Olympiad problems

2006 Estonia National Olympiad, 1

Tags: inequalities

Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real numbers x, for which this value is achieved.

2019 Mediterranean Mathematics Olympiad, 1

Tags: geometry , incircle , inradius

Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that \[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]

2009 Singapore Senior Math Olympiad, 4

Tags: inequalities , weighted am-gm , logarithm

Given that $ a,b,c, x_1, x_2, ... , x_5 $ are real positives such that $ a+b+c=1 $ and $ x_1.x_2.x_3.x_4.x_5 = 1 $. Prove that \[ (ax_1^2+bx_1+c)(ax_2^2+bx_2+c)...(ax_5^2+bx_5+c)\ge 1\]

2024-IMOC, A3

Tags: sequence , integer sequence

Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying \[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\] holds for all $n\geq 1$. Define $0^0=1$

2014 Contests, 2

Tags: modular arithmetic , number theory

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

2005 JHMT, 3

Tags: geometry

Isosceles triangle $ABC$ has angle $\angle BAC = 135^o$ and $AB = 2$. What is its area?

2012 China Team Selection Test, 1

Tags: floor function , function , number theory , logarithm

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2010 Sharygin Geometry Olympiad, 6

Tags: midpoint , square , angle chasing , angle , geometry

Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

2016 Saudi Arabia BMO TST, 4

Tags: coloring , combinatorics

Given six three-element subsets of the set $X$ with at least $5$ elements, show that it is possible to color the elements of $X$ in two colors such that none of the given subsets is all in one color.

2022 Caucasus Mathematical Olympiad, 4

Tags: combinatorial geometry , lattice points , combinatorics

Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?

2023 AMC 8, 25

Tags:

Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that $$1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace 241 \le a_{15}\le 250.$$ What is the sum of digits of $a_{14}$? $\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

1969 Canada National Olympiad, 9

Tags: trigonometry

Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.

2019 China Team Selection Test, 3

Tags: combinatorics , geometry , combinatorial geometry

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

2006 MOP Homework, 4

Tags: circumcircle , euler , power of a point , radical axis , geometry

Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a$, P_b$, and $P_c$ lie on a line.

2017 Princeton University Math Competition, 17

Tags: combinatorics

Zack keeps cutting the interval $[0, 1]$ of the number line, each time cutting at a uniformly random point in the interval, until the interval is cut into pieces, none of which have length greater than $\frac35$ . The expected number of cuts that Zack makes can be written as $\frac{p}{q}$ for $p$ and $q$ relatively prime positive integers. Find $p + q$.

1986 National High School Mathematics League, 5

Tags:

There is a point set on a plane, and seven circles $C_1,C_2,\cdots,C_7$, where $C_7$ passes exactly 7 points in $M$, $C_6$ passes exactly 6 points in $M$, ..., $C_1$ passes exactly 1 point in $M$. Then how many points do set $M$ have at least? $\text{(A)}11\qquad\text{(B)}12\qquad\text{(C)}21\qquad\text{(D)}28$

2021 India National Olympiad, 2

Tags: integer root , cubic , algebra , vieta

Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

2023 Belarusian National Olympiad, 11.6

Tags: algebra , polynomial

Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients

2002 AMC 10, 21

Tags:

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

1995 Tournament Of Towns, (466) 4

Tags: geometry , angle bisector , equal segments

From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$. (I Sharygin)

2012 Bosnia and Herzegovina Junior BMO TST, 1

Tags: pentagon , geometry , circles

On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$

2004 Finnish National High School Mathematics Competition, 5

Tags: number theory

Finland is going to change the monetary system again and replace the Euro by the Finnish Mark. The Mark is divided into $100$ pennies. There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able to pay for any purchase less than one mark should be minimal. Determine the coin denominations.

2017 All-Russian Olympiad, 6

Tags: combinatorics

In the $200\times 200$ table in some cells lays red or blue chip. Every chip "see" other chip, if they lay in same row or column. Every chip "see" exactly $5$ chips of other color. Find maximum number of chips in the table.

2011 IMO Shortlist, 1

Tags: algorithm , number theory , prime number , divisibility , number of divisors

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

2018 PUMaC Team Round, 1

Tags: probability , expected value

Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.