Olympiad problems

2023 Dutch IMO TST, 3

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

2013 Turkmenistan National Math Olympiad, 4

Tags: geometry , geometric inequality , convex quadrilateral

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

2016 239 Open Mathematical Olympiad, 1

Tags: number theory

A natural number $k>1$ is given. The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,where $a>k + 1$. Prove that at least one of the numbers $a - 1$ or $a + 1$ composite.

MBMT Guts Rounds, 2015.1

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Mr. Stein is ordering a two-course dessert at a restaurant. For each course, he can choose to eat pie, cake, rødgrød, and crème brûlée, but he doesn't want to have the same dessert twice. In how many ways can Mr. Stein order his meal? (Order matters.)

2008 Balkan MO, 3

Tags: geometry , rectangle , analytic geometry , graphing lines , slope , geometric transformation , reflection

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

1999 Harvard-MIT Mathematics Tournament, 2

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Stacy has $d$ dollars. She enters a mall with $10$ shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends $1024$ dollars. After that she alternates playing the lottery and getting her money doubled (Stacy always wins) then going into a new shop and spending $\$1024$. When she comes out of the last shop she has no money left. What is the minimum possible value of $d$?

1964 Polish MO Finals, 3

Tags: geometry , 3d geometry , tetrahedron , sphere

Given a tetrahedron $ ABCD $ whose edges $ AB, BC, CD, DA $ are tangent to a certain sphere. Prove that the points of tangency lie in the same plane.

1973 IMO Longlists, 5

Tags: parallelogram , geometry

Given a ball $K$. Find the locus of the vertices $A$ of all parallelograms $ABCD$ such that $ AC \leq BD$, and the diagonal $BD$ lies completely inside the ball $K$.

2005 Purple Comet Problems, 7

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The graph of the equation $y = 5x + 24$ intersects the graph of the equation $y = x^2$ at two points. The two points are a distance $\sqrt{N}$ apart. Find $N$.

1971 Bundeswettbewerb Mathematik, 3

Tags: geometry

Given five segments such that any three of them can be used to form a triangle. Show that at least one of these triangles is acute-angled. [i]Alternative formulation:[/i] Five segments have lengths such that any three of them can be sides of a triangle. Prove that there exists at least one acute-angled triangle among these triangles.

1996 AMC 12/AHSME, 6

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If $f(x) = x^{\left(x+1\right)}\times \left(x+2\right)^{\left(x+3\right)}$ then $f(0) + f(-1) + f(-2) + f(-3) =$ $\textbf{(A)}\ \frac{-8}{9} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{8}{9} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac{10}{9}$

2016 Online Math Open Problems, 14

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Let $ABC$ be a triangle with $BC=20$ and $CA=16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m+\sqrt{n}$ for positive integers $m$ and $n$. What is $100m+n$? [i] Proposed by Tristan Shin [/i]

2016 Postal Coaching, 6

Tags: combinatorics , combinatorial geometry

Consider a set of $2016$ distinct points in the plane, no four of which are collinear. Prove that there is a subset of $63$ points among them such that no three of these $63$ points are collinear.

2013 AIME Problems, 5

Tags: algebra , polynomial , geometry , 3d geometry , difference of squares , special factorizations , factorization

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

2012 Online Math Open Problems, 10

Tags: probability , number theory , relatively prime

A drawer has $5$ pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$? [i]Author: Ray Li[/i]

2015 ASDAN Math Tournament, 10

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Let $\sigma(n)$ be the sum of all the positive divisors of $n$. Let $a$ be the smallest positive integer greater than or equal to $2015$ for which there exists some positive integer $n$ satisfying $\sigma(n)=a$. Finally, let $b$ be the largest such value of $n$. Compute $a+b$.

2018 Purple Comet Problems, 6

Tags: geometry

Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^o$. Find the degree measure of $\angle BAC$.

2002 Czech-Polish-Slovak Match, 5

Tags: ratio , circumcircle , geometric transformation , geometry

In an acute-angled triangle $ABC$ with circumcenter $O$, points $P$ and $Q$ are taken on sides $AC$ and $BC$ respectively such that $\frac{AP}{PQ} = \frac{BC}{AB}$ and $\frac{BQ}{PQ} =\frac{AC}{AB}$ . Prove that the points $O, P,Q,C$ lie on a circle.

2003 Romania Team Selection Test, 17

Tags: inequalities , symmetry , combinatorics

A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called [i]straight[/i] if and only if for each integer $k$, $1\leq k\leq n-1$ the following inequality is fulfilled \[ |\sigma(k)-\sigma(k+1)|\leq 2. \] Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations. [i]Valentin Vornicu[/i]

1953 AMC 12/AHSME, 15

Tags: geometry , rectangle , rhombus

A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is: $ \textbf{(A)}\ \frac{1}{4} \text{ the area of the original square}\\ \textbf{(B)}\ \frac{1}{2} \text{ the area of the original square}\\ \textbf{(C)}\ \frac{1}{2} \text{ the area of the circular piece}\\ \textbf{(D)}\ \frac{1}{4} \text{ the area of the circular piece}\\ \textbf{(E)}\ \text{none of these}$

2008 Sharygin Geometry Olympiad, 8

Tags: perimeter , geometry

(B.Frenkin, A.Zaslavsky) A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?

2023 Indonesia TST, 1

Tags: number theory , divisibility , prime

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2006 Taiwan National Olympiad, 1

Tags: probability , expected value , combinatorics , random walk

There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes? (Once a safe is opened, the key inside the safe can be used to open another safe.)

2013 Princeton University Math Competition, 6

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What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23$?

1979 USAMO, 2

Tags: 3d geometry , sphere , geometric transformation , reflection , angle bisector , geometry

Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$. [b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.