Olympiad problems

2013 Abels Math Contest (Norwegian MO) Final, 4b

A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c \ge 2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

2021 LMT Fall, 2

Tags: combinatorics

How many ways are there to permute the letters $\{S,C,R, A,M,B,L,E\}$ without the permutation containing the substring $L AME$?

MathLinks Contest 2nd, 6.2

Tags: geometry

A triangle $ABC$ is located in a cartesian plane $\pi$ and has a perimeter of $3 + 2\sqrt3$. It is known that the triangle $ABC$ has the property that any triangle in the plane $\pi$, congruent with it, contains inside or on the boundary at least one lattice point (a point with both coordinates integers). Prove that the triangle $ABC$ is equilateral.

1965 Putnam, A2

Tags: binomial coefficient

Show that, for any positive integer $n$, \[ \sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1}, \] where $[x]$ means the greatest integer not exceeding $x$, and $\textstyle\binom nr$ is the binomial coefficient "$n$ choose $r$", with the convention $\textstyle\binom n0 = 1$.

2016 Postal Coaching, 4

Tags: number theory , inequalities , algebra

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

2009 Ukraine National Mathematical Olympiad, 3

Tags: inequalities , trigonometry , function , algebra , domain , geometry

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]

2016 District Olympiad, 1

Tags: equation , diophantine equation , algebra

Solve in $ \mathbb{N}^2: $ $$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$

1998 Mexico National Olympiad, 5

Tags: tangent , geometry

The tangents at points $B$ and $C$ on a given circle meet at point $A$. Let $Q$ be a point on segment $AC$ and let $BQ$ meet the circle again at $P$. The line through $Q $ parallel to $AB$ intersects $BC$ at $J$. Prove that $PJ$ is parallel to $AC$ if and only if $BC^2 = AC\cdot QC$.

2001 Tournament Of Towns, 1

Tags: induction , number theory

The natural number $n$ can be replaced by $ab$ if $a + b = n$, where $a$ and $b$ are natural numbers. Can the number $2001$ be obtained from $22$ after a sequence of such replacements?

2025 Caucasus Mathematical Olympiad, 5

Tags: combinatorics

Given a $20 \times 25$ board whose rows are numbered from $1$ to $20$ and whose columns are numbered from $1$ to $25$, Nikita wishes to place one precious stone in some cells of this board so that at least one stone is present and the following magical condition holds: for any $1 \leqslant i \leqslant 20$ and $1 \leqslant j \leqslant 25$, there is a stone in the cell at the intersection of the $i^\text{th}$ row and the $j^\text{th}$ column if and only if the cross formed by the union of the $i^\text{th}$ row and the $j^\text{th}$ column contains exactly $i + j$ stones. Determine whether Nikita's wish is achievable.

2024 SG Originals, Q5

Tags: combinatorics , grid , number theory , prime number , prime

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2000 AIME Problems, 9

Tags: algebra , system of equations , binomial theorem , logarithm

The system of equations \begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*} has two solutions $ (x_{1},y_{1},z_{1})$ and $ (x_{2},y_{2},z_{2}).$ Find $ y_{1} + y_{2}.$

1991 Arnold's Trivium, 92

Tags: group theory , abstract algebra , geometry , geometric transformation , rotation , 3d geometry

Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.

2019 Dutch Mathematical Olympiad, 2

Tags: combinatorics

There are $n$ guests at a party. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. What are the possible values of $n$?

2021 Sharygin Geometry Olympiad, 17

Tags: incenter , geometry

Let $ABC$ be an acute-angled triangle. Points $A_0$ and $C_0$ are the midpoints of minor arcs $BC$ and $AB$ respectively. A circle passing though $A_0$ and $C_0$ meet $AB$ and $BC$ at points $P$ and $S$ , $Q$ and $R$ respectively ([i]all these points are distinct[/i]). It is known that $PQ\parallel AC$. Prove that $A_0P+C_0S=C_0Q+A_0R$.

1969 AMC 12/AHSME, 13

Tags: geometry

A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals: $\textbf{(A) }\sqrt a:\sqrt b\qquad \textbf{(B) }\sqrt a:\sqrt{a-b}\qquad \textbf{(C) }\sqrt b:\sqrt{a-b}\qquad$ $\textbf{(D) }a:\sqrt{a-b}\qquad \textbf{(E) }b:\sqrt{a-b}$

2017 Polish Junior Math Olympiad Finals, 1.

Tags: number theory

Let $a$, $b$, and $c$ be positive integers for which the number \[\frac{a\sqrt2+b}{b\sqrt2+c}\] is rational. Show that the number $ab+bc+ca$ is divisible by $a+b+c$.

2002 Tuymaada Olympiad, 3

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

The points $D$ and $E$ on the circumcircle of an acute triangle $ABC$ are such that $AD=AE = BC$. Let $H$ be the common point of the altitudes of triangle $ABC$. It is known that $AH^{2}=BH^{2}+CH^{2}$. Prove that $H$ lies on the segment $DE$. [i]Proposed by D. Shiryaev[/i]

2024 Bangladesh Mathematical Olympiad, P2

Tags: geometry , cyclic quadrilateral , power of a point

In a cyclic quadrilateral $ABCD$, the diagonals intersect at $E$. $F$ and $G$ are on chord $AC$ and chord $BD$ respectively such that $AF = BE$ and $DG = CE$. Prove that, $A, G, F, D$ lie on the same circle.

2019 Ecuador NMO (OMEC), 3

Tags: divisible , divide , power of 2 , number theory

For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

1981 AMC 12/AHSME, 26

Tags: probability , geometric series , geometric sequence

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $ \frac{1}{6}$, independent of the outcome of any other toss.) $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{2}{9}\qquad \textbf{(C)}\ \frac{5}{18}\qquad \textbf{(D)}\ \frac{25}{91}\qquad \textbf{(E)}\ \frac{36}{91}$

2002 Iran Team Selection Test, 13

Tags: geometry , symmetry , geometric transformation , incenter , projective geometry , poles and polars

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.

2004 Harvard-MIT Mathematics Tournament, 2

Tags: number theory

What is the largest whole number that is equal to the product of its digits?

2013 Costa Rica - Final Round, F1

Tags: algebra , functional equation , functional

Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

2008 AMC 8, 19

Tags: probability

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart? [asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} $