Olympiad problems

2008 Argentina Iberoamerican TST, 2

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2021 Iran RMM TST, 2

Tags: function , algebra

Let $f : \mathbb{R}^+\to\mathbb{R}$ satisfying $f(x)=f(x+2)+2f(x^2+2x)$. Prove that if for all $x>1400^{2021}$, $xf(x) \le 2021$, then $xf(x) \le 2021$ for all $x \in \mathbb {R}^+$ Proposed by [i]Navid Safaei[/i]

2015 SDMO (High School), 1

Tags:

Let $\mathcal{C}$ be a circle and let $P$ and $Q$ be points inside $\mathcal{C}$. Prove that there are infinitely many circle through $P$ and $Q$ that are completely contained inside of $\mathcal{C}$.

2010 Peru MO (ONEM), 1

Tags: combinatorics

In each of the $9$ small circles of the following figure we write positive integers less than $10$, without repetitions. In addition, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/2db2c1ac7f45022606fb0099f24e6287977d10.png[/img]

1999 Baltic Way, 10

Tags: combinatorics

May the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance $1$?

2005 Austrian-Polish Competition, 1

Tags: polygon , enumeration , combinatorics

For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a [i]diagonal-quadrangle[/i] of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.

1962 AMC 12/AHSME, 2

Tags:

The expression $ \sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: $ \textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad \textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad \textbf{(C)}\ \frac{\sqrt{-3}}{6} \qquad \textbf{(D)}\ \frac{5 \sqrt{3}}{6} \qquad \textbf{(E)}\ 1$

2018 HMNT, 4

Tags:

Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.

2012 ITAMO, 3

Tags: induction , combinatorics

Let $n$ be an integer greater than or equal to $2$. There are $n$ people in one line, each of which is either a [i]scoundrel[/i] (who always lie) or a [i]knight[/i] (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.

2021 AMC 10 Fall, 15

Tags:

Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$ $\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi$

1999 Italy TST, 2

Tags: inequalities , geometry

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

1967 IMO Shortlist, 2

Tags: trigonometry , inequality , trigonometric inequality , minimum

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

2014 Polish MO Finals, 3

Tags: geometry , 3d geometry , tetrahedron , sphere

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

2001 AMC 10, 19

Tags:

Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 18$

2013 India Regional Mathematical Olympiad, 4

Tags: geometry , angle chasing , similar triangles

Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 1:2:\sqrt{3}$. Prove that $\angle DAE= 45^{\circ}$

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , fibonacci

The (Fibonacci) sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\ge1$. Prove that the area of the triangle with the sides $\sqrt{f_{2n+1}},\sqrt{f_{2n+2}},$ and $\sqrt{f_{2n+3}}$ is equal to $\frac12$.

2015 Mathematical Talent Reward Programme, MCQ: P 13

Tags: differentiable function , trigonometry , calculus

Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable? [list=1] [*] 0 [*] 2 [*] 4 [*] $\infty$ [/list]

2002 Romania National Olympiad, 3

Tags: linear algebra , matrix , vector , induction

Let $A\in M_4(C)$ be a non-zero matrix. $a)$ If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that: \[UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}\] where $I_r$ is the $r$-unit matrix. $b)$ Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.

2014 Romania National Olympiad, 1

Tags: inequalities

Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$

2008 Irish Math Olympiad, 4

Tags: combinatorics

How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$?

2016 CMIMC, 9

Tags: combinatorics

1007 distinct potatoes are chosen independently and randomly from a box of 2016 potatoes numbered $1, 2, \dots, 2016$, with $p$ being the smallest chosen potato. Then, potatoes are drawn one at a time from the remaining 1009 until the first one with value $q < p$ is drawn. If no such $q$ exists, let $S = 1$. Otherwise, let $S = pq$. Then given that the expected value of $S$ can be expressed as simplified fraction $\tfrac{m}{n}$, find $m+n$.

1993 IberoAmerican, 2

Tags: pigeonhole principle , geometry , circumcircle , rhombus , perpendicular bisector , combinatorics

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

2025 Junior Balkan Team Selection Tests - Romania, P5

Tags: combinatorics

Let $n\geqslant 3$ be a positive integer and $\mathcal F$ be a family of at most $n$ distinct subsets of the set $\{1,2,\ldots,n\}$ with the following property: we can consider $n$ distinct points in the plane, labelled $1,2,\ldots,n$ and draw segments connecting these points such that points $i$ and $j$ are connected if and only if $i{}$ belongs to $j$ subsets in $\mathcal F$ for any $i\neq j.$ Determine the maximal value that the sum of the cardinalities of the subsets in $\mathcal{F}$ can take.

2025 Sharygin Geometry Olympiad, 20

Tags: brocard s theorem , poles and polars , euler circle , geometry

Let $H$ be the orthocenter of a triangle $ABC$, and $M$, $N$ be the midpoints of segments $BC$, $AH$ respectively. The perpendicular from $N$ to $MH$ meets $BC$ at point $A^{\prime}$. Points $B^{\prime}$ and $C^{\prime}$ are defined similarly. Prove that $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ are collinear. Proposed by: F.Ivlev

2015 China Northern MO, 3

Tags: number theory , prime factorization

If $n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}$ then $\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)$. Find the smallest positive integer $n$ such that $\phi (n)=\frac{2^5}{47}n.$