Olympiad problems

2010 Kazakhstan National Olympiad, 1

Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

2025 India STEMS Category A, 3

Tags: geometry , a-hm point , reflection

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

2003 Alexandru Myller, 2

Tags: linear algebra , matrix

Let be two $ 3\times 3 $ real matrices that have the property that $$ AX=\begin{pmatrix}0\\0\\0\end{pmatrix}\implies BX=\begin{pmatrix}0\\0\\0\end{pmatrix} , $$ for any three-dimensional vectors $ X. $ Prove that there exists a $ 3\times 3 $ real matrix $ C $ such that $ B=CA. $

2023 BMT, 4

Tags: number theory

Given positive integers $a \ge 2$ and $k$, let $m_a(k)$ denote the remainder when $k$ is divided by $a$. Compute the number of positive integers, $n$, less than 500 such that $m_2(m_5(m_{11}(n))) = 1$.

2024 Miklos Schweitzer, 6

Tags:

During heat diffusion, we say that the evolution of temperature at a point $x \in \mathbb{R}^n$ is astonishing if it changes monotonicity infinitely many times. Can it happen that the temperature evolves astonishingly at every point $x \in \mathbb{R}^n$? More precisely, does there exist a nonnegative $u \in C^2((0, +\infty) \times \mathbb{R}^n)$ solving the heat equation $\partial_t u = \Delta u$, such that $u(t,x) \to 0$ for every $x$ as $t \to \infty$, and for every $x \in \mathbb{R}^n$, the function $t \mapsto u(t,x)$ changes monotonicity infinitely many times on $(0, \infty)$?

1988 Romania Team Selection Test, 1

Tags: 3d geometry , sphere , geometric transformation , rotation , geometry

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

2001 All-Russian Olympiad, 3

Tags: geometry , circumcircle , ratio , trigonometry , geometric transformation , homothety , power of a point

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

2001 Bulgaria National Olympiad, 2

Tags: parallelogram , circumcircle , power of a point , radical axis , geometry

Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic.

2023/2024 Tournament of Towns, 2

Tags: combinatorics

2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide? Alexandr Yuran

2023 AMC 10, 12

Tags:

How many three-digit positive integers $N$ satisfy the following properties? - The number $N$ is divisible by $7$. - The number formed by reversing the digits of $N$ is divisible by $5$. $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$

2005 Miklós Schweitzer, 4

Tags: abstract algebra , free group

Let F be a countable free group and let $F = H_1> H_2> H_3> \cdots$ be a descending chain of finite index subgroups of group F. Suppose that $\cap H_i$ does not contain any nontrivial normal subgroups of F. Prove that there exist $g_i\in F$ for which the conjugated subgroups $H_i^{g_i}$ also form a chain, and $\cap H_i^{g_i}=\{1\}$. [hide=Note]Nielsen-Schreier Theorem might be useful.[/hide]

2010 Contests, 3

Tags: circumcircle , geometric transformation , reflection , geometry

Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

2016 EGMO TST Turkey, 6

Tags: number theory , squarefree

Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying \[ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. \]

2011 Saudi Arabia Pre-TST, 4.1

Tags: ratio , fixed , geometry , semicircle

On a semicircle of diameter $AB$ and center $C$, consider variable points $M$ and $N$ such that $MC \perp NC$. The circumcircle of triangle $MNC$ intersects $AB$ for the second time at $P$. Prove that $\frac{|PM-PN|}{PC}$ constant and find its value.

2024 AMC 12/AHSME, 2

Tags: factorial

What is $10! - 7! \cdot 6!$? $ \textbf{(A) }-120 \qquad \textbf{(B) }0 \qquad \textbf{(C) }120 \qquad \textbf{(D) }600 \qquad \textbf{(E) }720 \qquad $

2019 Saint Petersburg Mathematical Olympiad, 3

Tags: inequalities

Let $a, b$ and $c$ be non-zero natural numbers such that $c \geq b$ . Show that $$a^b\left(a+b\right)^c>c^b a^c.$$

1995 Yugoslav Team Selection Test, Problem 2

Tags: number theory

A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.

2000 All-Russian Olympiad, 4

Tags: inequalities , algebra , combinatorics , sequence

Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$

2012 Romania Team Selection Test, 4

Tags: algebra , function , domain , inequalities

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

1964 Miklós Schweitzer, 3

Tags: superior algebra , abstract algebra , geometric series

Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.

1972 IMO Shortlist, 7

Tags: geometry , 3d geometry , tetrahedron

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

2016 Dutch BxMO TST, 5

Tags: number theory , divide

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

MOAA Gunga Bowls, 2023.24

Tags:

Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$. [i]Proposed by Harry Kim[/i]

2020 AMC 10, 18

Tags: probability

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? $\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$

1997 Israel National Olympiad, 8

Tags: parallel , tangent circle , geometry

Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.