Olympiad problems

2016 District Olympiad, 4

Let $ a\ge 2 $ be a natural number. Show that the following relations are equivalent: $ \text{(i)} \ a $ is the hypothenuse of a right triangle whose sides are natural numbers. $ \text{(ii)}\quad $ there exists a natural number $ d $ for which the polynoms $ X^2-aX\pm d $ have integer roots.

2010 Today's Calculation Of Integral, 631

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

1988 National High School Mathematics League, 12

Tags: inequalities

$a,b$ are real numbers, satisfying that $\frac{1}{a}+\frac{1}{b}=1$. Prove that for any $n\in\mathbb{Z}_+$, $(a+b)^{2n}-a^n-b^n\geq2^{2n}-2^{n+1}$.

2005 All-Russian Olympiad Regional Round, 9.7

Tags: arithmetic sequence , number theory

9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$? ([i]V. Senderov[/i])

May Olympiad L1 - geometry, 2004.4

Tags: square , geometry , find angle , area

In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.

2023 Iran Team Selection Test, 3

Tags: combinatorics

Arman, starting from a number, calculates the sum of the cubes of the digits of that number, and again calculates the sum of the cubes of the digits of the resulting number and continues the same process. Arman calls a number $Good$ if it reaches $1$ after performing a number of steps. Prove that there is an arithmetic progression of length $1402$ of good numbers. [i]Proposed by Navid Safaei [/i]

1994 IMC, 5

Tags: calculus , real analysis

a) Let $f\in C[0,b]$, $g\in C(\mathbb R)$ and let $g$ be periodic with period $b$. Prove that $\int_0^b f(x) g(nx)\,\mathrm dx$ has a limit as $n\to\infty$ and $$\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx$$ b) Find $$\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx$$

2016 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics

We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?

1997 India National Olympiad, 1

Tags: parallelogram , trigonometry , circumcircle , geometry

Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]

1986 Greece Junior Math Olympiad, 4

Tags: algebra

i) If $b=\dfrac{a^2+ \dfrac{1}{a^2}}{a^2-\dfrac{1}{a^2}}$ , express $c=\dfrac{a^4+\dfrac{1}{a^4}}{a^4-\dfrac{1}{a^4}}$ , in terms of $b$. ii) If $k= \frac{x^{n}+\dfrac{1}{x^{n}}}{x^{n}-\dfrac{1}{x^{n}}}$, express $m= \frac{x^{2n}+\dfrac{1}{x^{2n}}}{x^{2n}-\dfrac{1}{x^{2n}}}$ in terms of $k$.

2021 Science ON grade V, 4

Tags: number theory

Find the least positive integer which is a multiple of $13$ and all its digits are the same. [i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]

2022 BMT, 8

Tags: geometry

Anton is playing a game with shapes. He starts with a circle $\omega_1$ of radius $1$, and to get a new circle $\omega_2$, he circumscribes a square about $\omega_1$ and then circumscribes circle $\omega_2$ about that square. To get another new circle $\omega_3$, he circumscribes a regular octagon about circle $\omega_2$ and then circumscribes circle $\omega_3$ about that octagon. He continues like this, circumscribing a $2n$-gon about $\omega_{n-1}$ and then circumscribing a new circle $\omega_n$ about the $2n$-gon. As $n$ increases, the area of $\omega_n$ approaches a constant $A$. Compute $A$.

2025 District Olympiad, P2

Tags: linear algebra , matrix

Let $n\in\mathbb{Z}$, $n\geq 3$. A matrix $A\in\mathcal{M}_n(\mathbb{C})$ is said to have property $(\mathcal{P})$ if $\det(A+X_{ij})=\det(A+X_{ji})$, for all $i,j\in\{1,2,\dots ,n\}$, where $X_{ij}\in\mathcal{M}_n(\mathbb{C})$ is the matrix with $1$ on position $(i,j)$ and $0$ otherwise. [list=a] [*] Show that if $A\in\mathcal{M}_n(\mathbb{C})$ has property $(\mathcal{P})$ and $\det(A)\neq 0$, then $A=A^T$. [*] Give an example of a matrix $A\in\mathcal{M}_n(\mathbb{C})$ with property $(\mathcal{P})$ such that $A\neq A^T$.

1981 Austrian-Polish Competition, 6

Tags: recurrence relation , sequence , unbounded , bounded , algebra

The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$,$ y_{n+1}=z_n +\frac{1}{y_n}$,$z_{n+1}=x_n +\frac{1}{z_n} $ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.

2018 Balkan MO Shortlist, A1

Tags: algebra , inequality

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2021 New Zealand MO, 3

Tags: number theory , divisible

In a sequence of numbers, a term is called [i]golden [/i] if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, . . . , 2021$?

2018 AIME Problems, 10

Tags: function

Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.

2023-24 IOQM India, 4

Tags: number theory , divisibility , diophantine

Let $x, y$ be positive integers such that $$ x^4=(x-1)\left(y^3-23\right)-1 . $$ Find the maximum possible value of $x+y$.

2015 CCA Math Bonanza, L3.3

Tags:

Michael the Mouse stands in a circle with $11$ other mice. Eshaan the Elephant walks around the circle, squashing every other non-squashed mouse he comes across. If it takes Eshaan $1$ minute ($60$ seconds) to complete one circle and he walks at a constant rate, what is the maximum length of time in seconds from when the first mouse is squashed that Michael can survive? [i]2015 CCA Math Bonanza Lightning Round #3.3[/i]

2021 AMC 12/AHSME Fall, 3

Tags:

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 $N?$ $(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

2007 Grigore Moisil Intercounty, 2

Tags: function , real analysis

Prove that $ |f(x)|\le |f(0)| +\int_0^x |f(t) +f'(t)|dt , $ for any nonnegative real numbers $ x, $ and functions $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ of class $ \mathcal{C}^1. $

2007 ITest, 22

Tags: algebra

Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }223&\textbf{(U) }678\\\\ \textbf{(V) }2007 & &\end{array}$

2019 Tournament Of Towns, 5

Tags: algorithm , game theory , magician , combinatorics

A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.

2016 Argentina National Olympiad, 6

Tags: combinatorics , time

Let $AB$ be a segment of length $1$. Several particles start moving simultaneously at constant speeds from $A$ up to$ B$. As soon as a particle reaches $B$ , turns around and goes to $A$; when it reaches $A$, begins to move again towards $ B$ , and so on indefinitely. Find all rational numbers$ r>1$ such that there exists an instant $t$ with the following property: For each $n\ge 1$ , if $n+1$ particles with constant speeds $1,r,r^2,…,r^n$ move as described, at instant $t$, they all lie at the same interior point of segment $AB$.

2022 Greece Junior Math Olympiad, 3

Tags: combinatorics

On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold: (i) The sum of every $40$ consecutive numbers is equal to $0$. (ii) The sum of every $42$ consecutive numbers is not equal to $0$. We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$.