Olympiad problems

1999 Putnam, 6

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The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1,a_2=2,a_3=24,$ and, for $n\geq 4,$ \[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all $n$, $a_n$ is an integer multiple of $n$.

2020 Durer Math Competition Finals, 2

Tags: number theory

What number should we put in place of the question mark such that the following statement becomes true? $$11001_? = 54001_{10}$$ A number written in the subscript means which base the number is in.

2021 Harvard-MIT Mathematics Tournament., 9

Tags: geometry , circumcircle , incenter

Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.

2021 China Team Selection Test, 2

Tags: number theory , number sequence , divisibility

Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.

2016 Nigerian Senior MO Round 2, Problem 8

Tags: algebra , polynomial

If $a, b, c, d$ are the solutions of the equation $x^4-kx-15=0$, find the equation whose solutions are $\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}$.

2014 Harvard-MIT Mathematics Tournament, 8

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The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?

1975 Putnam, A6

Tags: 3d geometry , triangle

Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.

2003 Olympic Revenge, 1

Tags: geometry , circumcircle , arc midpoint , perpendicular , isosceles

Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.

2012 Belarus Team Selection Test, 2

Tags: circumcircle , polynomial , quadratic , complex number , geometry , algebra

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

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?