Olympiad problems

2005 MOP Homework, 3

Prove that the equation $a^3-b^3=2004$ does not have any solutions in positive integers.

2022 Romania Team Selection Test, 1

Tags: combinatorics , geometry

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

2011 Today's Calculation Of Integral, 755

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

2019 IFYM, Sozopol, 2

Tags: functional equation , number theory

Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$: $f(f(f(n)))=n+2f(n)$?

2019 USAMTS Problems, 5

Tags:

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for any reals $x, y,$ $$f(x + y)f(x - y) = (f(x))^2 - (f(y))^2$$. Additionally, suppose that $f(x + 2 \pi) = f(x)$ and that there does not exist a positive real $a < 2 \pi$ such that $f(x + a) = f(x)$ for all reals $x$. Show that for all reals $x$, $$|f(\frac{\pi}{2})| \geq f(x)$$.

2022-IMOC, A3

Tags: algebra , function , functional equation

Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$ [i]Proposed by USJL[/i]

2023 Sharygin Geometry Olympiad, 9.6

Tags: geometry , orthocenter , tangent circle

Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.

1999 Mongolian Mathematical Olympiad, Problem 5

Tags: combinatorics

Let $A_1,\ldots,A_m$ be three-element subsets of an $n$-element set $X$ such that $|A_i\cup A_j|\le1$ whenever $i\ne j$. Prove that there exists a subset $A$ of $X$ with $|A|\ge2\sqrt n$ such that it does not contain any of the $A_i$.

1980 IMO, 3

Tags: induction , combinatorial geometry , euclidean distance , geometry

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

2009 District Olympiad, 1

Tags: function , inequalities , integral , real analysis

Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality: $$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations: [b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $ [b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $

2018 Hanoi Open Mathematics Competitions, 2

Tags: geometry , triangle

In triangle $ABC,\angle BAC = 60^o, AB = 3a$ and $AC = 4a, (a > 0)$. Let $M$ be point on the segment $AB$ such that $AM =\frac13 AB, N$ be point on the side $AC$ such that $AN =\frac12AC$. Let $I$ be midpoint of $MN$. Determine the length of $BI$. A. $\frac{a\sqrt2}{19}$ B. $\frac{2a}{\sqrt{19}}$ C. $\frac{19a\sqrt{19}}{2}$ D. $\frac{19a}{\sqrt2}$ E. $\frac{a\sqrt{19}}{2}$

2017 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: geometry , angle

Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.

2005 MOP Homework, 6

Tags: geometry , 3d geometry , analytic geometry , combinatorics

A $10 \times 10 \times 10$ cube is made up up from $500$ white unit cubes and $500$ black unit cubes, arranged in such a way that every two unit cubes that shares a face are in different colors. A line is a $1 \times 1 \times 10$ portion of the cube that is parallel to one of cube’s edges. From the initial cube have been removed $100$ unit cubes such that $300$ lines of the cube has exactly one missing cube. Determine if it is possible that the number of removed black unit cubes is divisible by $4$.

2006 India IMO Training Camp, 3

Tags: modular arithmetic , invariant , combinatorics

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

2011 ISI B.Stat Entrance Exam, 1

Tags: calculus , inequalities , rearrangement inequality , n-variable inequality

Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that \[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]

1983 IMO, 3

Tags: modular arithmetic , number theory , frobenius , additive number theory , additive combinatorics

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

2014 Iran MO (2nd Round), 3

Tags: quadratic , function , algebra , quadratic formula , inequalities

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

1969 Poland - Second Round, 6

Tags: combinatorics , geometry , 3d geometry , polyhedron , combinatorial geometry

Prove that every polyhedron has at least two faces with the same number of sides.

2007 IMO Shortlist, 7

Tags: modular arithmetic , number theory , prime factorization , factorial , combinatorial number theory

For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$. [i]Author: Tejaswi Navilarekkallu, India[/i]

1976 Swedish Mathematical Competition, 1

Tags: combinatorics

In a tournament every team plays every other team just once. Each game is won by one of the teams (there are no draws). Each team loses at least once. Show that there must be three teams $A$, $B$, $C$ such that $A$ beat $B$, $B$ beat $C$ and $C$ beat $A$.

1999 IMO, 5

Tags: geometry , homothety , circumcircle

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

1987 IMO Longlists, 19

Tags: symmetry , combinatorics of words , counting , digit , recurrence relation , combinatorics

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

2021 Science ON all problems, 2

Tags: linear algebra

Consider $A,B\in\mathcal{M}_n(\mathbb{C})$ for which there exist $p,q\in\mathbb{C}$ such that $pAB-qBA=I_n$. Prove that either $(AB-BA)^n=O_n$ or the fraction $\frac{p}{q}$ is well-defined ($q \neq 0$) and it is a root of unity. [i](Sergiu Novac)[/i]

2006 AMC 12/AHSME, 6

Tags:

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? $ \textbf{(A) } 129 \qquad \textbf{(B) } 137 \qquad \textbf{(C) } 174 \qquad \textbf{(D) } 223 \qquad \textbf{(E) } 411$

2007 Argentina National Olympiad, 5

Tags: number theory , sum of digits

We will say that a positive integer is [i]lucky [/i ]if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive [i]lucky [/i ] numbers?