the incircle of triangle $ABC$. For any positive real number $\lambda$, let $\omega_{\lambda}$ be the circle concentric with $\omega$ that has radius $\lambda$ times that of $\omega$. Let $X$ be the intersection between a trisector of $\angle B$ closer to $BC$ and a trisector of $\angle C$ closer to $BC$. Similarly define $Y$ and $Z$. Let $\epsilon = \frac{1}{2024}$. Show that the circumcircle of triangle $XYZ$ lies inside $\omega_{1-\epsilon}$. [i]Note. Weaker results with smaller $\epsilon$ may be awarded points depending on the value of the constant $\epsilon <\frac{1}{2024}$.[/i]

2021 USMCA, 2

Tags:

A four-digit positive integer is called [i]doubly[/i] if its first two digits form some permutation of its last two digits. For example, $1331$ and $2121$ are both [i]doubly[/i]. How many four-digit [i]doubly[/i] positive integers are there?

1962 Putnam, A1

Tags: convex , point , geometry

Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

2016 HMNT, 10

Tags: hmmt

We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ . Find the minimum possible value of $n$ such that it is possible to get a chip on $A_{10}$ through a sequence of moves.

2014 China Team Selection Test, 3

Tags: diophantine equation , number theory

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2021 IMO Shortlist, A7

Tags: algebra , inequality , inequalities

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

1998 Czech And Slovak Olympiad IIIA, 5

Tags: fixed point , diagonal , trapezoid , geometry

A circle $k$ and a point $A$ outside it are given in the plane. Prove that all trapezoids, whose non-parallel sides meet at $A$, have the same intersection of diagonals.