Olympiad problems

2019 Argentina National Olympiad, 1

A set of distinct positive integers is called [i]singular [/i] if, for each of its elements, after crossing out that element, the remaining ones can be grouped into two sets with no common elements such that the sum of the elements in the two groups is the same. Find the smallest positive integer $n>1$ such that there exists a singular set $A$ with $n$ items.

2007 Tournament Of Towns, 1

Tags: greatest common divisor , least common multiple

[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other. [b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?

2008 Postal Coaching, 1

Tags: combination , least common multiple , lcm , number theory

Prove that for any $n \ge 1$, $LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$

2016 Thailand Mathematical Olympiad, 9

Tags: functional equation , functional , algebra

A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.

1994 Canada National Olympiad, 4

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Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $AB$. Suppose that the line through $PA$ cuts $\Omega$ again at $U$, and the line through $PB$ cuts $\Omega$ at $V$. Note that in case of tangency, $U$ may coincide with $A$ or $V$ might coincide with $B$. Also, if $P$ is on $\Omega$ then $P=U=V$. Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some $0\le s,t\in \mathbb{R}$. Determine $\cos \angle APB$ in terms of $s,t$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.1

Tags: algebra , functional

Let the function $f:R \to R$ satisfies the following conditions: 1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$ 2)$ f(1)=1$ 3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$ Prove that for all $x \in R$, $f(x) = x$.

2001 VJIMC, Problem 2

Tags: calculus , integration

Let $f:[0,1]\to\mathbb R$ be a continuous function. Define a sequence of functions $f_n:[0,1]\to\mathbb R$ in the following way: $$f_0(x)=f(x),\qquad f_{n+1}(x)=\int^x_0f_n(t)\text dt,\qquad n=0,1,2,\ldots.$$Prove that if $f_n(1)=0$ for all $n$, then $f(x)\equiv0$.

2007 Purple Comet Problems, 6

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The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum.

2009 ISI B.Math Entrance Exam, 5

Tags: number theory

Let $p$ be a prime number bigger than $5$. Suppose, the decimal expansion of $\frac{1}{p}$ looks like $0.\overline{a_1a_2\cdots a_r}$ where the line denotes a recurring decimal. Prove that $10^r$ leaves a remainder of $1$ on dividing by $p$.

2020 Stanford Mathematics Tournament, 8

Tags: geometry

Consider an acute angled triangle $\vartriangle ABC$ with side lengths $7$, $8$, and $9$. Let $H$ be the orthocenter of $ABC$. Let $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$ be the circumcircles of $\vartriangle BCH$, $\vartriangle CAH$, and $\vartriangle ABH$ respectively. Find the area of the region $\Gamma_A \cup \Gamma_B \cup \Gamma_C$ (the set of all points contained in at least one of $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$).

2010 Postal Coaching, 1

Tags: combinatorics , permutation , divisibility

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2013 Israel National Olympiad, 1

Tags: geometry , perimeter , coin

In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape? [img]https://i.imgur.com/aguQRVd.png[/img]

2019 Junior Balkan Team Selection Tests - Moldova, 7

Tags: geometry

Point $H$ is the orthocenter of the acute triangle $\Delta ABC$ and point $K$,situated on the line $(BC)$, is the foot of the perpendicular from point $A$ .The circle $\Omega$ passes through points $A$ and $K$ ,intersecting the sides $(AB)$ and $(AC)$ in points $M$ and $N$ .The line that passes through point $A$ and is parallel with $BC$ intersects again the circumcircles of triangles $\Delta AHM$ and $\Delta AHN$ in points $X$ and $Y$.Prove that $XY =BC$.

2007 Singapore Junior Math Olympiad, 1

Tags: trapezium , geometry , trapezoid , area

Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$. Suppose the area of $\vartriangle DOC$ is $2S/9$. Find the value of $a/b$.

2014 Kyiv Mathematical Festival, 2

Tags: rotation

Can an $8\times8$ board be covered with 13 equal 5-celled figures? It's alowed to rotate the figures or turn them over. [size=85](Kyiv mathematical festival 2014)[/size]

2005 Estonia Team Selection Test, 4

Tags: polynomial , algebra , real root

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

2016 ASDAN Math Tournament, 10

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Using the fact that $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6},$$ compute $$\int_0^1(\ln x)\ln(1-x)dx.$$

2011 ELMO Shortlist, 5

Tags: algorithm , combinatorics , logarithm

Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where a) $f(n)=n\ln{n}$; b) $f(n)=n$. [i]David Yang.[/i]

2003 Baltic Way, 10

Tags: calculus , integration , analytic geometry , combinatorics

A [i]lattice point[/i] in the plane is a point with integral coordinates. The[i] centroid[/i] of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$. Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.

2014 Turkey EGMO TST, 5

Tags: geometry , circumcircle

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.

2005 Greece Team Selection Test, 3

Tags: polynomial , algebra

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.

2016 USAMO, 4

Tags: function , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, $$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$

2016 Ecuador NMO (OMEC), 6

Tags: number theory , divisible

A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that: $\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$. $\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$. Find the largest positive "olympic" integer.

2023 Thailand Online MO, 1

Tags: combinatorics , enumerative combinatorics

Let $n$ be a positive integer. Chef Kao has $n$ different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the $2n$ ice cream cups into two rows of $n$ cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions: [list] [*]each row contains all ice cream flavors, and [*]each column has different sizes of ice cream cup. [/list]Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.

1998 Cono Sur Olympiad, 6

Tags: combinatorics

The mayor of a city wishes to establish a transport system with at least one bus line, in which: - each line passes exactly three stops, - every two different lines have exactly one stop in common, - for each two different bus stops there is exactly one line that passes through both. Determine the number of bus stops in the city.