Olympiad problems

2014 ELMO Shortlist, 4

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

2002 Moldova National Olympiad, 3

Tags: geometry

The sides $ AB$,$ BC$ and $ CA$ of the triangle $ ABC$ are tangent to the incircle of the triangle $ ABC$ with center $ I$ at the points $ C_1$,$ A_1$ and $ B_1$, respectively.Let $ B_2$ be the midpoint of the side $ AC$.Prove that the lines $ B_1I$, $ A_1C_1$ and $ BB_2$ are concurrent.

2004 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

Yet another trapezoid $ABCD$ has $AD$ parallel to $BC$. $AC$ and $BD$ intersect at $P$. If $[ADP]=[BCP] = 1/2$, find $[ADP]/[ABCD]$. (Here the notation $[P_1 ...P_n]$ denotes the area of the polygon $P_1 ...P_n$.)

1978 Romania Team Selection Test, 7

Tags: analytic geometry , linear algebra , geometry , barycenter

[b]a)[/b] Prove that for any natural number $ n\ge 1, $ there is a set $ \mathcal{M} $ of $ n $ points from the Cartesian plane such that the barycenter of every subset of $ \mathcal{M} $ has integral coordinates (both coordinates are integer numbers). [b]b)[/b] Show that if a set $ \mathcal{N} $ formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of $ \mathcal{N} , $ the barycenter of which has non-integral coordinates.

2018 CMIMC Algebra, 2

Tags: algebra

Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$?

1976 Miklós Schweitzer, 6

Tags: real analysis

Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

2022 Putnam, B3

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Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate the recoloring process, will we always end up with all the numbers red after a finite number of steps?

2018 Harvard-MIT Mathematics Tournament, 3

Tags: geometry

$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.

2024 Sharygin Geometry Olympiad, 3

Tags: geometry

Let $ABC$ be an acute-angled triangle, and $M$ be the midpoint of the minor arc $BC$ of its circumcircle. A circle $\omega$ touches the side $AB, AC$ at points $P, Q$ respectively and passes through $M$. Prove that $BP + CQ = PQ$.

2018 District Olympiad, 2

Tags: irrational number

Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.

2010 Belarus Team Selection Test, 1.1

Tags: sum , positive integer , number theory

Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ? (E. Barabanov)

2013 ELMO Shortlist, 9

Tags: rectangle , symmetry , circumcircle , cyclic quadrilateral , projective geometry , geometry

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

1966 All Russian Mathematical Olympiad, 076

Tags: algebra

A rectangle $ABCD$ is drawn on the cross-lined paper with its sides laying on the lines, and $|AD|$ is $k$ times more than $|AB|$ ($k$ is an integer). All the shortest paths from $A$ to $C$ coming along the lines are considered. Prove that the number of those with the first link on $[AD]$ is $k$ times more then of those with the first link on $[AB]$.

2011 Irish Math Olympiad, 4

Tags: inequalities

Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that (i) $$\frac{3}{4}\le xy+yz+zx<1$$ (ii) $$xyz\le \frac{1}{8}$$ Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.

1998 Tournament Of Towns, 4

Tags: algebra , system of equations

For some positive numbers $A, B, C$ and $D$, the system of equations $$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$ has $m$ solutions, while the system of equations $$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$ has $n$ solutions. If $m > n > 1$, find $m$ and $n$. ( G Galperin)

2017 USAMO, 5

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Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which: [list] [*] only finitely many distinct labels occur, and [*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$. [/list] [i]Proposed by Ricky Liu[/i]

2024 HMNT, 9

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Let $ABCDEF$ be a regular hexagon with center $O$ and side length $1.$ Point $X$ is placed in the interior of the hexagon such that $\angle BXC = \angle AXE = 90^\circ.$ Compute all possible values of $OX.$

2004 Tuymaada Olympiad, 1

Tags: polynomial , calculus , integration , algebra

Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ [i]Proposed by A. Golovanov[/i]

2025 Greece National Olympiad, 1

Tags: algebra , polynomial , root , integer

Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

1992 Vietnam National Olympiad, 1

Tags: polynomial , algebra

Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and \[ P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}.\] Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}$.

2014 Singapore Senior Math Olympiad, 13

Tags: algebra , polynomial

Suppose $a$ and $b$ are real numbers such that the polynomial $x^3+ax^2+bx+15$ has a factor of $x^2-2$. Find the value of $a^2b^2$.

1954 AMC 12/AHSME, 24

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The values of $ k$ for which the equation $ 2x^2\minus{}kx\plus{}x\plus{}8\equal{}0$ will have real and equal roots are: $ \textbf{(A)}\ 9 \text{ and }\minus{}7 \qquad \textbf{(B)}\ \text{only }\minus{}7 \qquad \textbf{(C)}\ \text{9 and 7} \\ \textbf{(D)}\ \minus{}9 \text{ and }\minus{}7 \qquad \textbf{(E)}\ \text{only 9}$

1945 Moscow Mathematical Olympiad, 098

Tags: locus , geometry

A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.

1981 National High School Mathematics League, 7

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The equation $x|x|+px+q=0$ is given. Which of the following is not true? $\text{(A)}$It has at most three real roots. $\text{(B)}$It has at least one real root. $\text{(C)}$Only if $p^2-4q\geq0 $,it has real roots. $\text{(D)}$If $p<0$ and $q>0$, it has three real roots.

CNCM Online Round 3, 2

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Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$? [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]