Olympiad problems

2010 National Olympiad First Round, 27

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

1968 IMO Shortlist, 7

Tags: trigonometry , triangle , geometric inequality

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?

2002 Vietnam Team Selection Test, 1

Tags: angle bisector , geometry , perpendicular bisector

Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.

2010 Tournament Of Towns, 2

Tags: parabola , function , conic , algebra

Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$

2016 PUMaC Individual Finals A, 3

Tags: geometry

On a cyclic quadrilateral $ABCD$, $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$. Let $E$ be the projection of $C$ onto $AB$ and $F$ the reflection of $N$ about the midpoint of $DE$. If $F$ is inside quadrilateral $ABCD$, show that $\angle BMF = \angle CBD$.

2016 Saudi Arabia IMO TST, 3

Tags: circles , geometry , tangent

Given two circles $(O_1)$ and $(O_2)$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $(O_1)$ and $(O_2)$ at $G, E$ ($\ne A$), respectively, the line $d_2$ cuts $(O_1)$ and $(O_2)$ at $F, H$ ($\ne A$), respectively, such that $E$ is between $A, G$ and $F$ is between $A, H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $(O_1), (O_2)$ at $K, L$ ($\ne B$), respectively. Let $N$ be the intersection of $O_1K$ and $O_2L$. Prove that the circle $(NLK)$ is tangent to $AB$.

2022 JHMT HS, 8

Tags: algebra , complex number

Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

2018 Centroamerican and Caribbean Math Olympiad, 4

Tags: number theory

Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$.

2009 Stars Of Mathematics, 5

Tags: coloring , color problem , linear algebra , matrix , combinatorics

The cells of a $(n^2-n+1)\times(n^2-n+1)$ matrix are coloured using $n$ colours. A colour is called [i]dominant[/i] on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called [i]extremal[/i] if its colour is [i]dominant [/i] both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no [i]extremal [/i] cells. Iurie Boreico (Moldova)

1946 Moscow Mathematical Olympiad, 117

Tags: diophantine equation , number theory

Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.

2021 Purple Comet Problems, 14

Tags:

Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.

1998 AIME Problems, 3

Tags: geometry , parallelogram , analytic geometry

The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions. What is the area of the bounded region?

2019 AMC 10, 19

Tags: algebra

What is the least possible value of $$(x+1)(x+2)(x+3)(x+4)+2019$$where $x$ is a real number? $\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

2024 Belarusian National Olympiad, 11.8

Tags: space , combinatorial geometry , geometry

Projector emits rays in space. Consider all acute angles between the rays. It is known that no matter what ray we remove, the number of acute angles decreases by exactly $2$ What is the maximal number of rays the projector can emit? [i]M. Karpuk, E. Barabanov[/i]

2021 Winter Stars of Mathematics, 3

Tags: quadratic residue , number theory

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

2022 Czech-Polish-Slovak Junior Match, 3

Tags: equal segments , geometry , parallel

The points $D, E, F$ lie respectively on the sides $BC$, $CA$, $AB$ of the triangle ABC such that $F B = BD$, $DC = CE$, and the lines $EF$ and $BC$ are parallel. Tangent to the circumscribed circle of triangle $DEF$ at point $F$ intersects line $AD$ at point $P$. Perpendicular bisector of segment $EF$ intersects the segment $AC$ at $Q$. Prove that the lines $P Q$ and $BC$ are parallel.

2011 IFYM, Sozopol, 1

Tags: numbers in a table , table , combinatorics

In the cells of a square table $n$ x $n$ the numbers $1,2,...,n^2$ are written in an arbitrary way. Prove that there exist two adjacent cells, for which the difference between the numbers written in them is no lesser than $n$.

2024 Pan-African, 4

Tags: combinatorics

Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other. What is the maximum value of $m$?

2012-2013 SDML (Middle School), 6

Tags:

What is the largest two-digit integer for which the product of its digits is $17$ more than their sum?

2019 Middle European Mathematical Olympiad, 7

Tags: inequalities , number theory

Let $a,b$ and $c$ be positive integers satisfying $a<b<c<a+b$. Prove that $c(a-1)+b$ does not divide $c(b-1)+a$. [i]Proposed by Dominik Burek, Poland[/i]

2021 Alibaba Global Math Competition, 8

Tags: complex analysis , function , inequalities

Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ($0<R<\infty$). Define \[M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, \quad A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.\] Show that \[M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \quad \forall 0 \le r<R.\]

2014 PUMaC Combinatorics B, 1

Tags: probability

A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 AM, also with equal probability. Let $\frac pq$ for $p$, $q$ coprime be the probability that they fall in love. Find $p + q$.

2024 Mathematical Talent Reward Programme, 9

Tags: algebra , polynomial

Find the number of integer polynomials $P$ such that $P(x)^2 = P(P(x)) \forall x$.

2016 South East Mathematical Olympiad, 4

Tags: combinatorics

For any four points on a plane, if the areas of four triangles formed are different positive integer and six distances between those four points are also six different positive integers, then the convex closure of $4$ points is called a "lotus design." (1) Construct an example of "lotus design". Also what are areas and distances in your example? (2) Prove that there are infinitely many "lotus design" which are not similar.

1989 Irish Math Olympiad, 2

Tags: magic square , combinatorics , matrix

A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is \[\left( \begin{array}{ccccc} 2 & 1 & 3\\ 3 & 2 & 1\\ 1 & 3 & 2 \end{array} \right)\]