Olympiad problems

1975 Bundeswettbewerb Mathematik, 3

For $n$positive integers $ x_1,x2,...,x_n$, $a_n$ is their arithmetic and $g_n$ the geometric mean. Consider the statement $S_n$: If $a_n/g_n$ is a positive integer, then $x_1 = x_2 = ··· = x_n$. Prove $S_2$ and disprove $S_n$ for all even $n > 2$.

2005 Turkey Team Selection Test, 2

Tags: circumcircle , geometric transformation , reflection , geometry

Let $ABC$ be a triangle such that $\angle A=90$ and $\angle B < \angle C$. The tangent at $A$ to its circumcircle $\Gamma$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ be the midpoint of $AX$. Let the line $BY$ meet $\Gamma$ again at $Z$. Prove that the line $BD$ is tangent to circumcircle of triangle $ADZ$ .

2011 Dutch BxMO TST, 3

Tags: algebra , equation , absolute value

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

1957 AMC 12/AHSME, 27

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The sum of the reciprocals of the roots of the equation $ x^2 \plus{} px \plus{} q \equal{} 0$ is: $ \textbf{(A)}\ \minus{}\frac{p}{q} \qquad \textbf{(B)}\ \frac{q}{p}\qquad \textbf{(C)}\ \frac{p}{q}\qquad \textbf{(D)}\ \minus{}\frac{q}{p}\qquad \textbf{(E)}\ pq$

2005 Iran MO (3rd Round), 3

Tags: logarithm , combinatorics , geometry , rectangle

$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino. Prove that $10000\leq f(1384)\leq960000$. Find some bound for $f(n)$

2001 AMC 10, 6

Tags: algebra

Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

2012 Purple Comet Problems, 4

Tags: complementary counting

How many two-digit positive integers contain at least one digit equal to 5?

1995 Moldova Team Selection Test, 4

Tags: function , induction , algebra

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following: $i)$ $f(1)=1$; $ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.

2023 Caucasus Mathematical Olympiad, 2

Tags: geometry

In a convex hexagon the value of each angle is $120^{\circ}$. The perimeter of the hexagon equals $2$. Prove that this hexagon can be covered by a triangle with perimeter at most $3$.

Russian TST 2017, P3

Tags: combinatorics , domino

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

2013 India Regional Mathematical Olympiad, 4

Tags: geometry , ratio

In a triangle $ABC$, points $D$ and $E$ are on segments $BC$ and $AC$ such that $BD=3DC$ and $AE=4EC$. Point $P$ is on line $ED$ such that $D$ is the midpoint of segment $EP$. Lines $AP$ and $BC$ intersect at point $S$. Find the ratio $BS/SD$.

2018 India PRMO, 28

Tags: combinatorics , sum of digits

Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.

2014 National Olympiad First Round, 6

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The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the $2014^{\text{th}}$ number in that sequence? $ \textbf{(A)}\ 66480 \qquad\textbf{(B)}\ 64096 \qquad\textbf{(C)}\ 62048 \qquad\textbf{(D)}\ 60288 \qquad\textbf{(E)}\ \text{None of the preceding} $

1986 AIME Problems, 15

Tags: area of a triangle , geometry , analytic geometry , graphing lines , slope , trigonometry , pythagorean theorem

Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.

2013 Singapore MO Open, 4

Tags: graph theory , combinatorics

Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that $\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$, $z \in F$; $\bullet$ If $1 \leq k \leq n$ and $x_1$, ..., $x_k \in F$, then $x_1+\cdots+x_k \neq 0$. Show that $F$ has at least $2n+2$ elements.

2020 Regional Competition For Advanced Students, 1

Tags: algebra

Let $a$ be a positive integer. Determine all $a$ such that the equation $$ \biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x$$ has at least one integer solution for $x$. For every such $a$ state the respective solutions. (Richard Henner)

2014 BMO TST, 3

Tags: geometry

From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.

1995 National High School Mathematics League, 4

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Equation $|x-2n|=k\sqrt{x}(n\in\mathbb{Z}_+)$ has two different real roots on $(2n-1,2n+1]$, then the range value of $k$ is $\text{(A)}k>0\qquad\text{(B)}0<k\leq\frac{1}{\sqrt{2n+1}}\qquad\text{(C)}\frac{1}{2n+1}<k\leq\frac{1}{\sqrt{2n+1}}$ $\text{(D)}$ none above

2021 Turkey Junior National Olympiad, 4

Tags: geometry , tangent

Let $X$ be a point on the segment $[BC]$ of an equilateral triangle $ABC$ and let $Y$ and $Z$ be points on the rays $[BA$ and $[CA$ such that the lines $AX, BZ, CY$ are parallel. If the intersection of $XY$ and $AC$ is $M$ and the intersection of $XZ$ and $AB$ is $N$, prove that $MN$ is tangent to the incenter of $ABC$.

2005 Purple Comet Problems, 6

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$ABCDE$ is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments $AC$ and $BD$?

2005 Purple Comet Problems, 8

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Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]

2005 Taiwan National Olympiad, 2

Tags: geometry , trigonometry

Given a line segment $AB=7$, $C$ is constructed on $AB$ so that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as a side. Find the length of the segment connecting their two circumcenters.

2014 Thailand TSTST, 3

Tags: combinatorics , algebra

Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$. Find $$\sum_{(a,b,c)\in S} abc.$$

2019 LIMIT Category C, Problem 3

Tags: series , summation , algebra

Which of the following series are convergent? $\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$ $\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$ $\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$ $\textbf{(D)}~\text{None of the above}$

2012 Kosovo Team Selection Test, 3

Tags: parallelogram , geometry

If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that \[4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)\]