Olympiad problems

2009 Princeton University Math Competition, 6

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

2019 NMTC Junior, 8

Tags: combinatorics

A circular disc is divided into $12$ equal sectors and one of $6$ different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colorings possible.

2001 JBMO ShortLists, 8

Tags: analytic geometry , geometry

Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.

2016 Macedonia JBMO TST, 1

Tags: number theory

Solve the following equation in the set of integers $x_{1}^4 + x_{2}^4 +...+ x_{14}^4=2016^3 - 1$.

2013 Stanford Mathematics Tournament, 10

Tags: geometry

A unit circle is centered at the origin and a tangent line to the circle is constructed in the first quadrant such that it makes an angle $5\pi/6$ with the $y$-axis. A series of circles centered on the $x$-axis are constructed such that each circle is both tangent to the previous circle and the original tangent line. Find the total area of the series of circles.

2021 Brazil National Olympiad, 3

Tags: perfect square , power , number theory , floor function , irrational number

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is a perfect square minus $k$ for every integer $n$ with $n>N$.

2009 AMC 12/AHSME, 23

Tags: probability , geometry , trigonometry , rotation

A region $ S$ in the complex plane is defined by \[ S \equal{} \{x \plus{} iy: \minus{} 1\le x\le1, \minus{} 1\le y\le1\}.\] A complex number $ z \equal{} x \plus{} iy$ is chosen uniformly at random from $ S$. What is the probability that $ \left(\frac34 \plus{} \frac34i\right)z$ is also in $ S$? $ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78$

2012 Indonesia TST, 2

Tags: induction , combinatorics

An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.

2011 Puerto Rico Team Selection Test, 1

Tags: number theory

The product of 22 integers is 1. Show that their sum can not be 0.

2017 CCA Math Bonanza, L5.2

Tags: question answered

Compute $e^{\pi}+\pi^e$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is). [i]2017 CCA Math Bonanza Lightning Round #5.2[/i]

KoMaL A Problems 2020/2021, A. 780

Tags: coloring , combinatorics

We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used? [i]Based on a problem of the Dürer Competition[/i]

1985 Polish MO Finals, 6

Tags: 3d geometry , geometry , polyhedron , sphere , inscribed

There is a convex polyhedron with $k$ faces. Show that if more than $k/2$ of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.

2007 Purple Comet Problems, 8

Tags:

Penelope plays a game where she adds $25$ points to her score each time she wins a game and deducts $13$ points from her score each time she loses a game. Starting with a score of zero, Penelope plays $m$ games and has a total score of $2007$ points. What is the smallest possible value for $m$?

2013 Princeton University Math Competition, 5

Tags:

Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers.

2003 Junior Balkan MO, 1

Tags:

Let $n$ be a positive integer. A number $A$ consists of $2n$ digits, each of which is 4; and a number $B$ consists of $n$ digits, each of which is 8. Prove that $A+2B+4$ is a perfect square.

2016 Irish Math Olympiad, 10

Tags: euler circle , geometry

Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$. Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.

2010 China National Olympiad, 2

Tags: number theory

Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have \[a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases} \] Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.

2007 Peru IMO TST, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers, such that: $a+b+c \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$ Prove that: \[a+b+c \geq \frac{3}{a+b+c}+\frac{2}{abc}. \]

1984 Spain Mathematical Olympiad, 1

Tags: construction , geometry

At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$.

2025 Kosovo EGMO Team Selection Test, P4

Tags: inequalities

Let $a,b$ be positive real numbers such that $a^3+b^3=2(a^2+b^2)$. Prove the following inequality: $$ \sqrt{a^3+1} + \sqrt{b^3+1} \leq a+b+2. $$ When is equality achieved?

1977 IMO Longlists, 20

Tags: function , trigonometry , trigonometric inequality , inequality

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

2000 Harvard-MIT Mathematics Tournament, 35

Tags: function

If $1+2x+3x^2 +...=9$, find $x$.

2010 Romania Team Selection Test, 1

Tags: calculus , integration , algebra , polynomial , modular arithmetic , induction , arithmetic sequence

A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. [i]AMM Magazine[/i]

1969 Swedish Mathematical Competition, 2

Tags: irrational , trigonometry , algebra

Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.

2016 India Regional Mathematical Olympiad, 1

Tags: geometry , circumcircle

Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.