Olympiad problems

1997 Slovenia National Olympiad, Problem 1

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

2004 Harvard-MIT Mathematics Tournament, 9

Tags: geometry

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

2020-21 KVS IOQM India, 26

Tags: number theory

Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$. Find this sum.

2019 CMI B.Sc. Entrance Exam, 2

Tags: complex number , diophantine equation , algebra

$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy \begin{align*} \vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}} \end{align*} $(b)$ Find all real numbers $x$ that satisfy following equation $:$ \begin{align*} \frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6} \end{align*}

2015 AIME Problems, 3

Tags: number theory , prime number , factoring polynomials

There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

1998 Slovenia National Olympiad, Problem 4

Tags: number theory

Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?

2019 IFYM, Sozopol, 6

Tags: function , algebra

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: $xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$

1954 AMC 12/AHSME, 21

Tags:

The roots of the equation $ 2\sqrt {x} \plus{} 2x^{ \minus{} \frac {1}{2}} \equal{} 5$ can be found by solving: $ \textbf{(A)}\ 16x^2 \minus{} 92x \plus{} 1 \equal{} 0 \qquad \textbf{(B)}\ 4x^2 \minus{} 25x \plus{} 4 \equal{} 0 \qquad \textbf{(C)}\ 4x^2 \minus{} 17x \plus{} 4 \equal{} 0 \\ \textbf{(D)}\ 2x^2 \minus{} 21x \plus{} 2 \equal{} 0 \qquad \textbf{(E)}\ 4x^2 \minus{} 25x \minus{} 4 \equal{} 0$

2023 Brazil National Olympiad, 6

Tags: sequence , prime number , number theory

For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties: • $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$; • $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$. Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.

2017 Hanoi Open Mathematics Competitions, 7

Tags: number theory , digit

Determine two last digits of number $Q = 2^{2017} + 2017^2$

2022 CMIMC, 2.6

Tags: geometry

A triangle $\triangle ABC$ satisfies $AB = 13$, $BC = 14$, and $AC = 15$. Inside $\triangle ABC$ are three points $X$, $Y$, and $Z$ such that: [list] [*] $Y$ is the centroid of $\triangle ABX$ [*] $Z$ is the centroid of $\triangle BCY$ [*] $X$ is the centroid of $\triangle CAZ$ [/list] What is the area of $\triangle XYZ$? [i]Proposed by Adam Bertelli[/i]

2010 Germany Team Selection Test, 3

Tags: function , trigonometry , functional equation , algebra

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

1983 Bundeswettbewerb Mathematik, 3

Tags: number theory , digit

A real number is called [i]triplex[/i] if it has a decimal representation in which none of $0$ and $3$ different digit occurs. Prove that every positive real number is the sum of nine triplex numbers.

2010 Contests, 1

Tags:

Bob has picked positive integer $1<N<100$. Alice tells him some integer, and Bob replies with the remainder of division of this integer by $N$. What is the smallest number of integers which Alice should tell Bob to determine $N$ for sure?

1997 Spain Mathematical Olympiad, 2

Tags: square , combinatorics , grid , square grid , maximum

A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ no three of which form an isosceles right triangle?

1964 IMO Shortlist, 5

Tags: combinatorial geometry , 3d geometry , counting , intersection , geometry

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

2008 Argentina Iberoamerican TST, 1

Tags: number theory

Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square It's a nice problem ...hope you enjoy it! Daniel

1997 All-Russian Olympiad, 2

Tags: combinatorics

An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells. [i]E. Poroshenko[/i]

2013 National Chemistry Olympiad, 43

Tags:

To whom is the discovery of the nuclear atom attributed? $ \textbf{(A) }\text{Neils Bohr}\qquad\textbf{(B) }\text{Louis deBroglie}\qquad$ $\textbf{(C) }\text{Robert Millikan}\qquad\textbf{(D) }\text{Ernest Rutherford}\qquad $

2016 NIMO Problems, 2

Tags:

Define the [i]hotel elevator cubic [/i]as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$? [i]Proposed by Evan Chen[/i]

2024 Princeton University Math Competition, B1

Tags: geometry

Jeff the delivery driver starts at the point $(5, 0)$ and has to make deliveries at all the other lattice points with distance $5$ from the origin before returning to his starting point. The length of a shortest possible path he can make is $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b.$ Find $a + b.$

2016 Greece Team Selection Test, 1

Tags: number theory , sequence , divisibility

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

2010 Contests, A2

Tags: function , integration , real analysis , slope

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

2012 India IMO Training Camp, 1

Tags: geometry , incenter , symmetry , reflection

Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$. [i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]

2023 BMT, 2

Tags: algebra

Compute $1 \times 4 - 2 \times 3 + 2 \times 5 - 3 \times 4 + 3 \times 6 - 4 \times 5 + 4 \times 7 - 5 \times 6 + 5 \times 8 - 6 \times 7.$