Olympiad problems

2002 Moldova National Olympiad, 1

Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| \plus{} |b| \plus{} |c|$.

2013 National Olympiad First Round, 22

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For how many integers $0\leq n < 2013$, is $n^4+2n^3-20n^2+2n-21$ divisible by $2013$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None of above} $

2014 AMC 12/AHSME, 3

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Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? $ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $

2022 Purple Comet Problems, 8

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Find the number of divisors of $20^{22}$ that are perfect squares.

2015 District Olympiad, 4

Tags: modular arithmetic , number theory , arithmetic

[b]a)[/b] Show that the three last digits of $ 1038^2 $ are equal with $ 4. $ [b]b)[/b] Show that there are infinitely many perfect squares whose last three digits are equal with $ 4. $ [b]c)[/b] Prove that there is no perfect square whose last four digits are equal to $ 4. $

1992 Poland - Second Round, 6

Tags: algebra , sequence , recurrence relation

The sequences $(x_n)$ and $(y_n)$ are defined as follows: $$ x_{n+1} = \frac{x_n+2}{x_n+1},\quad y_{n+1}=\frac{y_n^2+2}{2y_n} \quad \text{ for } n= 0,1,2,\ldots.$$ Prove that for every integer $ n\geq 0 $ the equality $ y_n = x_{2^n-1} $ holds.

2006 Taiwan National Olympiad, 1

Tags: number theory

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

1986 Balkan MO, 1

Tags: incenter , circumcircle , geometry

A line passing through the incenter $I$ of the triangle $ABC$ intersect its incircle at $D$ and $E$ and its circumcircle at $F$ and $G$, in such a way that the point $D$ lies between $I$ and $F$. Prove that: $DF \cdot EG \geq r^{2}$.

2005 IMO Shortlist, 4

Tags: algebra , functional equation

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

2022 Poland - Second Round, 6

Tags: combinatorics , graph theory

$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.

2010 Iran MO (3rd Round), 3

Tags: geometry , geometric transformation , reflection , circumcircle , homothety , ratio , power of a point

in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.(20 points)

2009 Jozsef Wildt International Math Competition, W. 7

Tags: integration , inequalities , calculus , logarithm

If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$

2016 Junior Regional Olympiad - FBH, 2

Tags: root , algebra

If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$

2002 Germany Team Selection Test, 1

Tags: function , quadratic , algebra

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

2000 Harvard-MIT Mathematics Tournament, 36

Tags: geometry , inradius , trigonometry , trig identities , law of cosines

If, in a triangle of sides $a, b, c$, the incircle has radius $\frac{b+c-a}{2}$, what is the magnitude of $\angle A$?

2017 QEDMO 15th, 6

Tags: number theory , diophantine , diophantine equation

Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.

2017 Iranian Geometry Olympiad, 3

Tags: geometry

In the regular pentagon $ABCDE$, the perpendicular at $C$ to $CD$ meets $AB$ at $F$. Prove that $AE+AF=BE$. [i]Proposed by Alireza Cheraghi[/i]

2008 Germany Team Selection Test, 2

Tags: rectangle , combinatorics , geometry

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

KoMaL A Problems 2018/2019, A. 745

Tags: combinatorics , geometry

A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.

2004 Purple Comet Problems, 7

Tags: geometry , rectangle , percent

A rectangle has area $1100$. If the length is increased by ten percent and the width is decreased by ten percent, what is the area of the new rectangle?

2007 Oral Moscow Geometry Olympiad, 6

Tags: geometry , fixed , tangent

A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle. (A. Zaslavsky)

1993 Romania Team Selection Test, 3

Tags: diagonal , symmetry , hexagon , geometry , equal area

Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?

1986 AMC 12/AHSME, 8

Tags: geometry

The population of the United States in 1980 was $226,504,825$. The area of the country is $3,615,122$ square miles. The are $(5280)^{2}$ square feet in one square mile. Which number below best approximates the average number of square feet per person? $ \textbf{(A)}\ 5,000\qquad\textbf{(B)}\ 10,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000 $

2015 Taiwan TST Round 2, 1

Tags: algebra , polynomial

Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero. Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$. Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$

2023 VN Math Olympiad For High School Students, Problem 11

Tags: geometry , ratio

Given a triangle $ABC$ inscribed in $(O)$ with $2$ symmedians $AD, CF(D,F$ are on the sides $BC, AB,$ respectively$).$ The ray $DF$ intersects $(O)$ at $P.$ The line passing through $P$ and perpendicular to $OA$ intersects $AB,AC$ at $Q,R,$ respectively$.$ Compute the ratio $\dfrac{PR}{PQ}.$