Olympiad problems

2025 All-Russian Olympiad, 11.4

Tags: combinatorics

A natural number $N$ is given. A cube with side length $2N + 1$ is made up of $(2N + 1)^3$ unit cubes, each of which is either black or white. It turns out that among any $8$ cubes that share a common vertex and form a $2 \times 2 \times 2$ cube, there are at most $4$ black cubes. What is the maximum number of black cubes that could have been used?

2016 Harvard-MIT Mathematics Tournament, 1

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Let $x$ and $y$ be complex numbers such that $x+y=\sqrt{20}$ and $x^2+y^2=15$. Compute $|x-y|$.

2009 Ukraine National Mathematical Olympiad, 3

Tags: geometry , rectangle

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

2001 District Olympiad, 3

Tags: function , real analysis

Let $f:\mathbb{R}\to \mathbb{R}$ a function which transforms any closed bounded interval in a closed bounded interval and any open bounded interval in an open bounded interval. Prove that $f$ is continuous. [i]Mihai Piticari[/i]

2011 Mongolia Team Selection Test, 2

Tags: incenter , circumcircle , geometry

Let $ABC$ be a scalene triangle. The inscribed circle of $ABC$ touches the sides $BC$, $CA$, and $AB$ at the points $A_1$, $B_1$, $C_1$ respectively. Let $I$ be the incenter, $O$ be the circumcenter, and lines $OI$ and $BC$ meet at point $D$. The perpendicular line from $A_1$ to $B_1 C_1$ intersects $AD$ at point $E$. Prove that $B_1 C_1$ passes through the midpoint of $EA_1$.

2006 Princeton University Math Competition, 9

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The curve $y=x^4+2x^3-11x^2-13x+35$ has a bitangent (a line tangent to the curve at two points). What is the equation of the bitangent?

2025 AIME, 10

Tags: combinatorics

Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.

2012 AMC 12/AHSME, 4

Tags: percent

Suppose that the euro is worth $1.30$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the value of Etienne's money greater than the value of Diana's money? ${{ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6.5\qquad\textbf{(D)}\ 8}\qquad\textbf{(E)}\ 13} $

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 1

Tags: linear algebra , matrix , algebra , polynomial

Let $A=\left( \begin{array}{ccc} 1 & 1& 0 \\ 0 & 1& 0 \\ 0 &0 & 2 \end{array} \right),\ B=\left( \begin{array}{ccc} a & 1& 0 \\ b & 2& c \\ 0 &0 & a+1 \end{array} \right)\ (a,\ b,\ c\in{\mathbb{C}}).$ (1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$ (2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.

2010 Indonesia TST, 2

Tags: modular arithmetic , number theory

Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] [i]Nanang Susyanto, Jogjakarta[/i]

2009 Kazakhstan National Olympiad, 2

Tags: perpendicular bisector , geometry

In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$. Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.

2007 Germany Team Selection Test, 2

Tags: induction , combinatorics

Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.

2004 National Olympiad First Round, 23

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What is the maximal possible value of $n$ such that no matter how $25$ squares are selected in an infinite chessboard one can find $n$ squares in which none of them share a common corner? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

2017 Iberoamerican, 1

Tags: number theory

For every positive integer $n$ let $S(n)$ be the sum of its digits. We say $n$ has a property $P$ if all terms in the infinite secuence $n, S(n), S(S(n)),...$ are even numbers, and we say $n$ has a property $I$ if all terms in this secuence are odd. Show that for, $1 \le n \le 2017$ there are more $n$ that have property $I$ than those who have $P$.

2016 Baltic Way, 6

Tags: algebra

The set $\{1, 2, . . . , 10\}$ is partitioned to three subsets $A, B$ and $C.$ For each subset the sum of its elements, the product of its elements and the sum of the digits of all its elements are calculated. Is it possible that $A$ alone has the largest sum of elements, $B$ alone has the largest product of elements, and $C$ alone has the largest sum of digits?

2024 Brazil Cono Sur TST, 4

Tags: geometry , power of a point

Let $ABC$ be a triangle, $O$ its circumcenter and $\Gamma$ its circumcircle. Let $E$ and $F$ be points on $AB$ and $AC$, respectively, such that $O$ is the midpoint of $EF$. Let $A'=AO\cap \Gamma$, with $A'\ne A$. Finally, let $P$ be the point on line $EF$ such that $A'P\perp EF$. Prove that the lines $EF,BC$ and the tangent to $\Gamma$ at $A'$ are concurrent and that $\angle BPA' = \angle CPA'$.

2003 VJIMC, Problem 1

Tags: matrix , linear algebra

Two real square matrices $A$ and $B$ satisfy the conditions $A^{2002}=B^{2003}=I$ and $AB=BA$. Prove that $A+B+I$ is invertible. (The symbol $I$ denotes the identity matrix.)

Kvant 2024, M2795

Tags: number theory , combinatorial geometry

Is it possible to release a ray on a plane from each point with rational coordinates so that no two rays have a common point and at the same time, among the lines containing these rays, no two are parallel and do not coincide? [i]Proposed by P. Kozhevnikov[/i]

2019 Korea Junior Math Olympiad., 6

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfies the followings. (Note that $\mathbb{R}$ stands for the set of all real numbers) (1) For each real numbers $x$, $y$, the equality $f(x+f(x)+xy) = 2f(x)+xf(y)$ holds. (2) For every real number $z$, there exists $x$ such that $f(x) = z$.

2004 Czech and Slovak Olympiad III A, 2

Tags: system of equations , combinatorics , algebra

Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression \[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]

2010 Tournament Of Towns, 3

Tags: ratio , absolute value

From a police station situated on a straight road innite in both directions, a thief has stolen a police car. Its maximal speed equals $90$% of the maximal speed of a police cruiser. When the theft is discovered some time later, a policeman starts to pursue the thief on a cruiser. However, he does not know in which direction along the road the thief has gone, nor does he know how long ago the car has been stolen. Is it possible for the policeman to catch the thief?

2024 Australian Mathematical Olympiad, P1

Tags: number theory

Determine all triples $(k, m, n) $ of positive integers satisfying $$k!+m!=k!n!.$$

2020 Ukrainian Geometry Olympiad - December, 5

Tags: parallel , centroid , circumcenter , circumcircle , geometry

Let $ABC$ be an acute triangle with $\angle ACB = 45^o$, $G$ is the point of intersection of the medians, and $O$ is the center of the circumscribed circle. If $OG =1$ and $OG \parallel BC$, find the length of $BC$.

2019 Philippine MO, 4

Tags: concyclic , equal angles , circumcircle , geometry

In acute triangle $ABC $with $\angle BAC > \angle BCA$, let $P$ be the point on side $BC$ such that $\angle PAB = \angle BCA$. The circumcircle of triangle $AP B$ meets side $AC$ again at $Q$. Point $D$ lies on segment $AP$ such that $\angle QDC = \angle CAP$. Point $E$ lies on line $BD$ such that $CE = CD$. The circumcircle of triangle $CQE$ meets segment $CD$ again at $F$, and line $QF$ meets side $BC$ at $G$. Show that $B, D, F$, and $G$ are concyclic

1998 Romania National Olympiad, 3

Tags: geometry , isosceles , right triangle

In the exterior of the triangle $ABC$ with $m(\angle B) > 45^o$, $m(\angle C) >45°^o$ one constructs the right isosceles triangles $ACM$ and $ABN$ such that $m(\angle CAM) = m(\angle BAN) = 90^o$ and, in the interior of $ABC$, the right isosceles triangle $BCP$, with $m(\angle P) = 90^o$. Show that the triangle $MNP$ is a right isosceles triangle.