Olympiad problems

2003 IMO Shortlist, 6

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

1998 Greece JBMO TST, 5

Tags: ceiling function , algebra

Let $I$ be an open interval of length $\frac{1}{n}$, where $n$ is a positive integer. Find the maximum possible number of rational numbers of the form $\frac{a}{b}$ where $1 \le b \le n$ that lie in $I$.

2007 Romania Team Selection Test, 2

Tags: parallelogram , trapezoid , geometry

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

CNCM Online Round 1, 2

Tags:

Akshar is reading a $500$ page book, with odd numbered pages on the left, and even numbered pages on the right. Multiple times in the book, the sum of the digits of the two opened pages are $18$. Find the sum of the page numbers of the last time this occurs. Proposed by Minseok Eli Park (wolfpack)

1965 All Russian Mathematical Olympiad, 061

Tags: combinatorics

A society created in the help to the police contains $100$ men exactly. Every evening $3$ men are on duty. Prove that you can not organise duties in such a way, that every couple will meet on duty once exactly.

2012 Pre-Preparation Course Examination, 2

Tags: linear algebra , vector

Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.

1977 Spain Mathematical Olympiad, 2

Tags: group theory , algebra

Prove that all square matrices of the form (with $a, b \in R$), $$\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$$ form a commutative field $K$ when considering the operations of addition and matrix product. Prove also that if $A \in K$ is an element of said field, there exist two matrices of $K$ such that the square of each is equal to $A$.

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