Olympiad problems

1967 Spain Mathematical Olympiad, 4

There is a bottle with a flat and circular bottom, closed and partially filled of wine, so that its level does not exceed the cylindrical part. Discuss in which cases the capacity of the bottle can be calculated without opening it, having only one double graduated decimeter; and if possible, describe how it would be calculated. (Problem of the Italian [i]Gara Mathematica[/i]).

2015 Postal Coaching, Problem 1

Tags: geometry , incenter

$O$ is the centre of the circumcircle of triangle $ABC$, and $M$ is its orthocentre. Point $A$ is reflected in the perpendicular bisector of the side $BC$,$ B$ is reflected in the perpendicular bisector of the side $CA$, and finally $C$ is reflected in the perpendicular bisector of the side $AB$. The images are denoted by $A_1, B_1, C_1$ respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that $O$ bisects the line segment $MK$.

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

Tags: combinatorial geometry , combinatorics

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?