Olympiad problems

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P1

Tags: combinatorics , board , Coloring , chesslike , maximum

A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board. [i]Proposed by Nikola Velov[/i]

2023 Brazil Cono Sur TST, 1

Tags:

A quadrilateral $ABCD$ is inscribed in a circle and the lenght of side $AD$ equals the sum of the lenghts of the sides $AB$ and $CD$. Prove that the angle bisectors of $\angle ABC$ and $\angle BCD$ meet on the side $AD$.

2016 ISI Entrance Examination, 2

Tags: algebra , polynomial

Consider the polynomial $ax^3+bx^2+cx+d$ where $a,b,c,d$ are integers such that $ad$ is odd and $bc$ is even.Prove that not all of its roots are rational..

2012 All-Russian Olympiad, 4

Tags: induction , number theory proposed , number theory

For a positive integer $n$ define $S_n=1!+2!+\ldots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$.

2010 China Northern MO, 2

Tags: geometry , equal segments

From a point $P$ exterior of circle $\odot O$, we draw tangents $PA$, $PB$ and the secant $PCD$ . The line passing through point $C$ parallel to $PA$ intersects chords $AB$, $AD$ at points $E$, $F$ respectively. Prove that $CE = EF$. [img]https://cdn.artofproblemsolving.com/attachments/8/c/bf15595bc341b917df30b3aa93067887317c65.png[/img]

2023 Kyiv City MO Round 1, Problem 3

Tags: geometry , circumcircle

You are given a right triangle $ABC$ with $\angle ACB = 90^\circ$. Let $W_A , W_B$ respectively be the midpoints of the smaller arcs $BC$ and $AC$ of the circumcircle of $\triangle ABC$, and $N_A , N_B$ respectively be the midpoints of the larger arcs $BC$ and $AC$ of this circle. Denote by $P$ and $Q$ the points of intersection of segment $AB$ with the lines $N_AW_B$ and $N_BW_A$, respectively. Prove that $AP = BQ$. [i]Proposed by Oleksiy Masalitin[/i]

2002 Pan African, 6

Tags: inequalities , induction

If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ and $a_1+a_2+\cdots+a_n=1$, then prove: \[a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1\]

2018 Saudi Arabia GMO TST, 4

Tags: graph , edges , combinatorics , graph theory

In a graph with $8$ vertices that contains no cycle of length $4$, at most how many edges can there be?

2020-2021 OMMC, 4

Tags: ommc

Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.

1973 IMO Longlists, 1

Tags: geometry , 3D geometry , sphere , tetrahedron

Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.

2016 Argentina National Olympiad Level 2, 1

Tags: combinatorics

In the cells of a $1 \times 100$ board, Julián writes all the integers from $1$ to $100$ (inclusive) in any order of his choice, without repeating numbers. For every three consecutive cells on the board, the cell containing the middle value of the three numbers in those cells is marked. For example, if the three numbers are $7$, $99$ and $22$, then the cell with $22$ is marked. Let $S$ be the sum of all the numbers in the marked cells. Determine the minimum value that $S$ can take. [b]Note:[/b] Each marked number contributes to the sum $S$ exactly once, but it can be marked multiple times.

2018 BMT Spring, 10

Tags: number theory

Evaluate the following $$\prod^{50}_{j=1} \left( 2 cos \left( \frac{4\pi j}{101} \right) + 1\right).$$

2001 Turkey MO (2nd round), 2

Tags: trigonometry , geometry , circumcircle , perpendicular bisector , geometric transformation , geometry proposed

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

2022 Assara - South Russian Girl's MO, 8

Tags: geometry , hexagon , concurrency , concurrent

About the convex hexagon $ABCDEF$ it is known that $AB = BC =CD = DE = EF = FA$ and $AD = BE = CF$. Prove that the diagonals $AD$, $BE$, $CF$ intersect at one point.

MOAA Team Rounds, TO5

Tags: algebra , theme , 2020

For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$

2015 AMC 12/AHSME, 24

Tags: probability , trigonometry , function , complex numbers , AMC

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number? $\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$

2006 Romania National Olympiad, 2

Tags: conics , parabola , analytic geometry

Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that \[ n = \sum_{1\leq i < j \leq k } a_ia_j . \]

1976 Putnam, 1

Tags: college contests

Evaluate $$lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} ([\frac{2n}{k}] -2[\frac{n}{k}])$$ and express your answer in the form $\log a-b,$ with $a$ and $b$ positive integers. Here $[x]$ is defined to be the integer such that $[x] \leq x <[x]+1$ and $\log x$ is the logarithm of $x$ to base $e.$

1983 All Soviet Union Mathematical Olympiad, 350

Tags: number theory , combinatorics , game strategy , blackboard

Three numbers were written with a chalk on the blackboard. The following operation was repeated several times: One of the numbers was cleared and the sum of two other numbers, decreased by $1$, was written instead of it. The final set of numbers is $\{17, 1967, 1983\}$.Is it possible to admit that the initial numbers were a) $\{2, 2, 2\}$? b) $\{3, 3, 3\}$?

2000 Belarusian National Olympiad, 2

Tags: number theory , factorial

Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$

2015 Princeton University Math Competition, 10

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Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and: \[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?

1968 IMO Shortlist, 16

Tags: algebra , polynomial , calculus , number theory , Divisibility , IMO Shortlist

A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$

2006 Polish MO Finals, 1

Tags: modular arithmetic , combinatorics proposed , combinatorics

Given a triplet we perform on it the following operation. We choose two numbers among them and change them into their sum and product, left number stays unchanged. Can we, starting from triplet $(3,4,5)$ and performing above operation, obtain again a triplet of numbers which are lengths of right triangle?

2002 AMC 12/AHSME, 18

Tags: inequalities , algebra

If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$. $\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$

Kvant 2024, M2804

Tags: geometry

There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$. (a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles. (b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.