Olympiad problems

1991 IMTS, 5

Tags: geometry , 3d geometry , tetrahedron , rotation , area of a triangle , heron's formula , algebra

The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?

2017 F = ma, 15

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An object starting from rest can roll without slipping down an incline. Which of the following four objects, each a uniform solid sphere released from rest, would have the largest speed after the center of mass has moved through a vertical distance h? $\textbf{(A)}\text{a sphere of mass M and radius R}$ $\textbf{(B)}\text{a sphere of mass 2M and radius} \frac{R}{2}$ $\textbf{(C)}\text{a sphere of mass }\frac{M}{2} \text{ and radius 2R}$ $\textbf{(D)}\text{a sphere of mass 3M and radius 3R}$ $\textbf{(E)}\text{All objects would have the same speed}$

2018 Math Prize for Girls Problems, 9

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How many 3-term geometric sequences $a$, $b$, $c$ are there where $a$, $b$, and $c$ are positive integers with $a < b < c$ and $c = 8000$?

2022 JBMO Shortlist, G6

Tags: geometry , tangent , collinear , shortlist

Let $ABC$ be a right triangle with hypotenuse $BC$. The tangent to the circumcircle of triangle $ABC$ at $A$ intersects the line $BC$ at $T$. The points $D$ and $E$ are chosen so that $AD = BD, AE = CE,$ and $\angle CBD = \angle BCE < 90^{\circ}$. Prove that $D, E,$ and $T$ are collinear. Proposed by [i]Nikola Velov, Macedonia[/i]

2017 Finnish National High School Mathematics Comp, 1

Tags: number theory , remainder , euclidean algorithm , division

By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder. As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$. Find $m$ and $n$.

2001 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry , equilateral , equal segments

Let the convex quadrilateral $ABCD$ with $AD = BC$ ¸and $\angle A + \angle B = 120^o$. Take a point $P$ in the plane so that the line $CD$ separates the points $A$ and $P$, and the $DCP$ triangle is equilateral. Show that the triangle $ABP$ is equilateral. It is the true statement for a non-convex quadrilateral?

2015 AMC 10, 23

Tags: factorial , modular arithmetic , floor function

Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? $ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $

2009 Bosnia and Herzegovina Junior BMO TST, 4

Tags: number , combinatorics , circles

On circle there are $2009$ positive integers which sum is $7036$. Show that it is possible to find two pairs of neighboring numbers such that sum of both pairs is greater or equal to $8$

2009 Iran MO (3rd Round), 3

Tags: trapezoid , circumcircle , geometry

3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.

2016 CentroAmerican, 2

Tags: geometry , circumcircle , cyclic , concurrency

Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$. Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$. Let $R$ be the midpoint of $AM$, $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$. Prove that $R$, $S$ and $T$ are collinear.

2004 District Olympiad, 4

Tags: complex number , algebra

If $x,y \in (0, \frac{\pi}{2})$ such as $ (cosx+isiny)^n=cos(nx)+isin(ny)$ for two consecutive positive integers, then the relation is true for all positive integers.

1999 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra

Consider the set $ \mathcal{M}=\left\{ \gcd(2n+3m+13,3n+5m+1,6n+6m-1) | m,n\in\mathbb{N} \right\} . $ Show that there is a natural $ k $ such that the set of its positive divisors is $ \mathcal{M} . $ [i]Dan Brânzei[/i]

1997 Slovenia National Olympiad, Problem 2

Tags: divisibility , number theory

Let $a$ be an integer and $p$ a prime number that divides both $5a-1$ and $a-10$. Show that $p$ also divides $a-3$.

1954 Polish MO Finals, 5

Tags: geometry , 3d geometry , tetrahedron

Prove that if in a tetrahedron $ ABCD $ opposite edges are equal, i.e. $ AB = CD $, $ AC = BD $, $ AD = BC $, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are the axes of symmetry of the tetrahedron.

2005 AMC 10, 22

Tags: floor function

Let $ S$ be the set of the $ 2005$ smallest multiples of $ 4$, and let $ T$ be the set of the $ 2005$ smallest positive multiples of $ 6$. How many elements are common to $ S$ and $ T$? $ \textbf{(A)}\ 166\qquad \textbf{(B)}\ 333\qquad \textbf{(C)}\ 500\qquad \textbf{(D)}\ 668\qquad \textbf{(E)}\ 1001$

2021 Miklós Schweitzer, 4

Tags: real analysis

Let $I$ be a nonempty open subinterval of the set of positive real numbers. For which even $n \in \mathbb{N}$ are there injective function $f: I \to \mathbb{R}$ and positive function $p: I \to \mathbb{R}$, such that for all $x_1 , \ldots , x_n \in I$, \[ f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)} \] holds?

2024 AIME, 2

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A list of positive integers has the following properties: - The sum of the items in the list is $30$. - The unique mode of the list is $9$. - The median of the list is a positive integer that does not appear in the list itself. Find the sum of the squares of all the items in the list.

2006 Korea National Olympiad, 4

Tags: geometry

On the circle $O,$ six points $A,B,C,D,E,F$ are on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$

1995 Italy TST, 3

Tags: algebra , inequalities , function , functional equation , bounding

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions \[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\] Prove that $f(x+1)=f(x)+1$ for all real $x$.

2021 DIME, 7

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In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\tfrac{p}{2^q}$, where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$. [i]Proposed by firebolt360[/i]

KoMaL A Problems 2020/2021, A. 800

Tags: probability , probabilistic method , combinatorics

In a finite, simple, connected graph $G$ we play the following game: initially we color all the vertices with a different color. In each step we choose a vertex randomly (with uniform distribution), and then choose one of its neighbors randomly (also with uniform distribution), and color it to the the same color as the originally chosen vertex (if the two chosen vertices already have the same color, we do nothing). The game ends when all the vertices have the same color. Knowing graph $G$ find the probability for each vertex that the game ends with all vertices having the same color as the chosen vertex.

2025 Nepal National Olympiad, 4

Tags: number theory , factorial

Find all pairs of positive integers $ n $ and $ x $ such that \[ 1^n + 2^n + 3^n + \cdots + n^n = x! \] [i](Petko Lazarov, Bulgaria)[/i]

2011 Akdeniz University MO, 2

Tags: plane geometry , geometry

Let $O$ is a point in a plane $P$ and let $[OX,[OY,[OZ$ is distinct ray in $P$. Prove that, if $A \in [OX$ , $B \in [OY$ and $C \in [OZ$ points such that $\triangle OAB$ , $\triangle OBC$ and $\triangle OCA$ 's perimeter is 2, there is only one $(A,B,C)$ triple

2020 BMT Fall, 7

Tags: number theory

Compute the number of ordered triples of positive integers $(a,b,c)$ such that $a + b + c + ab + bc + ac = abc + 1$.

2022 Saint Petersburg Mathematical Olympiad, 6

Tags: algebra

Find all pairs of nonzero rational numbers $x, y$, such that every positive rational number can be written as $\frac{\{rx\}} {\{ry\}}$ for some positive rational $r$.