Olympiad problems

2007 Italy TST, 1

Tags: rectangle , incenter , geometric transformation , homothety , reflection , geometry

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

2004 Germany Team Selection Test, 4

Tags: equation , integer equation

Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.

2023 Oral Moscow Geometry Olympiad, 2

Tags: geometry

Points $X_1$ and $X_2$ move along fixed circles with centers $O_1$ and $O_2$, respectively, so that $O_1X_1 \parallel O_2X_2$. Find the locus of the intersection point of lines $O_1X_2$ and $O_2X_1$.

2014 Math Prize for Girls Olympiad, 1

Tags: rhombus , geometry

Say that a convex quadrilateral is [i]tasty[/i] if its two diagonals divide the quadrilateral into four nonoverlapping similar triangles. Find all tasty convex quadrilaterals. Justify your answer.

2024 Stars of Mathematics, P2

Tags: number theory , double factorial

For any positive integer $n$ we define $n!!=\prod_{k=0}^{\lceil n/2\rceil -1}(n-2k)$. Prove that if the positive integers $a,b,c$ satisfy $a!=b!!+c!!$, then $b$ and $c$ are odd. [i]Proposed by Mihai Cipu[/i]

2008 Grigore Moisil Intercounty, 3

Tags: linear algebra , matrix

Let be a $ 2\times 2 $ real matrix $ A $ whose primary diagonal has positive elements and whose secondary diagonal has negative elements. If $ \det A>0, $ show that [b]a)[/b] for any $ 2\times 2 $ matrix $ X $ of positive real numbers there exists a $ 2\times 2 $ matrix of positive real numbers such that $ AY=X. $ [b]b)[/b] there is a $ 2\times 2 $ matrix $ Z $ of positive real numbers having the property that all elements of $ AZ $ are positive. [i]Vasile Pop[/i]

2017 Argentina National Math Olympiad Level 2, 1

Tags: combinatorics

On a table, there are $16$ weights of the same appearance, which have all the integer weights from $13$ to $28$ grams, that is, they weigh $13, 14, 15, \dots, 28$ grams. Determine the four weights that weigh $13, 14, 27, 28$ grams, using a two-pan balance at most $26$ times.

2022 BAMO, 4

Tags: probability , combinatorics

Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?

2007 Oral Moscow Geometry Olympiad, 4

Tags: geometry , hexagon , equal segments , equilateral

The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal. (M. Volchkevich)

2021 Taiwan TST Round 1, G

Tags: geometry , incenter , circumcircle

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$. A point $X$ on $\Omega$ which is different from $A$ satisfies $AI=XI$. The incircle touches $AC$ and $AB$ at $E, F$, respectively. Let $M_a, M_b, M_c$ be the midpoints of sides $BC, CA, AB$, respectively. Let $T$ be the intersection of the lines $M_bF$ and $M_cE$. Suppose that $AT$ intersects $\Omega$ again at a point $S$. Prove that $X, M_a, S, T$ are concyclic. [i]Proposed by ltf0501 and Li4[/i]

2020-21 KVS IOQM India, 10

Tags:

Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.

2017 OMMock - Mexico National Olympiad Mock Exam, 2

Tags: combinatorics , board , game , triangle

Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can achieve this no matter how Bob plays. [i]Proposed by Oriol Solé[/i]

2012 Morocco TST, 1

Tags: number theory , prime number

Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

2000 Moldova National Olympiad, Problem 7

Tags: triangle , geometry

In an isosceles triangle $ABC$ with $BC=AC$, $I$ is the incenter and $O$ the circumcenter. The line through $I$ parallel to $AC$ meets $BC$ at $D$. Prove that the lines $DO$ and $BI$ are perpendicular.

2016-2017 SDML (Middle School), 5

Tags:

What is the measure in degrees of the acute angle formed by the hands of a $12$-hour clock at $3:20$ PM? $\text{(A) }18\qquad\text{(B) }20\qquad\text{(C) }22\qquad\text{(D) }25\qquad\text{(E) }30$

2009 Kazakhstan National Olympiad, 1

Tags: binomial coefficient , algebra

Let $S_n$ be number of ordered sets of natural numbers $(a_1;a_2;....;a_n)$ for which $\frac{1}{a_1}+\frac{1}{a_2}+....+\frac{1}{a_n}=1$. Determine 1)$S_{10} mod(2)$. 2)$S_7 mod(2)$. (1) is first problem in 10 grade, (2)- third in 9 grade.

2000 National Olympiad First Round, 8

Tags: function

\[\begin{array}{rcl} (x+y)^5 &=& z \\ (y+z)^5 &=& x \\ (z+x)^5 &=& y \end{array}\] How many real triples $(x,y,z)$ are there satisfying above equation system? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None} $

1999 Slovenia National Olympiad, Problem 2

Tags: game , combinatorics

The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?

PEN S Problems, 14

Tags: inequalities

Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \le y$ and $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is nonnegative and as small as possible.

1979 AMC 12/AHSME, 24

Tags: trigonometry , trig identities , law of cosines , law of sines , geometry

Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length $\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$ [size=70]*A polygon is called “simple” if it is not self intersecting.[/size]

2022 Brazil Team Selection Test, 4

Tags: combinatorics , graph theory , perfect matching

Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.

2004 AMC 12/AHSME, 23

Tags: algebra , polynomial

A polynomial \[ P(x) \equal{} c_{2004}x^{2004} \plus{} c_{2003}x^{2003} \plus{} ... \plus{} c_1x \plus{} c_0 \]has real coefficients with $ c_{2004}\not \equal{} 0$ and $ 2004$ distinct complex zeroes $ z_k \equal{} a_k \plus{} b_ki$, $ 1\leq k\leq 2004$ with $ a_k$ and $ b_k$ real, $ a_1 \equal{} b_1 \equal{} 0$, and \[ \sum_{k \equal{} 1}^{2004}{a_k} \equal{} \sum_{k \equal{} 1}^{2004}{b_k}. \]Which of the following quantities can be a nonzero number? $ \textbf{(A)}\ c_0 \qquad \textbf{(B)}\ c_{2003} \qquad \textbf{(C)}\ b_2b_3...b_{2004} \qquad \textbf{(D)}\ \sum_{k \equal{} 1}^{2004}{a_k} \qquad \textbf{(E)}\ \sum_{k \equal{} 1}^{2004}{c_k}$

2019 LIMIT Category C, Problem 8

Tags: floor function , algebra

The value of $$\left\lfloor\frac1{3!}+\frac4{4!}+\frac9{5!}+\ldots\right\rfloor$$

2018 Middle European Mathematical Olympiad, 5

Tags: geometry , parallelogram , ohm

Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC,$ then $DHMO$ is a parallelogram.

1955 Moscow Mathematical Olympiad, 297

Tags: midpoint , geometry , locus , circles

Given two distinct nonintersecting circles none of which is inside the other. Find the locus of the midpoints of all segments whose endpoints lie on the circles.