Olympiad problems

2023 Spain Mathematical Olympiad, 1

Tags:

A $3\times 3\times 3$ cube is made of 27 unit cube pieces. Each piece contains a lamp, which can be on or off. Every time a piece is pressed (the center piece cannot be pressed), the state of that piece and the pieces that share a face with it changes. Initially all lamps are off. Determine which of the following states are achievable: (1) All lamps are on. (2) All lamps are on except the central one. (3) Only the central lamp is on.

2021 CMIMC Integration Bee, 13

Tags: calculus , integration

$$\int_0^1 x\ln(x^2)\ln(1+x)\,dx$$ [i]Proposed by Connor Gordon[/i]

2015 All-Russian Olympiad, 8

Tags: combinatorics

$N\geq9$ distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than $1$. It happens that for very $8$ distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values $N$ for which this is possible. [i](F. Nilov)[/i]

2014 JBMO Shortlist, 3

Tags: number theory

Find all integer solutions to the equation $x^2=y^2(x+y^4+2y^2)$ .

2021 Princeton University Math Competition, 6

Tags: combinatorics

Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.

KoMaL A Problems 2023/2024, A. 869

Tags: inequality , algebra

Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\ldots \le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le \ldots\le y_n$ be $B$. Find the largest possible value of \[\sum_{i=1}^n (x_i-y_i)^2.\] [i]Proposed by Péter Csikvári, Budapest[/i]

2011 Indonesia TST, 1

Tags: number theory , algebra

Find all real number $x$ which could be represented as $x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$

1989 Tournament Of Towns, (216) 4

Tags: combinatorial geometry , geometry , 3d geometry

Is it possible to mark a diagonal on each little square on the surface of a Rubik 's cube so that one obtains a non-intersecting path? (S . Fomin, Leningrad)

1999 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , function

$f$ is a continuous real-valued function such that $f(x+y)=f(x)f(y)$ for all real $x$, $y$. If $f(2)=5$, find $f(5)$.

2022 Thailand Online MO, 1

Tags: algebra , equation

Determine, with proof, all triples of real numbers $(x,y,z)$ satisfying the equations $$x^3+y+z=x+y^3+z=x+y+z^3=-xyz.$$

2019 Purple Comet Problems, 9

Tags: geometry

A semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/b/c/c10258e2e15cab74abafbac5ff50b1d0fd42e6.png[/img]

2016 Korea National Olympiad, 3

Tags: geometry , geometric inequality

Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$. Let $T$ be the area of $\triangle DEF$. Prove the following inequality. $$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$

2019 Dutch IMO TST, 3

Tags: geometry , parallelogram , circumcircle , concurrency

Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.

MOAA Gunga Bowls, 2023.16

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Compute the sum $$\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}$$ where $\varphi(n)$ returns the number of positive integers less than $n$ that are relatively prime to $n$. [i]Proposed by Andy Xu[/i]

2016 Online Math Open Problems, 23

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$S$ be the set of all $2017^2$ lattice points $(x,y)$ with $x,y\in \{0\}\cup\{2^{0},2^{1},\cdots,2^{2015}\}$. A subset $X\subseteq S$ is called BQ if it has the following properties: (a) $X$ contains at least three points, no three of which are collinear. (b) One of the points in $X$ is $(0,0)$. (c) For any three distinct points $A,B,C \in X$, the orthocenter of $\triangle ABC$ is in $X$. (d) The convex hull of $X$ contains at least one horizontal line segment. Determine the number of BQ subsets of $S$. [i] Proposed by Vincent Huang [/i]

2024 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics

In a very large City, they are building a subway: there are many stations, some pairs of which are connected by tunnels, and from any station you can get through tunnels to any other. All metro tunnels must be divided into "lines": each line consists of several consecutive tunnels, all stations in which are different (in particular, the line should not be circular); lines consisting of one tunnel are also allowed. By law, it is required that you can get from any station to any other station by making no more than $100$ transfers from line to line. At what is the largest $N$, any connected metro with $N$ stations can be divided into lines, observing the law?

2008 India Regional Mathematical Olympiad, 5

Tags: number theory

Three nonzero real numbers $ a,b,c$ are said to be in harmonic progression if $ \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}$. Find all three term harmonic progressions $ a,b,c$ of strictly increasing positive integers in which $ a \equal{} 20$ and $ b$ divides $ c$. [17 points out of 100 for the 6 problems]

2008 Romania National Olympiad, 1

Tags: circumcircle , geometric transformation , reflection , perpendicular bisector , geometry

Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.

2020 Dürer Math Competition (First Round), P2

Tags: cell , combinatorics , table

How many ways can you fill a table of size $n\times n$ with integers such that each cell contains the total number of even numbers in its row and column other than itself? Two tables are different if they differ in at least one cell.

2009 Iran Team Selection Test, 4

Tags: algebra , polynomial , number theory

Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition: \[ p \mid uv - 1 \] we always have $p \mid f(u)f(v) - 1$.

1962 AMC 12/AHSME, 16

Tags: geometry , rectangle , ratio

Given rectangle $ R_1$ with one side $ 2$ inches and area $ 12$ square inches. Rectangle $ R_2$ with diagonal $ 15$ inches is similar to $ R_1.$ Expressed in square inches the area of $ R_2$ is: $ \textbf{(A)}\ \frac92 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ \frac{135}{2} \qquad \textbf{(D)}\ 9 \sqrt{10} \qquad \textbf{(E)}\ \frac{27 \sqrt{10}}{4}$

2008 Cuba MO, 6

Tags: geometry , geometric inequality

We have an isosceles triangle $ABC$ with base $BC$. Through vertex $A$ draw a line $r$ parallel to $BC$. The points $P, Q$ are located on the perpendicular bisectors of $AB$ and $AC$ respectively, such that $PQ\perp BC$. They are points $M$ and $N$ on the line $r$ such that $\angle APM = \angle AQN = 90^o$. Prove that $$\frac{1}{AM} + \frac{1}{AN}\le \frac{2}{ AB}$$

2021 Greece Junior Math Olympiad, 4

Tags: geometry , equal circles

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

1991 AIME Problems, 5

Tags: number theory , prime number , prime factorization

Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?

2002 Czech and Slovak Olympiad III A, 6

Tags: function , algebra

Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality \[f(xf(y))=f(xy)+x\]