Olympiad problems

2010 Tuymaada Olympiad, 3

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

2013-2014 SDML (High School), 3

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In the following base-$10$ equation, each of the letter represents a unique digit: $AM\cdot PM=ZZZ$. Find the sum of $A+M+P+Z$. $\text{(A) }15\qquad\text{(B) }17\qquad\text{(C) }19\qquad\text{(D) }20\qquad\text{(E) }21$

Russian TST 2016, P3

Tags: algebra , inequalities

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]

2024 International Zhautykov Olympiad, 5

Tags: combinatorics

We are given $m\times n$ table tiled with $3\times 1$ stripes and we are given that $6 | mn$. Prove that there exists a tiling of the table with $2\times 1$ dominoes such that each of these stripes contains one whole domino.

2007 Federal Competition For Advanced Students, Part 2, 2

Tags: combinatorics

38th Austrian Mathematical Olympiad 2007, round 3 problem 5 Given is a convex $ n$-gon with a triangulation, that is a partition into triangles through diagonals that don’t cut each other. Show that it’s always possible to mark the $ n$ corners with the digits of the number $ 2007$ such that every quadrilateral consisting of $ 2$ neighbored (along an edge) triangles has got $ 9$ as the sum of the numbers on its $ 4$ corners.

2023 MOAA, 9

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Real numbers $x$ and $y$ satisfy $$xy+\frac{x}{y} = 3$$ $$\frac{1}{x^2y^2}+\frac{y^2}{x^2} = 4$$ If $x^2$ can be expressed in the form $\frac{a+\sqrt{b}}{c}$ for integers $a$, $b$, and $c$. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

2014 National Olympiad First Round, 2

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How many pairs of integers $(m,n)$ are there such that $mn+n+14=\left (m-1 \right)^2$? $ \textbf{a)}\ 16 \qquad\textbf{b)}\ 12 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 6 \qquad\textbf{e)}\ 2 $

2007 District Olympiad, 3

Tags: calculus , integration , function , limit , real analysis , inequalities

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.

2006 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , median , ratio , angle

Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.

2024 AMC 10, 5

Tags: factorial

What is the least value of $n$ such that $n!$ is a multiple of $2024$? $ \textbf{(A) }11 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22 \qquad \textbf{(D) }23 \qquad \textbf{(E) }253 \qquad $

2015 Danube Mathematical Competition, 2

Tags: combinatorics , number theory , absolute value

Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.

2013 AIME Problems, 7

Tags: geometry , ratio , trigonometry , vector , quadratic

A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.

2014 ELMO Shortlist, 3

Tags: parameterization , inequalities

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

ICMC 3, 2

Tags: complex number , sum , roots of unity

Find integers $a$ and $b$ such that \[a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.\] [i]proposed by the ICMC Problem Committee[/i]

2008 National Olympiad First Round, 6

Tags: modular arithmetic

A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ however its digits are rearranged. How many good numbers are there? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{Infinitely many} $

1986 China Team Selection Test, 1

Tags: geometry , perimeter , trigonometry , angle bisector , congruent triangles

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

LMT Guts Rounds, 2020 F29

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Find the number of pairs of integers $(a,b)$ with $0 \le a,b \le 2019$ where $ax \equiv b \pmod{2020}$ has exactly $2$ integer solutions $0 \le x \le 2019$. [i]Proposed by Richard Chen[/i]

1979 AMC 12/AHSME, 13

Tags: inequalities

The inequality $y-x<\sqrt{x^2}$ is satisfied if and only if $\textbf{(A) }y<0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad$ $\textbf{(B) }y>0\text{ or }y<2x\text{ (or both inequalities hold)}\qquad$ $\textbf{(C) }y^2<2xy\qquad\textbf{(D) }y<0\qquad\textbf{(E) }x>0\text{ and }y<2x$

2005 AMC 8, 4

Tags: perimeter , geometry

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters? $ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64 $

2008 Putnam, B3

Tags: trigonometry , symmetry , inequalities , geometry , 3d geometry , sphere

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

2009 All-Russian Olympiad Regional Round, 11.2

Tags: combinatorics

In some cells of the table $10\times 10$ arranged several $X$'s and a few $O$'s. It is known that there is no line (row or column) completely filled with identical symbols (crosses or zeros). However, if in any empty If you place any icon in a cell, this condition will be violated. What is the minimum number of icons that can appear in a table?

2006 Iran MO (3rd Round), 7

Tags: geometry

We have finite number of distinct shapes in plane. A "[i]convex Kearting[/i]" of these shapes is covering plane with convex sets, that each set consists exactly one of the shapes, and sets intersect at most in border. [img]http://aycu30.webshots.com/image/4109/2003791140004582959_th.jpg[/img] In which case Convex kearting is possible? 1) Finite distinct points 2) Finite distinct segments 3) Finite distinct circles

2010 Indonesia TST, 4

Tags: rectangle , incenter , ratio , geometry

Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that \[ r\plus{}r_0\equal{}R.\] [i]Soewono, Bandung[/i]

2023 HMNT, 25

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A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths $1, 24,$ and $3,$ and the segment of length $24$ is a chord of the circle. Compute the area of the triangle.

2018 China Girls Math Olympiad, 8

Tags: geometry , incenter

Let $I$ be the incenter of triangle $ABC$. The tangent point of $\odot I$ on $AB,AC$ is $D,E$, respectively. Let $BI \cap AC = F$, $CI \cap AB = G$, $DE \cap BI = M$, $DE \cap CI = N$, $DE \cap FG = P$, $BC \cap IP = Q$. Prove that $BC = 2MN$ is equivalent to $IQ = 2IP$.