Olympiad problems

2020 USMCA, 17

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Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b\in \{0,\ldots,2016\}$ and $(a,b)\neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$.

2021 Peru Cono Sur TST., P7

Tags: algebra

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

2002 Canada National Olympiad, 2

Tags: number theory

Call a positive integer $n$ [b]practical[/b] if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$. For example, the divisors of 6 are 1, 2, 3, and 6. Since \[ \centerline{1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+ {\bf 3}, ~~ 6={\bf 6},} \] we see that 6 is practical. Prove that the product of two practical numbers is also practical.

2011 Argentina National Olympiad Level 2, 3

Tags: geometry , altitude , incenter

Let $ABC$ be a triangle of sides $AB = 15$, $AC = 14$ and $BC = 13$. Let $M$ be the midpoint of side $AB$ and let $I$ be the incenter of triangle $ABC$. The line $MI$ intersects the altitude corresponding to the side $AB$ of triangle $ABC$ at point $P$. Calculate the length of the segment $PC$. Note: The incenter of a triangle is the intersection point of its angle bisectors.

2013 All-Russian Olympiad, 1

Tags: pigeonhole principle , absolute value , combinatorics

$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values of their pairwise differences, there are ten different numbers not exceeding $100$.

2017 Harvard-MIT Mathematics Tournament, 23

Tags: combinatorics

Five points are chosen uniformly at random on a segment of length $1$. What is the expected distance between the closest pair of points?

2012 Dutch IMO TST, 2

Tags: inequalities

Let $a, b, c$ and $d$ be positive real numbers. Prove that $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$

2013 Albania Team Selection Test, 2

Tags: rearrangement inequality , inequalities

Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds: \[a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\]

1987 Tournament Of Towns, (153) 4

Tags: line , line construction , geometry , arc , broken line , bisects area

We are given a figure bounded by arc $AC$ of a circle, and a broken line $ABC$, with the arc and broken line being on opposite sides of the chord $AC$. Construct a line passing through the mid-point of arc $AC$ and dividing the area of the figure into two regions of equal area.

2006 Cuba MO, 3

Tags: algebra , inequalities

Let $a, b, c$ be different real numbers. prove that $$\left(\frac{2a-b}{a-b}\right)^2+ \left(\frac{2b- c}{b-c}\right)^2+ \left(\frac{2c-a}{c-a}\right)^2 \ge 5. $$

2019 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

A polynomial $f(x)$ of degree $2000$ is given. It's known that $f(x^2-1)$ has exactly $3400$ real roots while $f(1-x^2)$ has exactly $2700$ real roots. Prove that there exist two real roots of $f(x)$ such that the difference between them is less that $0.002$. [i](А. Солынин)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

Kvant 2020, M2622

Tags: geometry , rhombus

The points $E, F, G$ and $H{}$ are located on the sides $DA, AB, BC$ and $CD$ of the rhombus $ABCD$ respectively, so that the segments $EF$ and $GH$ touch the circle inscribed in the rhombus. Prove that $FG\parallel HE$. [i]Proposed by V. Eisenstadt[/i]

2016 Purple Comet Problems, 4

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One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the perimeter of the rectangle.

2022 Junior Balkan Team Selection Tests - Romania, P1

Tags: number theory

Let $p$ be an odd prime number. Prove that there exist nonnegative integers $x,y,z,t$ not all of which are $0$ such that $t<p$ and \[x^2+y^2+z^2=tp.\]

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: algebra , inequalities , positive real

Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$. Prove that there are positive numbers $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$

2010 Finnish National High School Mathematics Competition, 3

Tags: polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

2008 JBMO Shortlist, 5

Tags: geometry

Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)

1998 India Regional Mathematical Olympiad, 3

Tags: inequalities

Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]

2007 Nicolae Păun, 3

Tags: limit , permutation , group theory , group of permutations , centralizer , real analysis

In the following exercise, $ C_G (e) $ denotes the centralizer of the element $ e $ in the group $ G. $ [b]a)[/b] Prove that $ \max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right| <\frac{n!}{2} , $ for any natural number $ n\ge 4. $ [b]b)[/b] Show that $ \lim_{n\to\infty} \left(\frac{1}{n!}\cdot\max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right|\right) =0. $ [i]Alexandru Cioba[/i]

2024 ELMO Shortlist, G7

Tags: geometry , rectangle

Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$. [i]Linus Tang[/i]

2021 AMC 12/AHSME Spring, 19

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How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$

2022 Argentina National Olympiad, 6

Tags: algebra , polynomial

For every positive integer $n$, we consider the polynomial of real coefficients, of $2n+1$ terms, $$P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$$ where all coefficients are real numbers satisfying $100 \le a_i \le 101$ for $0 \le i \le 2n$. Find the smallest possible value of $n$ such that the polynomial can have at least one real root.

2013 Purple Comet Problems, 17

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For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.

1966 IMO Shortlist, 1

Tags: combinatorial geometry , circles , geometry

Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?

1975 AMC 12/AHSME, 13

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The equation $ x^6 \minus{} 3x^5 \minus{} 6x^3 \minus{} x \plus{} 8$ has $ \textbf{(A)}\ \text{no real roots} \qquad$ $ \textbf{(B)}\ \text{exactly two distinct negative roots} \qquad$ $ \textbf{(C)}\ \text{exactly one negative root} \qquad$ $ \textbf{(D)}\ \text{no negative roots, but at least one positive root} \qquad$ $ \textbf{(E)}\ \text{none of these}$