Olympiad problems

2005 China Team Selection Test, 1

Tags: inequalities , algebra

Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.

2015 Romania Team Selection Test, 2

Tags: circles , triangle inequality , inradius , inequalities , geometry

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

2015 Postal Coaching, Problem 2

Tags: algebra , cubic equation

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

2001 Estonia National Olympiad, 1

Tags: algebra , system of equations , trigonometry

Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$

1956 Moscow Mathematical Olympiad, 344

Tags: graph , acute , geometry , combinatorial geometry

* Let $A, B, C$ be three nodes of a graph paper. Prove that if $\vartriangle ABC$ is an acute one, then there is at least one more node either inside $\vartriangle ABC$ or on one of its sides.

2017 Federal Competition For Advanced Students, 3

Tags: combinatorics

Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table. Determine the smallest number $N\ge100000$ which Berta has wining strategy. [i]proposed by Gerhard Woeginger[/i]

2023 Kyiv City MO Round 1, Problem 3

Tags: circumcircle , geometry

Let $I$ be the incenter of triangle $ABC$ with $AB < AC$. Point $X$ is chosen on the external bisector of $\angle ABC$ such that $IC = IX$. Let the tangent to the circumscribed circle of $\triangle BXC$ at point $X$ intersect the line $AB$ at point $Y$. Prove that $AC = AY$. [i]Proposed by Oleksiy Masalitin[/i]

2019 USA EGMO Team Selection Test, 4

Tags: algebra , number theory

For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that \[a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}\] for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.

1999 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1

Tags:

Compute: $ \frac{777^2 \minus{} 66^2}{777\plus{}66}$

2010 Brazil Team Selection Test, 3

Tags: three variable inequality , inequality , inequalities

Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that: \[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\] [i]Proposed by Juhan Aru, Estonia[/i]

2002 Junior Balkan MO, 2

Tags: geometry , trapezoid

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

1997 USAMO, 4

Tags: geometry , modular arithmetic , geometric transformation , homothety , combinatorial geometry

To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

2012 Today's Calculation Of Integral, 788

Tags: calculus , integration , function , analytic geometry , calculus computations , logarithm

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

1998 National Olympiad First Round, 18

Tags: modular arithmetic , number theory , prime number

Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$. $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$

2009 Estonia Team Selection Test, 4

Tags: geometry , geometric inequalities , inequalities , ratio

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

2002 APMO, 3

Tags: geometry

Let $ABC$ be an equilateral triangle. Let $P$ be a point on the side $AC$ and $Q$ be a point on the side $AB$ so that both triangles $ABP$ and $ACQ$ are acute. Let $R$ be the orthocentre of triangle $ABP$ and $S$ be the orthocentre of triangle $ACQ$. Let $T$ be the point common to the segments $BP$ and $CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that the triangle $TRS$ is equilateral.

1952 Polish MO Finals, 3

Tags: construction , geometric construction , geometry

Construct the quadrilateral $ ABCD $ given the lengths of the sides $ AB $ and $ CD $ and the angles of the quadrilateral.

2008 Danube Mathematical Competition, 4

Tags: combinatorics

Let $ n\geq 2$ be a positive integer. Find the [u]maximum[/u] number of segments with lenghts greater than $ 1,$ determined by $ n$ points which lie on a closed disc with radius $ 1.$

2000 National High School Mathematics League, 2

Tags: trigonometry

If $\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}$, then the range value of $\frac{\alpha}{3}$ is $\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$ $\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$

1985 Putnam, A4

Tags:

Define a sequence $\left\{a_{i}\right\}$ by $a_{1}=3$ and $a_{i+1}=3^{a_{i}}$ for $i \geq 1.$ Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{i} ?$

2016 Regional Olympiad of Mexico Northeast, 3

Tags: combinatorics

Consider a grid board of $n \times n$, with $n \ge 5$. Two unit squares are said to be far [i]apart [/i] if they are neither on the same row nor on consecutive rows and neither in the same column nor in consecutive columns. Take $3$ rectangles with vertices and sides on the points and lines of board so that if two unit squares belong to different rectangles, then they are [i]apart [/i]. In how many ways is it possible to do this?

2022 CMIMC, 2.2 1.1

Tags: combinatorics

Starting with a $5 \times 5$ grid, choose a $4 \times 4$ square in it. Then, choose a $3 \times 3$ square in the $4 \times 4$ square, and a $2 \times 2$ square in the $3 \times 3$ square, and a $1 \times 1$ square in the $2 \times 2$ square. Assuming all squares chosen are made of unit squares inside the grid. In how many ways can the squares be chosen so that the final $1 \times 1$ square is the center of the original $5 \times 5$ grid? [i]Proposed by Nancy Kuang[/i]

2014 Iran Team Selection Test, 5

Tags: combinatorics

Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number. Prove that at least $2^{n-3}$ subsets of $X$ are compact. [i]Proposed by Mahyar Sefidgaran[/i]

1997 Estonia National Olympiad, 1

Tags: number theory , greatest common divisor , inequalities

For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$ (a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$. (b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?

2015 Postal Coaching, Problem 1

Tags: geometry

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and $WY$ meet $AB$ at points $N$ and $M$. Prove that the length of segment $NM$ doesn’t depend on point $C$.