Olympiad problems

2009 Purple Comet Problems, 9

Tags:

One plant is now $44$ centimeters tall and will grow at a rate of $3$ centimeters every $2$ years. A second plant is now $80$ centimeters tall and will grow at a rate of $5$ centimeters every $6$ years. In how many years will the plants be the same height?

2012 NIMO Problems, 5

A number is called [i]purple[/i] if it can be expressed in the form $\frac{1}{2^a 5^b}$ for positive integers $a > b$. The sum of all purple numbers can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Eugene Chen[/i]

2006 Victor Vâlcovici, 2

Tags: geometry , circumcircle , analytic geometry

Consider a point $ B $ on a segment $ AC. $ Find the locus of the points $ M $ that have the property that the circumcircles of $ ABM $ and $ BCM $ have equal radii. [i]Nicolae Soare[/i]

Kyiv City MO Seniors Round2 2010+ geometry, 2018.10.3.1

Tags: circumcircle , perpendicular , geometry

The point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AC$ intersects the circumscribed circle $\Delta ABO$ for second time at the point $X$. Prove that $XO \bot BC$.

1990 Austrian-Polish Competition, 3

Tags: algebra , system of equations

Show that there are two real solutions to: $$\begin{cases} x + y^2 + z^4 = 0 \\ y + z^2 + x^4 = 0 \\ z + x^2 + y^5 = 0\end {cases}$$

2022 Dutch IMO TST, 2

Tags: algebra

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$

2018 AMC 10, 24

Tags: geometry

Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X, Y,$ and $Z$ the midpoints of $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$? $\textbf{(A) }\dfrac{3}{8}\sqrt{3}\qquad\textbf{(B) }\dfrac{7}{16}\sqrt{3}\qquad\textbf{(C) }\dfrac{15}{32}\sqrt{3}\qquad\textbf{(D) }\dfrac{1}{2}\sqrt{3}\qquad\textbf{(E) }\dfrac{9}{16}\sqrt{3}$

2003 Finnish National High School Mathematics Competition, 5

Tags: symmetry , arithmetic sequence , combinatorics

Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$ The one who obtains the progression wins. Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$

2024 Canadian Mathematical Olympiad Qualification, 1

Tags: algebra , functional equation

Find all functions $f : R \to R$ that satisfy the functional equation $$f(x + f(xy)) = f(x)(1 + y).$$

2013 Middle European Mathematical Olympiad, 3

Tags: perpendicular bisector , angle chasing , geometry

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $N$ be a point inside the triangle such that $2 \angle ANB = 180 ^\circ + \angle ACB $. Let $ D $ be the intersection of the line $BN$ and the line parallel to $AN$ that passes through $C$. Let $P$ be the intersection of the angle bisectors of the angles $CAN$ and $ABN$. Show that the lines $DP$ and $AN$ are perpendicular.

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..

2019 Bulgaria National Olympiad, 3

Tags: integer sequence , inequalities

Find all real numbers $a,$ which satisfy the following condition: For every sequence $a_1,a_2,a_3,\ldots$ of pairwise different positive integers, for which the inequality $a_n\leq an$ holds for every positive integer $n,$ there exist infinitely many numbers in the sequence with sum of their digits in base $4038,$ which is not divisible by $2019.$

2006 Bulgaria Team Selection Test, 3

Tags: function , inequalities , floor function , ceiling function , combinatorics

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

1962 Miklós Schweitzer, 10

Tags: geometry , probability , probability and stats

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]

2018 Tuymaada Olympiad, 7

Tags: combinatorics

A school has three senior classes of $M$ students each. Every student knows at least $\frac{3}{4}M$ people in each of the other two classes. Prove that the school can send $M$ non-intersecting teams to the olympiad so that each team consists of $3$ students from different classes who know each other. [i]Proposed by C. Magyar, R. Martin[/i]

2006 ITAMO, 1

Tags: combinatorics

Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$, the other player has a card in his hand. What is it's value?

2015 Middle European Mathematical Olympiad, 8

Tags: divisor , c.r.t , number theory

Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$.

2010 Iran Team Selection Test, 9

Tags: induction , inequalities , concavity

Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers: \[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]

2019 Brazil Undergrad MO, Problem 5

Tags: combinatorics , counting , limit , number theory

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$

2022 ELMO Revenge, 1

Tags: algebra

In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation $ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer. [i]Proposed by Alexander Wang[/i]

2023 Kazakhstan National Olympiad, 4

Tags: inequalities , algebra

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

2021-2022 OMMC, 13

Tags:

$ABCD$ is a rhombus where $\angle BAD = 60^\circ$. Point $E$ lies on minor arc $\widehat{AD}$ of the circumcircle of $ABD$, and $F$ is the intersection of $AC$ and the circle circumcircle of $EDC$. If $AF = 4$ and the circumcircle of $EDC$ has radius $14$, find the squared area of $ABCD$. [i]Proposed by Vivian Loh [/i]

2011 Spain Mathematical Olympiad, 2

Tags: function , calculus , derivative , algebra , inequalities

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.

1981 Vietnam National Olympiad, 3

Tags: ratio , geometry

A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.

2008 Sharygin Geometry Olympiad, 9

Tags: geometry , geometric transformation , reflection , circumcircle , trigonometry , projective geometry , cyclic quadrilateral

(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).