Olympiad problems

2013 Romania Team Selection Test, 4

Tags: algebra , polynomial , number theory , rational roots , real analysis

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

1969 IMO Longlists, 10

Tags: trigonometry , geometry , trigonometric equation , median

$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$

2025 AIME, 5

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There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 exactly once. Let N be the number of these integers that are divisible by $22$. Find the difference between $N$ and 2025.

2012 Flanders Math Olympiad, 1

Tags: combinatorics

Our class decides to have a alpha - beta - gamma tournament. This party game is always played in groups of three. Any possible combination of three players (three students or two students and the teacher) plays the game $1$ time. The player who wins gets $1$ point. The two losers get no points. At the end of the tournament, miraculously, all students have as many points. The teacher has $3$ points. How many students are there in our class?

2007 Peru MO (ONEM), 4

Tags: geometry , rhombus , equilateral , equal angles

Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.

2010 Contests, 2

Tags: algebra

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

2009 Stanford Mathematics Tournament, 4

Tags: function

Find all values of $x$ for which $f(x)+xf\left(\frac{1}{x}\right)=x$ for any function $f(x)$

2010 LMT, 20

Tags:

Three vertices of a parallelogram are $(2,-4),(-2,8),$ and $(12,7.)$ Determine the sum of the three possible x-coordinates of the fourth vertex.

1991 IMO, 1

Tags: geometry , incenter , triangle inequality , geometric inequality , angle bisector

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

2020 Harvard-MIT Mathematics Tournament, 9

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Let $P(x)=x^{2020}+x+2$, which has $2020$ distinct roots. Let $Q(x)$ be the monic polynomial of degree $\binom{2020}{2}$ whose roots are the pairwise products of the roots of $P(x)$. Let $\alpha$ satisfy $P(\alpha)=4$. Compute the sum of all possible values of $Q(\alpha^2)^2$. [i]Proposed by Milan Haiman.[/i]

2004 South East Mathematical Olympiad, 5

Tags: inequalities , trigonometry

For $\theta\in[0, \dfrac{\pi}{2}]$, the following inequality $\sqrt{2}(2a+3)\cos(\theta-\dfrac{\pi}{4})+\dfrac{6}{\sin\theta+\cos\theta}-2\sin2\theta<3a+6$ is always true. Determine the range of $a$.

1991 Arnold's Trivium, 50

Tags: integration , calculus

Calculate \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]

2019 HMNT, 1

Tags: combinatorics , geometry

Dylan has a $100\times 100$ square, and wants to cut it into pieces of area at least $1$. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?

2021 MOAA, 9

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William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back. [i]Proposed by William Yue[/i]

2024 Iran Team Selection Test, 4

Tags: algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real numbers $x , y$ this equality holds : $$f(yf(x)+f(x)f(y))=xf(y)+f(xy)$$ [i]Proposed by Navid Safaei[/i]

1996 Cono Sur Olympiad, 2

Tags: sequence

Consider a sequence of real numbers defined by: $a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$ Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.

2023 Romania National Olympiad, 2

Tags: algebra , sum of digits

We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order). a) Determine the largest special number $m$ whose sum of digits is equal to $2023$. b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.

2015 FYROM JBMO Team Selection Test, 3

Tags: inequalities , 3-variable inequality

Let $a, b$ and $c$ be positive real numbers. Prove that $\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)$.

2013 Dutch Mathematical Olympiad, 5

Tags: digit , sum , number theory

The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$ If one writes down the number $S$, how often does the digit `$5$' occur in the result?

1986 Bulgaria National Olympiad, Problem 6

Tags: inequalities , recurrence relation , algebra , sequence

Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.

KoMaL A Problems 2023/2024, A. 882

Tags: number theory , gcd , greatest common divisor

Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that \[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\] [i]Proposed by Dávid Matolcsi, Berkeley[/i]

2010 Baltic Way, 12

Tags: arithmetic sequence , geometry

Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides. $(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression. $(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.

2020 Taiwan TST Round 1, 3

Tags: number theory

Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number \[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\] has at least $k$ distinct prime factors. Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$. [i]Proposed by Evan Chen[/i]

1996 Putnam, 1

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Define a $\emph{selfish}$ set to be a set which has its own cardinality as its element. And a set is a $\emph{minimal }\text{ selfish}$ set if none of its proper subsets are $\emph{selfish}$. Find with proof the number of $\text{minimal selfish}$ subsets of $\{1,2,\cdots ,n\}$.

2014 Romania National Olympiad, 4

Tags: geometry

Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $ Show that $ AB=AD. $