Olympiad problems

2024 Tuymaada Olympiad, 6

Tags: geometry

Extension of angle bisector $BL$ of the triangle $ABC$ (where $AB < BC$) meets its circumcircle at $N$. Let $M$ be the midpoint of $BL$. Isosceles triangle $BDC$ with base $BC$ and angle equal to $ABC$ at $D$ is constructed outside the triangle $ABC$. Prove that $CM \perp DN$. [i]Proposed by А. Mardanov[/i]

2008 China Northern MO, 3

Tags: number theory , prime factor , divide

Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

2006 May Olympiad, 2

Tags: geometry , rectangle , area , pentagon , paper

A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

2024 Korea Junior Math Olympiad (First Round), 14.

Tags: number theory , algebra , number

Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

2017 Romanian Master of Mathematics Shortlist, G1

Tags: geometry , trapezoid , circumcircle , moving points , conic , projective geometry

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

2004 Greece National Olympiad, 2

Tags: induction , algebra

If $m\geq 2$ show that there does not exist positive integers $x_1, x_2, ..., x_m,$ such that \[x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.\]

2015 BMT Spring, P1

Tags: number theory , algebra

Find two disjoint sets $N_1$ and $N_2$ with $N_1\cup N_2=\mathbb N$, so that neither set contains an infinite arithmetic progression.

2012 Canadian Mathematical Olympiad Qualification Repechage, 1

Tags: pigeonhole principle , combinatorics

The front row of a movie theatre contains $45$ seats. [list] [*] (a) If $42$ people are sitting in the front row, prove that there are $10$ consecutive seats that are all occupied. [*] (b) Show that this conclusion doesn’t necessarily hold if only $41$ people are sitting in the front row.[/list]

2016 Romanian Masters in Mathematic, 1

Tags: geometry

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC, D\neq B$ and $D\neq C$. The circle $ABD$ meets the segment $AC$ again at an interior point $E$. The circle $ACD$ meets the segment $AB$ again at an interior point $F$. Let $A'$ be the reflection of $A$ in the line $BC$. The lines $A'C$ and $DE$ meet at $P$, and the lines $A'B$ and $DF$ meet at $Q$. Prove that the lines $AD, BP$ and $CQ$ are concurrent (or all parallel).

1998 Tuymaada Olympiad, 6

Tags: number theory , periodic , periodic sequence , digit

Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

2006 Mathematics for Its Sake, 2

Tags: linear algebra , linear algebra equation , matrix equation , induction , matrix

Let be a natural number $ n. $ Solve in the set of $ 2\times 2 $ complex matrices the equation $$ \begin{pmatrix} -2& 2007\\ 0&-2 \end{pmatrix} =X^{3n}-3X^n. $$ [i]Petru Vlad[/i]

2024 Romanian Master of Mathematics, 3

Tags: geometry , combinatorics , linear algebra , combinatorial geometry , set

Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$ Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$? [i]Ivan Frolov, Russia[/i]

2016 Peru MO (ONEM), 2

Tags: combinatorics , tiles , tiling , domino

How many dominoes can be placed on a at least $3 \times 12$ board, such so that it is impossible to place a $1\times 3$, $3 \times 1$, or $ 2 \times 2$ tile on what remains of the board? Clarification: Each domino covers exactly two squares on the board. The chips cannot overlap.

2005 iTest, 2

Tags: number theory

When $1^0 + 2^1 + 3^2 + ...+ 100^{99}$ is divided by $5$, a remainder of $N$ is obtained such that $N$ is between $0$ and $4$ inclusive. Find $N$. [i](.1 point)[/i]

2018 Tajikistan Team Selection Test, 6

Tags:

Problem 6. Let H be orthocenter of an acute-angled triangle ABC. Points E,F are on the segments AB,AC respectively, such that BE=BH,CF=CH. The lines EH,FH meet BC in X,Y respectively. Draw the perpendicular HZ from H to EF. Prove that the circumcircle of triangle XYZ is tangent to the circle with diameter BC.

2018 Harvard-MIT Mathematics Tournament, 9

Tags: probability , expected value

$20$ players are playing in a Super Mario Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

1998 Bundeswettbewerb Mathematik, 4

Tags: greatest integer function , number theory

Prove that $n + \big[ (\sqrt{2} + 1)^n\big] $ is odd for all positive integers $n$. $\big[ x \big]$ denotes the greatest integer function.

2012 Serbia Team Selection Test, 3

Tags: circumcircle , radical axis , geometry

Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$. a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic. b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.

1956 Moscow Mathematical Olympiad, 324

Tags: point , combinatorics , combinatorial geometry , distance , geometry , circles

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

2010 Korea National Olympiad, 3

Tags: geometry , incenter , inradius , trigonometry , circumcircle , angle bisector , power of a point

Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.

2016 JBMO Shortlist, 1

Tags: inequality , algebra

Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6$.

Indonesia MO Shortlist - geometry, g3

Tags: geometry , congruent triangles , circles

Suppose $L_1$ is a circle with center $O$, and $L_2$ is a circle with center $O'$. The circles intersect at $ A$ and $ B$ such that $\angle OAO' = 90^o$. Suppose that point $X$ lies on the circumcircle of triangle $OAB$, but lies inside $L_2$. Let the extension of $OX$ intersect $L_1$ at $Y$ and $Z$. Let the extension of $O'X$ intersect $L_2$ at $W$ and $V$ . Prove that $\vartriangle XWZ$ is congruent with $\vartriangle XYV$.

2017 Auckland Mathematical Olympiad, 3

Tags: number theory , divisible , digit , divide

The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

2012 Philippine MO, 1

Tags: probability , combinatorics

A computer generates even integers half of the time and another computer generates even integers a third of the time. If $a_i$ and $b_i$ are the integers generated by the computers, respectively, at time $i$, what is the probability that $a_1b_1 +a_2b_2 +\cdots + a_kb_k$ is an even integer.

2001 Flanders Math Olympiad, 1

Tags:

may be challenge for beginner section, but anyone is able to solve it if you really try. show that for every natural $n > 1$ we have: $(n-1)^2|\ n^{n-1}-1$