Olympiad problems

2005 Taiwan TST Round 2, 2

Tags: geometry , circumcircle , homothety , Triangle , IMO Shortlist , Hi

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

2017 USAMO, 6

Tags: inequalities , AMC , USA(J)MO , USAMO(2017) , Tangent line trick , Hi

Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\] given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$. [i]Proposed by Titu Andreescu[/i]

2020 Taiwan TST Round 2, 1

Tags: number theory , IMO Shortlist , IMO Shortlist 2019 , functional equation , Hi

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2011 IMO, 4

Tags: IMO , IMO 2011 , combinatorics , weights , IMO Shortlist , Morteza Saghafiyan , Hi

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. [i]Proposed by Morteza Saghafian, Iran[/i]

2018 China Team Selection Test, 2

Tags: number theory , arithmetic series , number of divisors , China TST , Hi

A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

2013 APMO, 2

Tags: modular arithmetic , inequalities , quadratics , function , algebra , floor function , Hi

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2017 Ukraine Team Selection Test, 2

Tags: number theory , IMO Shortlist , function , Divisibility , Hi

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2020 Estonia Team Selection Test, 1

Tags: combinatorics , binomial coefficients , IMO Shortlist , IMO Shortlist 2019 , trivial , Hi

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

2016 China Team Selection Test, 4

Tags: number theory , floor function , Hi

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

2002 IMO Shortlist, 1

Tags: number theory , IMO Shortlist , modular arithmetic , sum of cubes , Hi

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]

1999 APMO, 1

Tags: arithmetic sequence , number theory unsolved , number theory , Hi

Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.

2012 USA TSTST, 6

Tags: inequalities , algebra , polynomial , Vieta , quadratics , USA TSTST , Hi

Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]

2018 Brazil Team Selection Test, 6

Tags: algebra , inequalities , IMO Shortlist , Hi

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

2014 France Team Selection Test, 3

Tags: number theory , prime divisor , polynomial , IMO Shortlist , imo shortlist n3 , Hi

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2018 Turkey Team Selection Test, 2

Tags: functional equation , TST , Turkey , 2018 , Hi

Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that $$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.

2000 IMO Shortlist, 2

Tags: geometry , circle , IMO , imo 2000 , Hi

Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.

2013 Germany Team Selection Test, 3

Tags: IMO Shortlist , combinatorics , Extremal combinatorics , Doublecounting , Hi

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

2017 ELMO Shortlist, 1

Tags: number theory , greatest common divisor , ELMO 2017 , Elmo , Hi

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

1992 USAMO, 4

Tags: geometry , 3D geometry , sphere , Hi

Chords $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$ of a sphere meet at an interior point $P$ but are not contained in a plane. The sphere through $A$, $B$, $C$, $P$ is tangent to the sphere through $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $P$. Prove that $\, AA' = BB' = CC'$.

2004 IMO Shortlist, 3

Tags: geometry , circumcircle , homothety , Triangle , IMO Shortlist , Hi

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

1997 IMO, 2

Tags: geometry , circumcircle , perpendicular bisector , IMO , IMO 1997 , isosceles , Hi

It is known that $ \angle BAC$ is the smallest angle in the triangle $ ABC$. The points $ B$ and $ C$ divide the circumcircle of the triangle into two arcs. Let $ U$ be an interior point of the arc between $ B$ and $ C$ which does not contain $ A$. The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AU$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$. Show that $ AU \equal{} TB \plus{} TC$. [i]Alternative formulation:[/i] Four different points $ A,B,C,D$ are chosen on a circle $ \Gamma$ such that the triangle $ BCD$ is not right-angled. Prove that: (a) The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AD$ at certain points $ W$ and $ V,$ respectively, and that the lines $ CV$ and $ BW$ meet at a certain point $ T.$ (b) The length of one of the line segments $ AD, BT,$ and $ CT$ is the sum of the lengths of the other two.

2015 EGMO, 4

Tags: number theory , Sequence , EGMO , Integer sequence , Hi

Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.

2008 IMO Shortlist, 1

Tags: algebra , functional equation , IMO Shortlist , IMO , IMO 2008 , Hi

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

2017 Estonia Team Selection Test, 12

Tags: combinatorics , IMO Shortlist , Extremal combinatorics , Hi

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

2012 India IMO Training Camp, 3

Tags: combinatorics , IMO Shortlist , Hi , discrete continous

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]