Olympiad problems

1975 IMO Shortlist, 13

Tags: geometry , point set , euclidean distance , Intersection , angles , IMO Shortlist

Let $A_0,A_1, \ldots , A_n$ be points in a plane such that (i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and (ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$ where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

2023 India IMO Training Camp, 2

Tags: logarithms , modular arithmetic , number theory unsolved , number theory

For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.

1991 China National Olympiad, 3

Tags: combinatorics unsolved , combinatorics

There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.

2005 Tournament of Towns, 1

Tags: coordinates , algebra

The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two. [i](3 points)[/i]

2010 Ukraine Team Selection Test, 9

Tags: combinatorics , game , IMO Shortlist , invariant , Gerhard Woeginger

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2011 Peru MO (ONEM), 1

Tags: number theory , multiple , consecutive

We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.

2001 Iran MO (3rd Round), 2

Tags: search , algebra proposed , algebra

Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]

2010 Stanford Mathematics Tournament, 1

Tags: quadratics , algebra , quadratic formula

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

KoMaL A Problems 2023/2024, A. 872

Tags: algebra , combinatorics , Sequence , komal

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

1978 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3D geometry , pyramid , inequalities , Geometric Inequalities

The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$. [i]Jordan Tabov[/i]

2019 Philippine TST, 6

Tags: IMO Shortlist , combinatorics , IMO shortlist 2018

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

2014 China Girls Math Olympiad, 3

Tags: combinatorics unsolved , combinatorics

There are $ n$ students; each student knows exactly $d $ girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $ of integers .

Indonesia MO Shortlist - geometry, g5

Tags: geometry , equal angles , cyclic quadrilateral

Let $ABCD$ be quadrilateral inscribed in a circle. Let $M$ be the midpoint of the segment $BD$. If the tangents of the circle at $ B$, and at $D$ are also concurrent with the extension of $AC$, prove that $\angle AMD = \angle CMD$.

2013 Cuba MO, 2

Tags: geometry , trisection

Two equal isosceles triangles $ABC$ and $ADB$, with $C$ and $D$ located in different halfplanes with respect to the line $AB$, share the base $AB$. The midpoints of $AC$ and $BC$ are denoted by $E$ and $F$ respectively. Show that $DE$ and $DF$ divide $AB$ into three equal parts length.

III Soros Olympiad 1996 - 97 (Russia), 11.6

Tags: logarithms , inequalities , analytic geometry

On the coordinate plane, draw a set of points $M(x,y)$, the coordinates of which satisfy the inequality $$\log_{x+y} (x^2+y^2) \le 1.$$

1990 AMC 12/AHSME, 21

Tags: geometry , 3D geometry , pyramid , trigonometry , Pythagorean Theorem

Consider a pyramid $P-ABCD$ whose base $ABCD$ is a square and whose vertex $P$ is equidistant from $A$, $B$, $C$, and $D$. If $AB=1$ and $\angle APD=2\theta$ then the volume of the pyramid is $\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}$

1998 IMO Shortlist, 7

Tags: rectangle , combinatorics , matrix , IMO Shortlist , game , invariant

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

2024 Chile Classification NMO Seniors, 1

Tags: number theory , Chile

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.

2004 Harvard-MIT Mathematics Tournament, 9

Tags: geometry

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

1996 South africa National Olympiad, 5

Tags: geometry , geometry unsolved

$ABC$ is a triangle with sides $1$, $2$ and $\sqrt3$. Determine the smallest possible area of an equilateral triangle with a vertex on each side of triangle $ABC$.

1974 IMO Shortlist, 1

Tags: combinatorics , game , algebra , system of equations , IMO , IMO 1974

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

2015 China Team Selection Test, 3

Tags: number theory , greatest common divisor , modular arithmetic

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

1989 Chile National Olympiad, 3

Tags: geometry , areas , right triangle , semicircle

In a right triangle with legs $a$, $b$ and hypotenuse $c$, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values $X,Y$ . Calculate $X + Y$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png[/img]

2012 Dutch IMO TST, 2

Tags: combinatorics , game

There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

2010 Turkey MO (2nd round), 3

Tags: inequalities , analytic geometry , email , combinatorics proposed , combinatorics

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]