Olympiad problems

2012 BAMO, 1

Hugo places a chess piece on the top left square of a $20 \times 20$ chessboard and makes $10$ moves with it. On each of these $10$ moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $X$ on the square that the piece occupies. When Hugo plays the game over and over again, what is the largest possible number of squares that could eventually be marked with an $X$? Prove that your answer is correct.

1997 All-Russian Olympiad Regional Round, 8.5

Tags: geometry , geometric inequality , square

Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

2016 Bangladesh Mathematical Olympiad, 2

Tags: number theory , modular arithmetic , divisibility

(a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?

2011 Serbia JBMO TST, 1

Tags: combinatorics , number theory , algebra

A $tetromino$ is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into $4\times 7$ rectangle without overlapping. [/list]

2014 Vietnam National Olympiad, 1

Tags: symmetry , geometry , gauss , trigonometry , circumcircle , radical axis , perpendicular bisector

Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$ a) Prove that $A,P,Q$ are collinear. b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.

Russian TST 2021, P1

Tags: excircle , geometry

A point $P{}$ is considered on the incircle of the triangle $ABC$. We draw the tangent segments from $P{}$ to the three excircles of $ABC$. Prove that from the obtained three tangent segments it is possible to make a right triangle if and only if the point $P{}$ lies on one of the lines connecting two of the midpoints of the sides of $ABC$.

2018 MIG, 5

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A fair six sided die is rolled to give a number $n$. A fair two sided coin is then flipped $n^2$ times. Find the expected number of heads flipped. Express your answer as a common fraction.

2022 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry

Given is a trapezoid $ABCD$, $AD \parallel BC$. The angle bisectors of the two pairs of opposite angles meet at $X, Y$. Prove that $AXYD$ and $BXYC$ are cyclic.

2016 IFYM, Sozopol, 6

Tags: algebra , polynomial

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 5

Tags: geometry , collinearity , collinear , circumcenter , appolonius circle , homothety , perpendicular bisector

Let $\triangle ABC$ be a triangle with circumcenter $O$. The perpendicular bisectors of the segments $OA,OB$ and $OC$ intersect the lines $BC,CA$ and $AB$ at $D,E$ and $F$, respectively. Prove that $D,E,F$ are collinear.

2007 Romania Team Selection Test, 2

Tags: parallelogram , trapezoid , geometry

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

2014 Cono Sur Olympiad, 1

Tags: inequalities , number theory , least common multiple , invariant , greatest common divisor , combinatorics

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

1979 All Soviet Union Mathematical Olympiad, 273

Tags: sequence , algebra , inequalities

For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $$x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$$ Prove that for every $n$, it also satisfies $$x_1+x_2/2+x_3/3+...+x_n/n\le 3$$

1979 IMO Longlists, 50

Tags: number theory , additive number theory , additive combinatorics , system of equations

Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.

2022 VTRMC, 1

Tags: algebra

Give all possible representations of $2022$ as a sum of at least two consecutive positive integers and prove that these are the only representations.

2005 Purple Comet Problems, 4

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A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?

2021 Belarusian National Olympiad, 10.3

Tags: number theory

Odd numbers $x,y,z$ such that $gcd(x,y,z)=1$ are given. It turned out that $x^2+y^2+z^2 \vdots x+y+z$ Prove that $x+y+z-2$ is not divisible by $3$

2022 Mexican Girls' Contest, 5

Tags: algebra

A biologist found a pond with frogs. When classifying them by their mass, he noticed the following: [i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass. [/i]As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?

2010 LMT, 11

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Al and Bob are playing Rock Paper Scissors. Al plays rock. What is the probability that Al wins, given that Bob plays randomly and has an equal probability of playing rock, paper, and scissors?

2007 JBMO Shortlist, 3

Tags: geometry

Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at $N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such that $\frac{DP}{DN}=\frac{BD}{CD}$ . Show that $DM \perp PN$ .

2013 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory

Given are positive integers $a, b, c$ such that $a$ is odd, $b>c$, $a, b, c$ are coprime and $a(b-c) =2bc$. Prove that $abc$ is square

2001 Taiwan National Olympiad, 4

Tags: geometry , similar triangles , incenter , locus , circumcircle

Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.

2012 Online Math Open Problems, 23

Tags: function , geometry , 3d geometry

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]

1969 IMO Shortlist, 52

Tags: polygon , dissection , geometry

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

2014 German National Olympiad, 5

Tags: combinatorics , coin , weight

There are $9$ visually indistinguishable coins, and one of them is fake and thus lighter. We are given $3$ indistinguishable balance scales to find the fake coin; however, one of the scales is defective and shows a random result each time. Show that the fake coin can still be found with $4$ weighings.