Olympiad problems

2006 Taiwan TST Round 1, 1

Tags: trigonometry , inequalities

Let the three sides of $\triangle ABC$ be $a,b,c$. Prove that $\displaystyle \frac{\sin^2A}{a}+\frac{\sin^2B}{b}+\frac{\sin^2C}{c} \le \frac{S^2}{abc}$ where $\displaystyle S=\frac{a+b+c}{2}$. Find the case where equality holds.

KoMaL A Problems 2023/2024, A. 883

Tags: algebra , polynomial , analysis , series

Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions: [list] [*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$. [/list] Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$? [i]Proposed by András Imolay, Budapest[/i]

2020-21 KVS IOQM India, 4

Tags: incenter , geometry

Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?

2023 HMNT, 5

Tags: combinatorics

On an $8\times 8$ chessboard, 6 black rooks and k white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of $k$. (Two rooks attack each other if they are in the same row or column and no rooks are between them.)

2022 AMC 12/AHSME, 20

Tags: trapezoid

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$ $\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$

1964 AMC 12/AHSME, 25

Tags: algebra , polynomial , parameterization , conic

The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: $ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $

2022 China Team Selection Test, 3

Tags: functional equation , algebra

Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$. [i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$.

2008 Hong kong National Olympiad, 4

Tags: combinatorics

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.

1965 All Russian Mathematical Olympiad, 065

Tags: combinatorics , algebra

Quasi-rounding is a substitution one of the two closest integers instead of the given number. Given $n$ numbers. Prove that you can quasi-round them in such a way, that a sum of every subset of quasi-rounded numbers will deviate from the sum of the same subset of initial numbers not greater than $(n+1)/4$ .

2001 National Olympiad First Round, 18

Tags: inequalities , triangle inequality , geometry , perpendicular bisector

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

PEN H Problems, 37

Tags: diophantine equation , modular arithmetic

Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.

Kyiv City MO Seniors Round2 2010+ geometry, 2015.11.2

Tags: geometry , equilateral , collinear , symmetric

The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $. (Serdyuk Nazar)

2007 Germany Team Selection Test, 2

Tags: function , functional equation , algebra

Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]

2007 IMC, 4

Tags: induction , function

Let $ G$ be a finite group. For arbitrary sets $ U, V, W \subset G$, denote by $ N_{UVW}$ the number of triples $ (x, y, z) \in U \times V \times W$ for which $ xyz$ is the unity . Suppose that $ G$ is partitioned into three sets $ A, B$ and $ C$ (i.e. sets $ A, B, C$ are pairwise disjoint and $ G = A \cup B \cup C$). Prove that $ N_{ABC}= N_{CBA}.$

PEN P Problems, 1

Tags: additive number theory

Show that any integer can be expressed as a sum of two squares and a cube.

2009 F = Ma, 7

Tags:

A bird is flying in a straight line initially at $\text{10 m/s}$. It uniformly increases its speed to $\text{15 m/s}$ while covering a distance of $\text{25 m}$. What is the magnitude of the acceleration of the bird? (A) $\text{5.0 m/s}^2$ (B) $\text{2.5 m/s}^2$ (C) $\text{2.0 m/s}^2$ (D) $\text{0.5 m/s}^2$ (E) $\text{0.2 m/s}^2$

2008 JBMO Shortlist, 11

Tags: geometry

Consider $ABC$ an acute-angled triangle with $AB \ne AC$. Denote by $M$ the midpoint of $BC$, by $D, E$ the feet of the altitudes from $B, C$ respectively and let $P$ be the intersection point of the lines $DE$ and $BC$. The perpendicular from $M$ to $AC$ meets the perpendicular from $C$ to $BC$ at point $R$. Prove that lines $PR$ and $AM$ are perpendicular.

Indonesia Regional MO OSP SMA - geometry, 2014.4

Tags: product , ratio , geometry

Let $\Gamma$ be the circumcircle of triangle $ABC$. One circle $\omega$is tangent to $\Gamma$ at $A$ and tangent to $BC$ at $N$. Suppose that the extension of $AN$ crosses $\Gamma$ again at $E$. Let $AD$ and $AF$ be respectively the line of altitude $ABC$ and diameter of $\Gamma$, show that $AN \times AE = AD \times AF = AB \times AC$

2000 Iran MO (3rd Round), 3

Tags: vector , geometry

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.

2023 Romania Team Selection Test, P2

Tags: geometry , reflection , polygon

A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon. [i]The Problem Selection Committee[/i]

Cono Sur Shortlist - geometry, 2018.G1.1

Tags: geometry

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

2007 Alexandru Myller, 3

Tags: geometry , pentagon

The convex pentagon $ ABCDE $ has the following properties: $ \text{(i)} AB=BC $ $ \text{(ii)} \angle ABE+\angle CBD =\angle DBE $ $ \text{(iii)} \angle AEB +\angle BDC=180^{\circ} $ Prove that the orthocenter of $ BDE $ lies on $ AC. $

2020 Princeton University Math Competition, A4/B6

Tags: geometry

Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?

2020 Regional Olympiad of Mexico Southeast, 1

Tags: number theory

Find all natural numbers $n$, such that $3$ divides the number $n\cdot 2^n+1$.

2013 China Second Round Olympiad, 1

Tags: modular arithmetic , number theory

For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.