Olympiad problems

1965 IMO, 1

Tags: inequalities , trigonometry

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

2013 National Chemistry Olympiad, 53

Tags: geometry

On the basis of VSEPR theory, what geometry is predicted for the central sulfur atom in $\ce{SOCl2}$? $ \textbf{(A) }\text{bent}\qquad\textbf{(B) }\text{T-shaped}\qquad\textbf{(C) }\text{trigonal planar} \qquad\textbf{(D) }\text{trigonal pyramidal} \qquad$

2003 Austria Beginners' Competition, 2

Tags: algebra

Find all real solutions of the equation $(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3$.

2004 Cuba MO, 9

Tags: geometry , geometric inequality

The angle $\angle XOY =\alpha $ and the points $A$ and $B$ on OY are given such that $OA = a$ and $OB = b$ with $a > b$. A circle passes through the points $A$ and $B$ and is tangent to $OX$. a) Calculate the radius of that circle in terms of $a, b$ and $\alpha $. b) If $a$ and $b$ are constants and $\alpha $ varies, show that the minimum value of the radius of the circle is $\frac{a-b}{2}$.

1996 Yugoslav Team Selection Test, Problem 2

Tags: geometry , combinatorics , combinatorial geometry

Let there be given a set of $1996$ equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.

2012 Indonesia TST, 1

Tags: polynomial , trigonometry , algebra

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

2000 Saint Petersburg Mathematical Olympiad, 11.2

Tags: vector , geometric inequality , inequalities , vector geometry , 3d geometry , analytic geometry , geometry

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

1987 Mexico National Olympiad, 7

Tags: number theory , irreducible

Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.

OIFMAT I 2010, 7

Tags: combinatorics

$ 15 $ teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive $ 3 $ points, while the loser receives $ 1 $ point. In the event of a tie, both teams receive $ 2 $ points. When all possible league matches are held, the following can be observed: $\bullet$ No two teams have finished with the same amount of points. $\bullet$ Each team finished the league with at least $ 21 $ points. Let $W$ be the team that finished the league with the highest score. Determine how many points $W$ scored and show that there were at least four ties in the league.

2019-2020 Fall SDPC, 6

Tags: geometry , trapezoid

Let $ABCD$ be an isosceles trapezoid inscribed in circle $\omega$, such that $AD \| BC$. Point $E$ is chosen on the arc $BC$ of $\omega$ not containing $A$. Let $BC$ and $DE$ intersect at $F$. Show that if $E$ is chosen such that $EB = EC$, the area of $AEF$ is maximized.

2005 JBMO Shortlist, 5

Tags: maximum , concentric , equilateral triangle , geometry

Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.

2014 AMC 12/AHSME, 13

Tags: geometry

Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$? ${ \textbf{(A)}\ \dfrac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \dfrac52\qquad\textbf{(C)}\ \dfrac{3+\sqrt{5}}{2}\qquad\textbf{(D)}}\ \dfrac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3 $

1951 Poland - Second Round, 4

Tags: algebra , trinomial

Prove that if equations $$x^2 + mx + n = 0 \,\,\,\, and\,\, \,\, x^2 + px + q = 0$$ have a common root, there is a relationship between the coefficients of these equations $$ (n - q)^2 - (m - p) (np - mq) = 0.$$

2017 BMT Spring, 20

Tags: algebra

Evaluate $\sum^{15}_{k=0}\left(2^{560}(-1)^k \cos^{560}\left( \frac{k\pi}{16}\right)\right) \pmod{17}.$

2024 CCA Math Bonanza, L3.3

Tags:

Define a [i]small[/i] prime to be a prime under $1$ billion. Find the sum of all [i]small[/i] primes of the form $20^n + 1$, given that the answer is greater than $1000$. [i]Lightning 3.3[/i]

2021 MOAA, 23

Tags:

Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

2013 China Team Selection Test, 3

Tags: inequalities , algebra

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

2014 Macedonia National Olympiad, 4

Tags: inequalities , calculus , three variable inequality

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]

2006 Iran Team Selection Test, 2

Tags: combinatorics

Suppose $n$ coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a [i]comparison[/i] will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least $n-1$ [i]comparisons[/i] are required.

2017 CCA Math Bonanza, T2

Tags: geometry

A square of side length $s$ is inscribed in circle $C_1$ and circumscribed about circle $C_2$. The area of the region in $C_1$ but outside $C_2$ is $25\pi$. What is $s$? [i]2017 CCA Math Bonanza Team Round #2[/i]

2014 Contests, 1

Tags: function , algebra , polynomial , number theory , greatest common divisor

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.

2016 Greece Junior Math Olympiad, 4

Tags: combinatorics

Find the number ot 6-tuples $(x_1, x_2,...,x_6)$, where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even

1993 Baltic Way, 13

Tags: combinatorics

An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of small triangles that can be chosen so that no two of them lie on a line that is parallel to any of the sides of the triangle $ABC$?

2010 Peru IMO TST, 4

Tags: inequalities

Let $ \displaystyle{a,b,c}$ be positive real numbers such that $\displaystyle{a+b+c=1.}$ Prove that $$ \displaystyle{\frac{1+ab}{a+b}+\frac{1+bc}{b+c}+\frac{1+ca}{c+a}\geq 5.}$$

2010 Postal Coaching, 3

Tags: divisibility theory

Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.