Olympiad problems

Kyiv City MO Seniors 2003+ geometry, 2007.10.3

The points $ P, Q$ are given on the plane, which are the points of intersection of the angle bisector $AL$ of some triangle $ABC$ with an inscribed circle, and the point $W$ is the intersection of the angle bisector $AL$ with a circumscribed circle other than the vertex $A$. a) Find the geometric locus of the possible location of the vertex $A$ of the triangle $ABC$. b) Find the geometric locus of the possible location of the vertex $B$ of the triangle $ABC$.

2023 Junior Macedonian Mathematical Olympiad, 2

Tags: number theory , prime number

A positive integer is called [i]superprime[/i] if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers. [i]Authored by Nikola Velov[/i]

2001 All-Russian Olympiad, 1

Tags: combinatorics

Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?

2008 Germany Team Selection Test, 3

Tags: number theory

Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$

Durer Math Competition CD Finals - geometry, 2022.D4

Tags: geometry , trapezoid

The longer base of trapezoid $ABCD$ is $AB$, while the shorter base is $CD$. Diagonal $AC$ bisects the interior angle at $A$. The interior bisector at $B$ meets diagonal $AC$ at $E$. Line $DE$ meets segment $AB$ at $F$. Suppose that $AD = FB$ and $BC = AF$. Find the interior angles of quadrilateral $ABCD$, if we know that $\angle BEC = 54^o$.

2008 Tournament Of Towns, 1

Tags: geometry , rectangle , ratio , combinatorics

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

1990 Federal Competition For Advanced Students, P2, 3

Tags: inequalities , geometry

In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that: $ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$

2022 Kyiv City MO Round 2, Problem 4

Tags: combinatorics , game , gcd

Fedir and Mykhailo have three piles of stones: the first contains $100$ stones, the second $101$, the third $102$. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have $a$ and $b$ stones left correspondently, and remove $gcd(a, b)$ stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy? As a reminder, $gcd(a, b)$ denotes the greatest common divisor of $a, b$. [i](Proposed by Oleksii Masalitin)[/i]

2024 Chile National Olympiad., 3

Tags: power of a point , radical axis , geometry

Let $ AD $ and $ BE $ be altitudes of triangle $ \triangle ABC $ that meet at the orthocenter $ H $. The midpoints of segments $ AB $ and $ CH $ are $ X $ and $ Y $, respectively. Prove that the line $ XY $ is perpendicular to line $ DE $.

2000 Harvard-MIT Mathematics Tournament, 4

Tags: analytic geometry , geometry , circumcircle

Let $ABC$ be a triangle and $H$ be its orthocenter. If it is given that $B$ is $(0,0)$, $C$ is $(1,2)$ and $H$ is $(5,0)$, find $A$.

2008 Singapore Junior Math Olympiad, 3

Tags: midpoint , geometry , right

In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.

2005 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: geometry , circumcircle , inequalities

Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that \[ \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.\]

2024 Kyiv City MO Round 1, Problem 2

Tags: arranging , square , parity , combinatorics

Is it possible to write the numbers from $1$ to $100$ in the cells of a of a $10 \times 10$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $10 \times 10$ square, the numbers in them must have the same parity. The figure below shows examples of such pairs of cells, in which the numbers must have the same parity. [img]https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

1990 Romania Team Selection Test, 11

Tags: combinatorics

In a group of $n$ persons, (i) each person is acquainted to exactly $k$ others, (ii) any two acquainted persons have exactly $l$ common acquaintances, (iii) any two non-acquainted persons have exactly $m$ common acquaintances. Prove that $m(n-k -1) = k(k -l -1)$.

2019 CMIMC, 10

Tags: geometry

Suppose $ABC$ is a triangle, and define $B_1$ and $C_1$ such that $\triangle AB_1C$ and $\triangle AC_1B$ are isosceles right triangles on the exterior of $\triangle ABC$ with right angles at $B_1$ and $C_1$, respectively. Let $M$ be the midpoint of $\overline{B_1C_1}$; if $B_1C_1 = 12$, $BM = 7$ and $CM = 11$, what is the area of $\triangle ABC$?

2009 Romania National Olympiad, 3

Tags: abstract algebra , ring theory , number theory

Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares.

1998 Baltic Way, 5

Tags: modular arithmetic , induction , number theory

Let $a$ be an odd digit and $b$ an even digit. Prove that for every positive integer $n$ there exists a positive integer, divisible by $2^n$, whose decimal representation contains no digits other than $a$ and $b$.

1963 Swedish Mathematical Competition., 6

Tags: analysis , algebra , inequalities , function , functional inequality

The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?

2021 JHMT HS, 6

Tags: number theory

A sequence of positive integers $a_0, a_1, a_2, \dots$ satisfies $a_0 = 83$ and $a_n = (a_{n-1})^{a_{n-1}}$ for all positive integers $n$. Compute the remainder when $a_{2021}$ is divided by $60$.

2013 Purple Comet Problems, 13

Tags: logarithm

There are relatively prime positive integers $m$ and $n$ so that \[\frac{m}{n}=\log_4\left(32^{\log_927}\right).\] Find $m+n$.

2020 AMC 8 -, 1

Tags:

Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Russian TST 2014, P2

Tags: number theory , prime divisor , polynomial

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2008 AMC 10, 6

Tags:

A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

1975 Canada National Olympiad, 3

Tags: function

For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] \equal{} 6, [\pi] \equal{} 3, [\minus{}1.5] \equal{} \minus{}2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 \plus{} [y]^2 \equal{} 4$.

2005 Federal Math Competition of S&M, Problem 3

Tags: geometry , triangle

In a triangle $ABC$, $D$ is the orthogonal projection of the incenter $I$ onto $BC$. Line $DI$ meets the incircle again at $E$. Line $AE$ intersects side $BC$ at point $F$. Suppose that the segment IO is parallel to $BC$, where $O$ is the circumcenter of $\triangle ABC$. If $R$ is the circumradius and $r$ the inradius of the triangle, prove that $EF=2(R-2r)$.