Olympiad problems

2023 Sharygin Geometry Olympiad, 8.2

The bisectors of angles $A$, $B$, and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$, $B_1$, $C_1$ respectively. Let $A_2$, $B_2$, $C_2$ be the midpoints of segments $AA_1$, $BB_1$, $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

2019 India PRMO, 18

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What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors ?

2010 All-Russian Olympiad Regional Round, 10.4

Tags: number theory , divisible

We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.

2023 MOAA, 5

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Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

2024 Sharygin Geometry Olympiad, 9.6

Tags: geo , geometry

The incircle of a triangle $ABC$ centered at $I$ touches the sides $BC, CA$, and $AB$ at points $A_1, B_1, $ and $C_1$ respectively. The excircle centered at $J$ touches the side $AC$ at point $B_2$ and touches the extensions of $AB, BC$ at points $C_2, A_2$ respectively. Let the lines $IB_2$ and $JB_1$ meet at point $X$, the lines $IC_2$ and $JC_1$ meet at point $Y$, the lines $IA_2$ and $JA_1$ meet at point $Z$. Prove that if one of points $X, Y, Z$ lies on the incircle then two remaining points also lie on it.

2013 Hanoi Open Mathematics Competitions, 2

Tags: function , minimum , algebra

The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

2021 China Team Selection Test, 1

Tags: combinatorial geometry , combinatorics , graph theory

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

1988 IMO Longlists, 74

Tags: linear recurrence , inequality system , inequality , sequence

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

2015 CHMMC (Fall), 5

Tags: combinatorics

Felix is playing a card-flipping game. $n$ face-down cards are randomly colored, each with equal probability of being black or red. Felix starts at the $1$st card. When Felix is at the $k$th card, he guesses its color and then flips it over. For $k < n$, if he guesses correctly, he moves onto the $(k + 1)$-th card. If he guesses incorrectly, he gains $k$ penalty points, the cards are replaced with newly randomized face-down cards, and he moves back to card $1$ to continue guessing. If Felix guesses the $n$th card correctly, the game ends. What is the expected number of penalty points Felix earns by the end of the game?

2011 Swedish Mathematical Competition, 6

Tags: algebra , functional , functional equation

How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and $$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$ for all non-negative integers $x$, $y$?

2021 AMC 10 Fall, 4

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At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ $(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

2014 Iran MO (3rd Round), 5

Tags: ratio , algebra , polynomial , number theory

Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )

1997 Tournament Of Towns, (525) 2

Tags: geometry , combinatorics , combinatorial geometry

Baron Munchausen plays billiards on a table with the shape of an equilateral triangle. He claims to have shot a ball from one of the sides of this table so that it passed through a certain point three times in three different directions and then returned to the original point on the side. Can that be true, assuming that the usual law of reflection holds? (Μ Evdokimov)

1976 Putnam, 6

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Suppose $f(x)$ is a twice continuously differentiable real valued function defined for all real numbers $x$ and satisfying $$|f(x)| \leq 1$$ for all x and $$(f(0))^2+(f'(0))^2=4.$$ Prove that there exists a real number $x_0$ such that $$f(x_0)+f''(x_0)=0.$$

1982 IMO Longlists, 7

Tags: modular arithmetic , algorithm , floor function , number theory , greatest common divisor , euclidean algorithm

Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

1988 Tournament Of Towns, (176) 2

Tags: diagonal , isosceles trapezoid , trapezoid , geometry , parallel

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

2024 Assara - South Russian Girl's MO, 6

Tags: geometry

In the regular hexagon $ABCDEF$, a point $X$ was marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$? [i]A.V.Smirnov, I.A.Efremov[/i]

2010 Tournament Of Towns, 1

Tags: ratio , number theory

Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?

2003 China Team Selection Test, 3

Tags: number theory

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.

2022 Saudi Arabia BMO + EGMO TST, 2.3

Tags: number theory , combinatorics

Let $n$ be an even positive integer. On a board n real numbers are written. In a single move we can erase any two numbers from the board and replace each of them with their product. Prove that for every $n$ initial numbers one can in finite number of moves obtain $n$ equal numbers on the board.

2009 Indonesia TST, 3

Tags: circumcircle , geometry

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

2019 Greece National Olympiad, 2

Tags: geometry

Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $\overarc{AB}$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$.

2023 MMATHS, 7

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A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x,y).$ For every distinct pair of lights $(x_1, y_1), (x_2, y_2),$ with $x_1<x_2$ and $y_1>y_2,$ all lights strictly between them (i.e. $x_1<x<x_2$ and $y_2<y<y_1$) are toggled. After this procedure is done, how many lights are on?

1986 ITAMO, 1

Tags: parallel , circles , tangent , geometry

Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$.

2012 239 Open Mathematical Olympiad, 3

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There are $n$ points in the space such that none $4$ of them lie on a plane. You can select two points $A$ and $B$ and move point $A$ to the midpoint of line segment $AB$. It turned out that, after several moves, the points took the same places (possibly in a different order). What is the smallest value of $n$ that this could happen for some $n$ points?