Olympiad problems

2014 Costa Rica - Final Round, 2

Find all positive integers $n$ such that $n!+2$ divides $(2n)!$.

1997 APMO, 2

Tags: number theory

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[ \frac{2^n+2}{n} \] is also an integer.

2020 Online Math Open Problems, 7

Tags: Online Math Open

On a $5 \times 5$ grid we randomly place two \emph{cars}, which each occupy a single cell and randomly face in one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequence of moves such that, afterward, both cars occupy the same cell is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$. [i]Proposed by Sean Li[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 9.4

Tags: algebra , inequalities

For real numbers $x \ge 0$ and $y \ge 0$, prove the inequality $$x^4+y^3+x^2+y+1 >\frac92 xy.$$

1982 Austrian-Polish Competition, 4

Tags: number theory , Sequence , recurrence relation , bounded , unbounded , Digits , product of digits

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

2005 IMO Shortlist, 2

Tags: geometry , ratio , IMO , Triangle , IMO 2005 , concurrence

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent. [i]Bogdan Enescu, Romania[/i]

1995 IMO, 1

Tags: geometry , circumcircle , concurrency , IMO , imo 1995

Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.

1983 Miklós Schweitzer, 6

Tags: search , Functional Analysis , linear algebra , linear algebra unsolved

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

2025 Belarusian National Olympiad, 9.5

Tags: combinatorics

Polina and Yan have $n$ cards, on the first card on one side $1$ is written, on the other side $n+1$, on the second card on one side $2$ is written, on the other side $n+2$, etc. Polina laid all cards in a circle in some order. Yan wants to turn some cards such that the numbers on the top sides of adjacent cards were not coprime. For every positive integer $n \geq 3$ determine can Yan accomplish that regardless of the actions of Polina. [i]M. Shutro[/i]

1978 AMC 12/AHSME, 6

Tags: AMC

The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \begin{align*}x&=x^2+y^2, \\ y&=2xy\end{align*} is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

2016 IberoAmerican, 3

Tags: geometry , circumcircle , Iberoamerican , Iberoamerican 2016

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.

1999 Akdeniz University MO, 3

Tags: inequalities , algebra , inequalities proposed

Let $a$,$b$,$c$ and $d$ positive reals. Prove that $$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$

2023 Baltic Way, 14

Tags: geometry , orthogonalaty , Centroid

Let $ABC$ be a triangle with centroid $G$. Let $D, E, F$ be the circumcenters of triangles $BCG, CAG, ABG$. Let $X$ be the intersection of the perpendiculars from $E$ to $AB$ and from $F$ to $AC$. Prove that $DX$ bisects $EF$.

2016 Postal Coaching, 4

Tags: number theory , combinatorics

Let $n \in \mathbb N$. Prove that for each factor $m \ge n$ of $n(n + 1)/2$, one can partition the set $\{1,2, 3,\cdots , n\}$ into disjoint subsets such that the sum of elements in each subset is equal to $m$.

2024 Putnam, A1

Tags: Putnam

Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying \[ 2a^n+3b^n=4c^n. \]

2010 Saint Petersburg Mathematical Olympiad, 3

Tags: combinatorics

There are $2009$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $75$ different cities?

The Golden Digits 2024, P1

Tags: algebra , inequalities

Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\][i]Proposed by Andrei Vila[/i]

2005 USAMTS Problems, 2

Tags: probability , expected value

Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops. One such sequence is \[ 12, 6, 6, 3, 3, 3, 1. \] What is the expected value of the number of terms in Anna’s sequence?

2008 AIME Problems, 9

Tags: analytic geometry , geometry , geometric transformation , rotation , linear algebra , matrix , AMC

A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.

1951 Putnam, B4

Tags: Putnam

Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.

1993 National High School Mathematics League, 9

Tags: complex numbers

If $z\in\mathbb{C},\arg{(z^2-4)}=\frac{5}{6}\pi,\arg{(z^2+4)}=\frac{\pi}{3}$, then the value of $z$ is________.

2013-2014 SDML (High School), 8

Tags:

Twenty-four congruent squares are arranged as shown in the figure. In how many ways can we select $12$ of the squares so that no two are diagonally adjacent? Directly adjacent spaces are acceptable.

1974 Spain Mathematical Olympiad, 3

Tags: linear algebra , algebra

We will designate by $Z_{(5)}$ a certain subset of the set $Q$ of the rational numbers . A rational belongs to $Z_{(5)}$ if and only if there exist equal fraction to this rational such that $5$ is not a divisor of its denominator. (For example, the rational number $13/10$ does not belong to $Z_{(5)}$ , since the denominator of all fractions equal to $13/10$ is a multiple of $5$. On the other hand, the rational $75/10$ belongs to $Z_{(5)}$ since that $75/10 = 15/12$). Reasonably answer the following questions: a) What algebraic structure (semigroup, group, etc.) does $Z_{(5)}$ have with respect to the sum? b) And regarding the product? c) Is $Z_{(5)}$ a subring of $Q$? d) Is $Z_{(5)}$ a vector space?

1999 Korea Junior Math Olympiad, 1

Tags: geometry , convex quadrilateral , inscribed quadrilateral , KJMO

There exists point $O$ inside a convex quadrilateral $ABCD$ satisfying $OA=OB$ and $OC=OD$, and $\angle AOB = \angle COD=90^{\circ}$. Consider two squares, (1)square having $AC$ as one side and located in the opposite side of $B$ and (2)square having $BD$ as one side and located in the opposite side of $E$. If the common part of these two squares is also a square, prove that $ABCD$ is an inscribed quadrilateral.

2006 All-Russian Olympiad Regional Round, 9.4

Tags: geometry , tangent

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect the circumcircle of this triangle at points $A_0$ and $C_0$, respectively. A straight line passing through the center of the inscribed circle of a triangle $ABC$ is parallel to side $AC$ and intersects line $A_0C_0$ at point $P$. Prove that line $PB$ is tangent to the circumcircle of the triangle $ABC$.