Olympiad problems

2022 BMT, 1

Tags: number theory

For lunch, Lamy, Botan, Nene, and Polka each choose one of three options: a hot dog, a slice of pizza, or a hamburger. Lamy and Botan choose different items, and Nene and Polka choose the same item. In how many ways could they choose their items?

2024 Mozambican National MO Selection Test, P1

Tags: counting

A school security guard works from Monday to Saturday from $7:30 am$ to $12:00 pm$ ($7:30$ to $12:00$). He also works the night shift, from Monday to Friday from $6pm$to $10pm$ ($18:00$ to $22:00$) . He receives $75MT$ per hour, up to $40$ hours of work per week. For the remaining hours of weekly work, he receives $95MT$ per hour. So, considering that a month has four weeks, what will be this security guard's monthly salary?

2023 Balkan MO Shortlist, G2

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ lying in the interior. Let $E$ and $F$ be the midpoints of the segments $BC$ and $AD$, respectively. Let $X$ be the point lying on the same side of the line $EF$ as the vertex $C$ such that $\triangle EXF$ and $\triangle BOA$ are similar. Prove that $XC = XD$.

2022 China Team Selection Test, 5

Tags: algebra , inequality , combinatorics

Let $n$ be a positive integer, $x_1,x_2,\ldots,x_{2n}$ be non-negative real numbers with sum $4$. Prove that there exist integer $p$ and $q$, with $0 \le q \le n-1$, such that \[ \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, \] where the indices are take modulo $2n$. [i]Note:[/i] If $q=0$, then $\sum_{i=1}^q x_{p+2i-1}=0$; if $q=n-1$, then $\sum_{i=q+1}^{n-1} x_{p+2i}=0$.

2023 HMNT, 27

Tags:

Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color.

2013 USAMTS Problems, 2

Tags: geometry , trapezoid , quadratic , algebra , function , domain

Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.

1998 Baltic Way, 18

Tags: combinatorics

Determine all positive integers $n$ for which there exists a set $S$ with the following properties: (i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$; (ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.

2022 Princeton University Math Competition, A1 / B3

Tags: geometry , analytic geometry

Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt3$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{m\pi}{n}$ forrelatively prime positive integers $m, n$, find $m + n$. (Two circles intersect orthogonally at a point $P$ if the tangent lines at $P$ form a right angle.)

2014 NIMO Problems, 8

Tags: complex number

Let $a$, $b$, $c$, $d$ be complex numbers satisfying \begin{align*} 5 &= a+b+c+d \\ 125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\ 1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\ 25 &= a^4+b^4+c^4+d^4 \end{align*} Compute $abcd$. [i]Proposed by Evan Chen[/i]

PEN S Problems, 36

Tags: modular arithmetic , induction , limit

For every natural number $n$, denote $Q(n)$ the sum of the digits in the decimal representation of $n$. Prove that there are infinitely many natural numbers $k$ with $Q(3^{k})>Q(3^{k+1})$.

1967 IMO Shortlist, 4

Tags: modular arithmetic , number theory , perfect cube , diophantine equation

Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

2005 Czech-Polish-Slovak Match, 4

Tags: floor function , algebra , polynomial , combinatorics

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

1976 IMO Longlists, 15

Tags: euler , ratio , symmetry , geometry

Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.

2012 Morocco TST, 1

Tags: modular arithmetic , number theory

Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.

2005 Postal Coaching, 25

Tags: algebra

Find all pairs of cubic equations $x^3 +ax^2 +bx +c =0$ and $x^3 +bx^2 + ax +c = 0$ where $a,b,c$ are integers, such that each equation has three integer roots and both the equations have exactly one common root.

2018 PUMaC Number Theory A, 1

Tags: number theory

Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.

2011 Oral Moscow Geometry Olympiad, 1

Tags: geometry , circumcircle , tangent , symmetry

$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

2013 Korea National Olympiad, 2

Tags: inequalities , algebra

Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that \[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]

2016 Switzerland - Final Round, 2

Tags: algebra , geometric inequalities , inequalities

Let $a, b$ and $c$ be the sides of a triangle, that is: $a + b > c$, $b + c > a$ and $c + a > b$. Show that: $$\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32$$

2018 Malaysia National Olympiad, A1

Tags: geometry

A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$?

LMT Speed Rounds, 2011.5

Tags:

The unit of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed with the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to only make one purchase, what is the maximum possible number of screws he can buy?

LMT Team Rounds 2021+, 8

Tags: number theory

An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.

Ukrainian TYM Qualifying - geometry, II.2

Tags: geometry , congruent triangles , 3d geometry , tetrahedron

Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?

2010 IFYM, Sozopol, 2

Tags: geometry

Let $ABCD$ be a quadrilateral, with an inscribed circle with center $I$. Through $A$ are constructed perpendiculars to $AB$ and $AD$, which intersect $BI$ and $DI$ in points $M$ and $N$ respectively. Prove that $MN\perp AC$.

2002 China Team Selection Test, 3

Tags: combinatorics

Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ rounds so that each contestant has played a game with all the $ 2m\minus{}1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.