Olympiad problems

2022 AMC 10, 8

Tags: arithmetic mean

A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$? $\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

1999 Italy TST, 2

Tags: inequalities , geometry

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

2020 Estonia Team Selection Test, 2

Tags: excircle , equal segments , circumcircle , geometry

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

2019 Indonesia MO, 3

Tags: geometry

Given that $ABCD$ is a rectangle such that $AD > AB$, where $E$ is on $AD$ such that $BE \perp AC$. Let $M$ be the intersection of $AC$ and $BE$. Let the circumcircle of $\triangle ABE$ intersects $AC$ and $BC$ at $N$ and $F$. Moreover, let the circumcircle of $\triangle DNE$ intersects $CD$ at $G$. Suppose $FG$ intersects $AB$ at $P$. Prove that $PM = PN$.

2002 South africa National Olympiad, 2

Tags: ratio , geometric sequence

Find all triples of natural numbers $(a,b,c)$ such that $a$, $b$ and $c$ are in geometric progression and $a + b + c = 111$.

PEN K Problems, 28

Tags: functional equation , function

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]

2012 IFYM, Sozopol, 4

Tags: algebra , inequality

Prove that if $x$, $y$, and $z$ are non-negative numbers and $x^2+y^2+z^2=1$, then the following inequality is true: $\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2 }\geq \frac{3\sqrt{3}}{2}$

2016 BMT Spring, 15

Tags: algebra

Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$. $$\prod_{i=1}^{3}(4s^4_i + 81)$$ can be written as $2^a3^b5^c$. Find $a + b + c$.

2008 AMC 12/AHSME, 4

Tags:

Which of the following is equal to the product \[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}? \]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$

2019 IFYM, Sozopol, 8

Tags: number theory

Solve the following equation in integers: $4n^4+7n^2+3n+6=m^3$.

2024 Thailand October Camp, 5

Tags: function , algebra

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.2

Tags: algebra

A bus and a cyclist left town $A$ at $10$ o'clock in the same direction, and a motorcyclist left town $B$ to meet them $15$ minutes later. The bus drove past the pedestrian at $10$ o'clock $30$ minutes, met the motorcyclist at $11$ o'clock and arrived in the city of $B$ at $12$ o'clock. The motorcyclist met the cyclist $15$ minutes after meeting the bus and another $15$ minutes later caught up with the pedestrian. At what time did the cyclist and the pedestrian meet? (The speeds and directions of movement of all participants were equal, the pedestrian and the motorcyclist were moving in the direction of city $A$.)

2010 Romania Team Selection Test, 2

Tags: function , parameterization , induction , inequalities , rearrangement inequality , algebra

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

2017 Turkey MO (2nd round), 1

Tags: combinatorics

A wedding is going to be held in a city with $25$ types of meals, to which some of the $2017$ citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "$suitable$ $list$" is a set of citizens, such that each meal is liked by at least one person in the set. A "$kamber$ $group$" is a set that contains at least one person from each "$suitable$ $list$". Given a "$kamber$ $group$", which has no subset (other than itself) that is also a "$kamber$ $group$", prove that there exists a meal, which is liked by everyone in the group.

2017 Federal Competition For Advanced Students, 4

Tags: number theory

Find all pairs $(a,b)$ of non-negative integers such that: $$2017^a=b^6-32b+1$$ [i]proposed by Walther Janous[/i]

2019 Sharygin Geometry Olympiad, 1

Tags: geometry

Given a triangle $ABC$ with $\angle A = 45^\circ$. Let $A'$ be the antipode of $A$ in the circumcircle of $ABC$. Points $E$ and $F$ on segments $AB$ and $AC$ respectively are such that $A'B = BE$, $A'C = CF$. Let $K$ be the second intersection of circumcircles of triangles $AEF$ and $ABC$. Prove that $EF$ bisects $A'K$.

2005 iTest, 3

Tags: probability , number theory , perfect square , pascal triangle

Find the probability that any given row in Pascal’s Triangle contains a perfect square. [i] (.1 point)[/i]

2021 Malaysia IMONST 1, 14

Tags: algebra , polynomial

Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. Each coefficient $a, b, c, d, e$, and$ f$ is equal to either $ 1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?

Oliforum Contest IV 2013, 8

Tags: combinatorics

Two distinct real numbers are written on each vertex of a convex $2012-$gon. Show that we can remove a number from each vertex such that the remaining numbers on any two adjacent vertices are different.

2011 Oral Moscow Geometry Olympiad, 3

Tags: ruler , geometry , construction , diagonal

A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

2003 China Team Selection Test, 3

Tags: induction , modular arithmetic , binomial theorem , algebra

Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

2016 Taiwan TST Round 2, 3

Tags: combinatorial geometry , combinatorics

There is a grid of equilateral triangles with a distance 1 between any two neighboring grid points. An equilateral triangle with side length $n$ lies on the grid so that all of its vertices are grid points, and all of its sides match the grid. Now, let us decompose this equilateral triangle into $n^2$ smaller triangles (not necessarily equilateral triangles) so that the vertices of all these smaller triangles are all grid points, and all these small triangles have equal areas. Prove that there are at least $n$ equilateral triangles among these smaller triangles.