Olympiad problems

2016 Germany Team Selection Test, 2

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

2021 Latvia Baltic Way TST, P4

Tags: inequalities

Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds: $$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$ where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points.

2012 EGMO, 7

Tags: geometry , circumcircle , geometric transformation

Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$. Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.) [i]Luxembourg (Pierre Haas)[/i]

2012 AMC 10, 13

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An [i]iterative average[/i] of the numbers $1$, $2$, $3$, $4$, and $5$ is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure? $ \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} $

2021 Sharygin Geometry Olympiad, 8.3

Tags: locus , centroid , geometry , speed

Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.

2022 Durer Math Competition Finals, 4

Tags: cyclic , concurrency , geometry

$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.

2024 AMC 12/AHSME, 10

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A list of 9 real numbers consists of $1$, $2.2 $, $3.2 $, $5.2 $, $6.2 $, $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) infinitely many}\qquad $

2021 China Team Selection Test, 3

Tags: sum of digits , decimal representation , number theory

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

1997 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divisor

Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.

2002 Indonesia MO, 5

Tags: modular arithmetic , algebra

Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table: $\begin{array} {|c|c|c|} \cline{1-3} 10 & & \\ \cline{1-3} & & 9 \\ \cline{1-3} & 3 & \\ \cline{1-3} 11 & & 17 \\ \cline{1-3} & 20 & \\ \cline{1-3} \end{array}$ such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.

1989 IMO Shortlist, 32

Tags: circumcircle , trigonometry , parallelogram , rhombus , angle , geometry

The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$

2012 USAMTS Problems, 5

Tags: algebra , polynomial

Let $P$ and $Q$ be two polynomials with real coeficients such that $P$ has degree greater than $1$ and \[P(Q(x)) = P(P(x)) + P(x).\]Show that $P(-x) = P(x) + x$.

2020 Tournament Of Towns, 4

Tags: combinatorics , square grid , table

For which integers $N$ it is possible to write real numbers into the cells of a square of size $N \times N$ so that among the sums of each pair of adjacent cells there are all integers from $1$ to $2(N-1)N$ (each integer once)? Maxim Didin

2006 Austrian-Polish Competition, 10

Tags: 3d geometry , pyramid , geometry

Let $ABCDS$ be a (not neccessarily straight) pyramid with a rectangular base $ABCD$ and acute triangular faces $ABS,BCS,CDS,DAS$. We consider all cuboids which are inscribed inside the pyramid with its base being in the plane $ABCD$ and its upper vertexes are in the triangular faces (one in each). Find the locus of the midpoints of these cuboids.

2021 Dutch IMO TST, 4

Tags: domino , combinatorics

On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.

2001 Tuymaada Olympiad, 1

Tags: inequalities , combinatorics

Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$. [i]Proposed by D. Teryoshin[/i]

2014 PUMaC Combinatorics A, 4

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Amy has a $2 \times 10$ puzzle grid which she can use $1 \times 1$ and $1 \times 2$ (1 vertical, 2 horizontal) tiles to cover. How many ways can she exactly cover the grid without any tiles overlapping and without rotating the tiles?

2017 Turkey Team Selection Test, 2

Tags: combinatorics

There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.

2016 Thailand TSTST, 3

Tags: perfect power , number theory

Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.

1969 Spain Mathematical Olympiad, 7

Tags: geometric inequality , geometry , regular polygon

A convex polygon $A_1A_2 . . .A_n$ of $n$ sides and inscribed in a circle, has its sides that satisfy the inequalities $$A_nA_1 > A_1A_2 > A_2A_3 >...> A_{n-1}A_n$$ Show that its interior angles satisfy the inequalities $$\angle A_1 < \angle A_2 < \angle A_3 < ... < \angle A_{n-1}, \angle A_{n-1} > \angle A_n> \angle A_1.$$

2013 IFYM, Sozopol, 1

Tags: divisibility , sequence , algebra

Let $u_1=1,u_2=2,u_3=24,$ and $u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$ Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.

2002 AMC 12/AHSME, 3

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According to the standard convention for exponentiation, \[2^{2^{2^2}} \equal{} 2^{\left(2^{\left(2^2\right)}\right)} \equal{} 2^{16} \equal{} 65,\!536.\] If the order in which the exponentiations are performed is changed, how many [u]other[/u] values are possible? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

2017 Turkey Team Selection Test, 8

Tags: geometry

In a triangle $ABC$ the bisectors through vertices $B$ and $C$ meet the sides $\left [ AC \right ]$ and $\left [ AB \right ]$ at $D$ and $E$ respectively. Let $I_{c}$ be the center of the excircle which is tangent to the side $\left [ AB \right ]$ and $F$ the midpoint of $\left [ BI_{c} \right ]$. If $\left | CF \right |^2=\left | CE \right |^2+\left | DF \right |^2$, show that $ABC$ is an equilateral triangle.

2008 CHKMO, 2

Tags: algebra , polynomial , modular arithmetic , number theory

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

2005 Austrian-Polish Competition, 5

Tags: geometry , radical axis

Given is a convex quadrilateral $ABCD$ with $AB=CD$. Draw the triangles $ABE$ and $CDF$ outside $ABCD$ so that $\angle{ABE} = \angle{DCF}$ and $\angle{BAE}=\angle{FDC}$. Prove that the midpoints of $\overline{AD}$, $\overline{BC}$ and $\overline{EF}$ are collinear.