Olympiad problems

1991 AMC 8, 12

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If $\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}$, then $N=$ $\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 1990 \qquad \text{(D)}\ 1991 \qquad \text{(E)}\ 1992$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.2

Given a triangle $ABC$ and a point $O$ inside it, it is known that $AB\le BC\le CA$. Prove that $$OA+OB+OC<BC+CA.$$

2018 Yasinsky Geometry Olympiad, 3

Tags: geometry , altitude , midpoint

In the triangle $ABC$, $\angle B = 2 \angle C$, $AD$ is altitude, $M$ is the midpoint of the side $BC$. Prove that $AB = 2DM$.

2010 Princeton University Math Competition, 4

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Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

PEN E Problems, 23

Tags: induction , number theory , prime number

Let $p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}$ be the first $n$ prime numbers, where $n \ge 3$. Prove that \[\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.\]

2015 Iran Team Selection Test, 3

Tags: combinatorics

Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes such that each two have an area equal to 1 in common.

1998 Tournament Of Towns, 3

Tags: tangent , geometry , tangential , circles

Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral. (P Kozhevnikov)

1973 Chisinau City MO, 68

Tags: geometry , equilateral , similar

Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.

2005 All-Russian Olympiad Regional Round, 8.6

Tags: geometry , angle bisector

In quadrilateral $ABCD$, angles $A$ and $C$ are equal. Angle bisector of $B$ intersects line $AD$ at point $P$. Perpendicular on $BP$ passing through point $A$ intersects line $BC$ at point $Q$. Prove that the lines $PQ$ and $CD$ are parallel.

2007 Grigore Moisil Intercounty, 4

Tags: function , group theory , complex number

Consider the group $ \{f:\mathbb{C}\setminus\mathbb{Q}\longrightarrow\mathbb{C}\setminus\mathbb{Q} | f\text{ is bijective}\} $ under the composition of functions. Find the order of the smallest subgroup of it that: $ \text{(1)} $ contains the function $ z\mapsto \frac{z-1}{z+1} . $ $ \text{(2)} $ contains the function $ z\mapsto \frac{z-3}{z+1} . $ $ \text{(3)} $ contain both of the above functions.

2010 AMC 12/AHSME, 10

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The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$? $ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$

2002 Germany Team Selection Test, 1

Tags: quadratic , number theory

Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.

2013 Federal Competition For Advanced Students, Part 1, 4

Tags: ratio , perpendicular bisector , geometry

Let $A$, $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$, let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$. Further, let $E$ be the second point of intersection of the line $AD$ with $k$. Show that for each circle $k$, the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.

2009 China Team Selection Test, 3

Tags: induction , number theory , prime number , relatively prime , prime factorization , greatest common divisor , mobius function

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

2021 New Zealand MO, 2

Tags: algebra , inequalities

Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.

2020 Purple Comet Problems, 3

Tags: number theory

Find the number of perfect squares that divide $20^{20}$.

2000 ITAMO, 3

Tags: ratio , geometry , 3d geometry , pyramid , volume , sphere

A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.

1997 Taiwan National Olympiad, 7

Tags: function , number theory

Find all positive integers $k$ for which there exists a function $f: \mathbb{N}\to\mathbb{Z}$ satisfying $f(1997)=1998$ and $f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b$.

2002 Iran MO (3rd Round), 10

Tags: geometry , incenter , circumcircle , vector , symmetry , geometric transformation , homothety

$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.

1996 India Regional Mathematical Olympiad, 3

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Solve for real numbers $x$ and $y$, \begin{eqnarray*} \\ xy^2 &=& 15x^2 + 17xy +15y^2 ; \\ \\ x^2y &=& 20x^2 + 3y^2. \end{eqnarray*}

2015 Princeton University Math Competition, A1/B1

Tags: combinatorics

Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.) [list=a] [*]Find, with proof, the maximum possible value of $n$. [*](A1 only) For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard. [/list]

2008 IberoAmerican, 5

Tags: geometry , geometric transformation , homothety , ratio , circumcircle , analytic geometry , inequalities

Let $ ABC$ a triangle and $ X$, $ Y$ and $ Z$ points at the segments $ BC$, $ AC$ and $ AB$, respectively.Let $ A'$, $ B'$ and $ C'$ the circuncenters of triangles $ AZY$,$ BXZ$,$ CYX$, respectively.Prove that $ 4(A'B'C')\geq(ABC)$ with equality if and only if $ AA'$, $ BB'$ and $ CC'$ are concurrents. Note: $ (XYZ)$ denotes the area of $ XYZ$

2023 LMT Fall, 9

Tags: geometry

In triangle $ABC$, let $O$ be the circumcenter and let $G$ be the centroid. The line perpendicular to $OG$ at $O $ intersects $BC$ at $M$ such that $M$, $G$, and $A$ are collinear and $OM = 3$. Compute the area of $ABC$, given that $OG = 1$.

2005 District Olympiad, 3

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We denote with $m_a$, $m_g$ the arithmetic mean and geometrical mean, respectively, of the positive numbers $x,y$. a) If $m_a+m_g=y-x$, determine the value of $\dfrac xy$; b) Prove that there exists exactly one pair of different positive integers $(x,y)$ for which $m_a+m_g=40$.

II Soros Olympiad 1995 - 96 (Russia), 10.5

Tags: number theory , perfect square

Find all pairs of natural numbers $x$ and $y$ for which $x^2+3y$ and $y^2+3x$ are simultaneously squares of natural numbers.