Olympiad problems

1998 Austrian-Polish Competition, 8

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)

2021 Sharygin Geometry Olympiad, 5

Tags: similar triangles , geometry

Five points are given in the plane. Find the maximum number of similar triangles whose vertices are among those five points.

2025 Ukraine National Mathematical Olympiad, 8.4

Tags: combinatorics

What is the maximum number of knights that can be placed on a chessboard of size $8 \times 8$ such that any knight, after making 1 or 2 arbitrary moves, does not land on a square occupied by another knight? [i]Proposed by Bogdan Rublov[/i]

2014 ASDAN Math Tournament, 6

Tags:

Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords.

2013 All-Russian Olympiad, 3

Tags: combinatorics

The head of the Mint wants to release 12 coins denominations (each - a natural number rubles) so that any amount from 1 to 6543 rubles could be paid without having to pass, using no more than 8 coins. Can he do it? (If the payment amount you can use a few coins of the same denomination.)

2010 Contests, 4

Tags: circumcircle , geometry

The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.

2012 Iran Team Selection Test, 3

Tags: inequalities , geometric transformation , reflection , limit , analytic geometry , complex number , geometry

The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that \[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\] where $S_X$ denotes the surface of figure $X$. [i]Proposed by Morteza Saghafian, Ali khezeli[/i]

2019 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , perfect square

Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.

1966 Spain Mathematical Olympiad, 7

Tags: algebra , geometric sequence

Determine a geometric progression of seven terms, knowing the sum, $7$, of the first three, and the sum, $112$, of the last three.

2019 Bosnia and Herzegovina EGMO TST, 4

Tags: combinatorics , combinatorial geometry

Let $n$ be a natural number. There are $n$ blue points , $n$ red points and one green point on the circle . Prove that it is possible to draw $n$ lengths whose ends are in the given points, so that a maximum of one segment emerges from each point, no more than two segments intersect and the endpoints of none of the segments are blue and red points. [hide=original wording]Нека je ? природан број. На кружници се налази ? плавих, ? црвених и једна зелена тачка. Доказати да је могуће повући ? дужи чији су крајеви у датим тачкама, тако да из сваке тачке излази максимално једна дуж, никоје две дужи се не сијеку и крајње тачке ниједне од дужи нису плава и црвена тачка.[/hide]

2016 LMT, 17

Tags:

Find the minimum possible value of \[\left\lfloor \dfrac{a+b}{c}\right\rfloor+2 \left\lfloor \dfrac{b+c}{a}\right\rfloor+ \left\lfloor \dfrac{c+a}{b}\right\rfloor\] where $a,b,c$ are the sidelengths of a triangle. [i]Proposed by Nathan Ramesh

2016 Nigerian Senior MO Round 2, Problem 2

Tags: ratio , trigonometry , geometry , function

$PQ$ is a diameter of a circle. $PR$ and $QS$ are chords with intersection at $T$. If $\angle PTQ= \theta$, determine the ratio of the area of $\triangle QTP$ to the area of $\triangle SRT$ (i.e. area of $\triangle QTP$/area of $\triangle SRT$) in terms of trigonometric functions of $\theta$

PEN P Problems, 5

Tags: 3d geometry , complex analysis , additive number theory , geometry

Show that any positive rational number can be represented as the sum of three positive rational cubes.

1988 IMO Longlists, 61

Tags: partial order , combinatorics , sperner

Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students $ A$ and $ B$ such that, for each problem, $ A$ will score at least as many points as $ B.$

2024 Baltic Way, 16

Tags: perfect power , number theory , divisor

Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.

2010 VTRMC, Problem 4

Tags: triangle , geometry

Let $\triangle ABC$ be a triangle with sides $a,b,c$ and corresponding angles $A,B,C$ (so $a=BC$ and $A=\angle BAC$ etc.). Suppose that $4A+3C=540^\circ$. Prove that $(a-b)^2(a+b)=bc^2$.

1990 Greece National Olympiad, 1

Tags: angle bisector , geometric inequality , right triangle , geometry

Let $ABC$ be a right triangle with $\angle A=90^o$ and $AB<AC$. Let $AH,AD,AM$ be altitude, angle bisector and median respectively. Prove that $\frac{BD}{CD}<\frac{HD}{MD}.$

1992 Chile National Olympiad, 6

Tags: combinatorics , game , algebra

A Mathlon is a competition where there are $M$ athletic events. $A, B$ and $C$ were the only participants of a Mathlon. In each event, $p_1$ points were given to the first place, $p_2$ points to the second place and $p_3$ points to third place, with $p_1> p_2> p_3> 0$ where $p_1$, $p_2$ and $p_3$ are integer numbers. The final result was $22$ points for $A$, $9$ for $B$, and $9$ for $C$. $B$ won the $100$ meter dash. Determine $M$ and who was the second in high jump.

2008 Ukraine Team Selection Test, 9

Tags: parallelogram , ratio , geometric transformation , homothety , geometry

Given $ \triangle ABC$ with point $ D$ inside. Let $ A_0\equal{}AD\cap BC$, $ B_0\equal{}BD\cap AC$, $ C_0 \equal{}CD\cap AB$ and $ A_1$, $ B_1$, $ C_1$, $ A_2$, $ B_2$, $ C_2$ are midpoints of $ BC$, $ AC$, $ AB$, $ AD$, $ BD$, $ CD$ respectively. Two lines parallel to $ A_1A_2$ and $ C_1C_2$ and passes through point $ B_0$ intersects $ B_1B_2$ in points $ A_3$ and $ C_3$respectively. Prove that $ \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}$.

2021 Durer Math Competition Finals, 8

Tags: algebra , polynomial

John found all real numbers $p$ such that in the polynomial $g(x) = (x -1)^2(p + 2x)^2$ , the quadratic term has coefficient $2021$. What is the sum of all of these values $p$?

2002 All-Russian Olympiad Regional Round, 9.3

Tags: geometry , concyclic

In an isosceles triangle $ABC$ ($AB = BC$), point $O$ is the center of the circumcircle. Point $M$ lies on the segment $BO$, point $M' $ is symmetric to $M$ wrt the midpoint of $AB$. Point K is the intersection point of of $M'O$ and $AB$. Point $L$ lies on side BC such that $\angle CLO = \angle BLM$. Prove that points $O, K,B,L$ lie on the same circle

2024 Portugal MO, 4

Tags: geometry

A circle inscribed in the square $ABCD$, with side $10$ cm, intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$, and $P$ is the orthogonal projection of $I$ into $MN$. Find the value of segment $PI$.

2014 Contests, 1

Tags: induction , analytic geometry , graphing lines , slope , combinatorics

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.

2012 European Mathematical Cup, 4

Tags: induction , strong induction , combinatorics , tournament graphs , hamiltonian path

Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against. [i]Proposed by Matija Bucić.[/i]

2012 Indonesia TST, 3

Tags: geometry

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.