Olympiad problems

2017 Mathematical Talent Reward Programme, MCQ: P 2

Tags: limit , calculus

$\lim \limits_{x\to \infty} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}=$ [list=1] [*] $\sqrt{e}$ [*] $\infty$ [*] Does not exists [*] None of these [/list]

2011 Dutch IMO TST, 1

Tags: combinatorics

Let $n \ge 2$ and $k \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$ .

2014 National Olympiad First Round, 26

Tags: modular arithmetic , number theory

Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

2003 Costa Rica - Final Round, 2

Tags: geometry

Let $AB$ be a diameter of circle $\omega$. $\ell$ is the tangent line to $\omega$ at $B$. Take two points $C$, $D$ on $\ell$ such that $B$ is between $C$ and $D$. $E$, $F$ are the intersections of $\omega$ and $AC$, $AD$, respectively, and $G$, $H$ are the intersections of $\omega$ and $CF$, $DE$, respectively. Prove that $AH=AG$.

2015 Baltic Way, 5

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation \[|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))\] for all real numbers $x$ and $y$.

2001 Federal Competition For Advanced Students, Part 2, 3

Tags: geometry

A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.

LMT Guts Rounds, 6

Tags:

Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?

2014 China Team Selection Test, 4

Tags: geometry , circumcircle , geometric transformation , reflection , trigonometry , trig identities , law of sines

Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$. (Edited)

2020 Czech-Austrian-Polish-Slovak Match, 1

Tags: geometry , parallelogram , circumcircle , computer problems

Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear. (Patrik Bak, Slovakia)

2023 Thailand TSTST, 5

Tags: combinatorics

Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.

2023 BMT, 10

Tags: geometry

Let $\vartriangle ABC$ be a triangle with $G$ as its centroid, which is the intersection of the three medians of the triangle, as shown in the diagram. If $\overline{GA} \perp \overline{GB}$ and $AB = 7$, compute $AC^2 + BC^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/1/240be132c6adcfde0334a000e1f916a6292907.png[/img]

2012-2013 SDML (Middle School), 7

Tags:

Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?

2018 Junior Balkan MO, 1

Tags: number theory , algebra

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

2015 Caucasus Mathematical Olympiad, 1

Tags: algebra , equation

Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given).

2009 Today's Calculation Of Integral, 472

Tags: calculus , integration , parabola , geometry , analytic geometry , calculus computations , conic

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

2025 Euler Olympiad, Round 1, 6

Tags: ratio , geometry

There are seven rays emanating from a point $A$ on a plane, such that the angle between the two consecutive rays is $30 ^{\circ}$. A point $A_1$ is located on the first ray. The projection of $A_1$ onto the second ray is denoted as $A_2$. Similarly, the projection of $A_2$ onto the third ray is $A_3$, and this process continues until the projection of $A_6$ onto the seventh ray is $A_7$. Find the ratio $\frac{A_7A}{A_1A}$. [img]https://i.imgur.com/oxixe5q.png[/img] [i]Proposed by Giorgi Arabidze, Georgia[/i]

2022 German National Olympiad, 6

Tags: function , functional equation , functional inequality , functional inequalities , algebra

Consider functions $f$ satisfying the following four conditions: (1) $f$ is real-valued and defined for all real numbers. (2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$. (3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$. (4) We have $f(2)=4$. Prove that: a) There is a function $f$ with $f(3)=9$ satisfying the four conditions. b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.

2012 Paraguay Mathematical Olympiad, 5

Tags: circumcircle , trigonometry , geometry

Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$. Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.

2014 Contests, 3

Tags: geometry , equal segments , right triangle , semicircle , angle

(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$. (ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

2016 PUMaC Algebra Individual A, A8

Tags:

Define the function $f:\mathbb{R} \backslash \{-1,1\} \to \mathbb{R}$ to be \[f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} .\] Suppose that $f\left(y\right)-f\left(\tfrac{1}{y}\right)=2016$. Then $y$ can be written in simplest form as $\tfrac{p}{q}$. Find $p+q$. ($\mathbb{R} \backslash \{-1,1\}$ refers to the set of real numbers excluding $-1$ and $1$.)

2011 ISI B.Math Entrance Exam, 4

Tags: function , inequalities , algebra

Let $t_1 < t_2 < t_3 < \cdots < t_{99}$ be real numbers. Consider a function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|$ . Show that $f(x)$ will attain minimum value at $x=t_{50}$.

2025 Kyiv City MO Round 1, Problem 4

Tags: number theory , functional equation

Find all functions $ f : \mathbb{N} \to \mathbb{N} $ that satisfy the following condition: for any positive integers $ m $ and $ n $ such that $ m > n $ and $ m $ is not divisible by $ n $, if we denote by $ r $ the remainder of the division of $ m $ by $ n $, then the remainder of the division of $ f(m) $ by $ n $ is $ f(r) $. [i]Proposed by Mykyta Kharin[/i]

2006 South East Mathematical Olympiad, 1

Tags: function , algebra

Suppose $a>b>0$, $f(x)=\dfrac{2(a+b)x+2ab}{4x+a+b}$. Show that there exists an unique positive number $x$, such that $f(x)=\left(\dfrac{a^{\frac{1}{3}}+b^{\frac{1}{3}}}{2} \right)^3$.

2007 Tuymaada Olympiad, 4

Tags: combinatorics

Determine maximum real $ k$ such that there exist a set $ X$ and its subsets $ Y_{1}$, $ Y_{2}$, $ ...$, $ Y_{31}$ satisfying the following conditions: (1) for every two elements of $ X$ there is an index $ i$ such that $ Y_{i}$ contains neither of these elements; (2) if any non-negative numbers $ \alpha_{i}$ are assigned to the subsets $ Y_{i}$ and $ \alpha_{1}+\dots+\alpha_{31}=1$ then there is an element $ x\in X$ such that the sum of $ \alpha_{i}$ corresponding to all the subsets $ Y_{i}$ that contain $ x$ is at least $ k$.

2003 Iran MO (3rd Round), 22

Tags: induction , number theory

Let $ a_1\equal{}a_2\equal{}1$ and \[ a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2\]for each $ n\in\mathbb N$. Find all $ n$ such that $ a_n\in\mathbb N$.