Olympiad problems

1999 Tournament Of Towns, 4

$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$. (V Proizvolov)

2013 Tournament of Towns, 1

Tags: point , combinatorics , combinatorial geometry

There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?

2007 Indonesia TST, 1

Tags: circumcircle , incenter , geometry

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

2023 Junior Balkan Team Selection Tests - Moldova, 7

Tags: combinatorics

Every point on a circle is coloured in blue or yellow such there is at least a point of each color. Prove that for every colouring of the circle there is always an isosceles triangle inscribed inside the circle 1) with all vertexes of the same colour. 2) with vertexes of both colours. For every colouring of the circle is there an equilateral triangle inscribed inside the circle 3) with all vertexes of the same colour? 4) with vertexes of both colours?

2025 Junior Balkan Team Selection Tests - Romania, P2

Tags: geometry

Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$

2006 Taiwan TST Round 1, 3

Tags: modular arithmetic , number theory , divisibility , chinese remainder theorem

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

2016 May Olympiad, 2

Tags: combinatorics

How many squares must be painted at least on a $5 \times 5$ board such that in each row, in each column and in each $2 \times 2$ square is there at least one square painted?

2016 Saudi Arabia BMO TST, 3

Tags: number theory , divide , divisible

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

2011 Today's Calculation Of Integral, 766

Tags: calculus , integration , function , trigonometry , inequalities , calculus computations

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

1994 ITAMO, 5

Tags: area , minimum , maximum , cube , plane , geometry

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

2004 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , decimal representation , perfect square

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

2012 Indonesia TST, 1

Tags: induction , algebra , binary representation , sequence , recurrence relation

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

Russian TST 2019, P2

Tags: permutation , combinatorics

For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.

2018 Iran MO (1st Round), 18

Tags: geometry

Three rods of lengths $1396, 1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?

2021 Kyiv Mathematical Festival, 1

Tags: number theory

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

2000 China National Olympiad, 1

Tags: permutation , combinatorics , permutation statistics

Given an ordered $n$-tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$, we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the “[i]innovated tuple[/i]” of $A$. The number of distinct elements in $B$ is called the “[i]innovated degree[/i]” of $A$. Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$

2019 ASDAN Math Tournament, 2

Tags:

Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$?

2015 USAMTS Problems, 3

Tags:

Let $P$ be a convex n-gon in the plane with vertices labeled $V_1,...,V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles the blue and green regions are the same. Suppose $P$ has exactly one balancing point/ Show that the balancing point must be a vertex of $P$

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , root , trinomial

Let $a$ and $b$ be arbitrary distinct numbers. Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.

2024 Brazil Cono Sur TST, 3

Tags: combinatorics , set

For a pair of integers $a$ and $b$, with $0<a<b<1000$, a set $S\subset \begin{Bmatrix}1,2,3,...,2024\end{Bmatrix}$ $escapes$ the pair $(a,b)$ if for any elements $s_1,s_2\in S$ we have $\left|s_1-s_2\right| \notin \begin{Bmatrix}a,b\end{Bmatrix}$. Let $f(a,b)$ be the greatest possible number of elements of a set that escapes the pair $(a,b)$. Find the maximum and minimum values of $f$.

2023 Bulgaria EGMO TST, 2

Tags: number theory , greatest common divisor , function

Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.

2010 Miklós Schweitzer, 10

Tags: topology

Consider the space $ \{0,1 \} ^{N} $ with the product topology (where $\{0,1 \}$ is a discrete space). Let $ T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}} $ be the left-shift, ie $ (Tx) (n) = x (n+1) $ for every $ n \in \mathbb {N} $. Can a finite number of Borel sets be given: $ B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N} $ such that $$ \left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \} $$the $ \sigma $-algebra generated by the set system coincides with the Borel set system?

2005 France Pre-TST, 4

Tags: inequalities

Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2 = 25.$ Find the minimum of $\frac {xy} z + \frac {yz} x + \frac {zx} y .$ Pierre.

2021 AMC 12/AHSME Fall, 7

Tags: important announcement

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$ $\textbf{(B)}\: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

1955 Moscow Mathematical Olympiad, 314

Tags: polynomial , root , algebra

Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0$, cannot have two positive roots.