Olympiad problems

2014 NIMO Problems, 1

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ with the property that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. Then the length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by Michael Ren[/i]

1979 IMO Longlists, 31

Tags: combinatorics , set systems , extremal combinatorics

Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ; (ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$ (iii) $\bigcup_{X \in F} X = R$

1989 AMC 12/AHSME, 17

Tags: geometry , perimeter

The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \ \text{cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? $\text{(A)} \ 0 \qquad \text{(B)} \ 9 \qquad \text{(C)} \ 221 \qquad \text{(D)} \ 663 \qquad \text{(E)} \ \text{infinitely many}$

2015 Korea - Final Round, 1

Tags: function , algebra , functional equation

Find all functions $f: R \rightarrow R$ such that $f(x^{2015} + (f(y))^{2015}) = (f(x))^{2015} + y^{2015}$ holds for all reals $x, y$

1995 Tournament Of Towns, (448) 4

Tags: number theory

Can the number $a + b + c + d$ be prime if $a, b, c$ and $d$ are positive integers and $ab = cd$?

1984 Swedish Mathematical Competition, 2

Tags: coloring , combinatorial geometry , combinatorics

The squares in a $3\times 7$ grid are colored either blue or yellow. Consider all $m\times n$ rectangles in this grid, where $m \in \{2,3\}$, $n \in \{2,...,7\}$. Prove that at least one of these rectangles has all four corner squares the same color.

2019 HMNT, 4

Tags: geometry

In $\vartriangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\vartriangle ABC$ so that each $n$-gon shares a side with a distinct side of $\vartriangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?

2018 Greece Junior Math Olympiad, 4

Tags: geometry , tangent circle

Let $ABC$ with $AB<AC<BC$ be an acute angled triangle and $c$ its circumcircle. Let $D$ be the point diametrically opposite to $A$. Point $K$ is on $BD$ such that $KB=KC$. The circle $(K, KC)$ intersects $AC$ at point $E$. Prove that the circle $(BKE)$ is tangent to $c$.

2025 Kyiv City MO Round 1, Problem 1

Tags: geometry

Lines $ FD $ and $ BE $ intersect at point $ O $. Rays $ OA $ and $ OC $ are drawn from point $ O $. You are given the following information about the angles: \[ \angle DOC = 36^\circ, \quad \angle AOC = 90^\circ, \quad \angle AOB = 4x, \quad \angle FOE = 5x, \] as shown in the figure below. What is the degree measure of $ x $? [img]https://i.ibb.co/m5rwmXm/Kyiv-MO-2025-R1-7.png[/img]

2016 Harvard-MIT Mathematics Tournament, 2

Tags:

Sherry is waiting for a train. Every minute, there is a $75\%$ chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75\%$ chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?

1949-56 Chisinau City MO, 3

Tags: perfect cube , geometry , 3d geometry , number theory

Prove that the number $N = 10 ...050...01$ (1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.

1986 All Soviet Union Mathematical Olympiad, 433

Tags: chessboard , diagonal , geometry

Find the relation of the black part length and the white part length for the main diagonal of the a) $100\times 99$ chess-board; b) $101\times 99$ chess-board.

2002 India National Olympiad, 2

Tags: number theory

Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ . Find all $a$, $b$, $c$ which give the smallest value

2019 China Western Mathematical Olympiad, 8

Tags: combinatorics , set

We call a set $S$ a [i]good[/i] set if $S=\{x,2x,3x\}(x\neq 0).$ For a given integer $n(n\geq 3),$ determine the largest possible number of the [i]good[/i] subsets of a set containing $n$ positive integers.

2011 Harvard-MIT Mathematics Tournament, 2

Tags: hmmt

Let $H$ be a regular hexagon of side length $x$. Call a hexagon in the same plane a "distortion" of $H$ if and only if it can be obtained from $H$ by translating each vertex of $H$ by a distance strictly less than $1$. Determine the smallest value of $x$ for which every distortion of $H$ is necessarily convex.

2020 Princeton University Math Competition, A6/B8

Tags: algebra

Given integer $n$, let $W_n$ be the set of complex numbers of the form $re^{2qi\pi}$, where $q$ is a rational number so that $q_n \in Z$ and $r$ is a real number. Suppose that p is a polynomial of degree $ \ge 2$ such that there exists a non-constant function $f : W_n \to C$ so that $p(f(x))p(f(y)) = f(xy)$ for all $x, y \in W_n$. If $p$ is the unique monic polynomial of lowest degree for which such an $f$ exists for $n = 65$, find $p(10)$.

2014 PUMaC Algebra B, 4

Tags: princeton , college

Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?

2009 CHKMO, 3

Tags: trigonometry , ratio , incenter , inradius , circumcircle , parallelogram , geometry

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$

2023 CMIMC Algebra/NT, 10

Tags: algebra , number theory

For a given $n$, consider the points $(x,y)\in \mathbb{N}^2$ such that $x\leq y\leq n$. An ant starts from $(0,1)$ and, every move, it goes from $(a,b)$ to point $(c,d)$ if $bc-ad=1$ and $d$ is maximized over all such points. Let $g_n$ be the number of moves made by the ant until no more moves can be made. Find $g_{2023} - g_{2022}$. [i]Proposed by David Tang[/i]

2011 Macedonia National Olympiad, 4

Tags: function , search , algebra

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]

2016 Turkmenistan Regional Math Olympiad, Problem 2

Tags: inequalities , geometry

If $a,b,c$ are triangle sides then prove that $(\sum_{cyc}\sqrt{\frac{a}{-a+b+c}} \geq 3$

2004 Regional Competition For Advanced Students, 3

Tags: geometry

Given is a convex quadrilateral $ ABCD$ with $ \angle ADC\equal{}\angle BCD>90^{\circ}$. Let $ E$ be the point of intersection of the line $ AC$ with the parallel line to $ AD$ through $ B$ and $ F$ be the point of intersection of the line $ BD$ with the parallel line to $ BC$ through $ A$. Show that $ EF$ is parallel to $ CD$

2017 Balkan MO Shortlist, C6

Tags: combinatorics , combinatorial geometry , minimum , diagonal

What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

2009 Harvard-MIT Mathematics Tournament, 7

Tags: function

Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]

1980 IMO Shortlist, 1

Tags: trigonometry , geometry , triangle , perpendicular bisector

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.