Olympiad problems

2014 Dutch IMO TST, 4

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

2007 Bulgaria National Olympiad, 2

Tags: combinatorics

Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i<j\leq n$ there exists a positive integer $\overline{a_{1}a_{2}\ldots a_{n}}$ from the chosen integers for which $a_{j}\geq a_{i}+2$. [i]A. Ivanov, E. Kolev[/i]

2004 France Team Selection Test, 1

Tags: inequalities , calculus , derivative , function

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

2023 NMTC Junior, P1

Tags: number theory

Find integers $m,n$ such that the sum of their cubes is equal to the square of their sum.

1998 Romania Team Selection Test, 1

Tags: function , algebra , functional equation

Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+y)=f(x)u(x)+f(y) \] for all $x,y\in\mathbb{R}$. [i]Vasile Pop[/i]

2019 Math Prize for Girls Problems, 14

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Devah draws a row of 1000 equally spaced dots on a sheet of paper. She goes through the dots from left to right, one by one, checking if the midpoint between the current dot and some remaining dot to its left is also a remaining dot. If so, she erases the current dot. How many dots does Devah end up erasing?

2008 National Olympiad First Round, 30

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In a sequence with the first term is positive integer, the next term is generated by adding the previous term and its largest digit. At most how many consequtive terms of this sequence are odd? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

2007 Hong kong National Olympiad, 4

Tags: floor function , number theory

find all positive integer pairs $(m,n)$,satisfies: (1)$gcd(m,n)=1$,and $m\le\ 2007$ (2)for any $k=1,2,...2007$,we have $[\frac{nk}{m}]=[\sqrt{2}k]$

2021 JHMT HS, 10

Tags: combinatorics , geometry

Let $P$ be a set of nine points in the Cartesian coordinate plane, no three of which lie on the same line. Call an ordering $\{Q_1, Q_2, \ldots, Q_9\}$ of the points in $P$ [i]special[/i] if there exists a point $C$ in the same plane such that $CQ_1 < CQ_2 < \cdots < CQ_9$. Over all possible sets $P,$ what is the largest possible number of distinct special orderings of $P?$

2022 Mexican Girls' Contest, 8

Tags: geometry , trapezoid

Let $n$ be a positive integer. Consider a figure of a equilateral triangle of side $n$ and splitted in $n^2$ small equilateral triangles of side $1$. One will mark some of the $1+2+\dots+(n+1)$ vertices of the small triangles, such that for every integer $k\geq 1$, there is [b]not[/b] any trapezoid(trapezium), whose the sides are $(1,k,1,k+1)$, with all the vertices marked. Furthermore, there are [b]no[/b] small triangle(side $1$) have your three vertices marked. Determine the greatest quantity of marked vertices.

2024 Moldova Team Selection Test, 1

Tags: number theory

If $ \frac{a }{b}+ \frac{b}{c}+ \frac{c}{a}$ is integer. show that $ abc$ is perfect cube.

1990 IMO Longlists, 61

Tags: 3d geometry , tetrahedron , octahedron , ratio , geometry

Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.

1985 ITAMO, 10

Tags: floor function , calculus , integration , function

How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

1996 Greece National Olympiad, 3

Tags: pigeonhole principle , number theory

Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.

2020-2021 Fall SDPC, 8

Tags: algebra

Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that \[x^2f(x^2+y^2)+y^4=(xf(x+y)+y^2)(xf(x-y)+y^2)\] for all $x,y \in \mathbb{R}$.

2022 LMT Fall, 2

Tags: combinatorics

Ada rolls a standard $4$-sided die $5$ times. The probability that the die lands on at most two distinct sides can be written as $ \frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$

2020 Belarusian National Olympiad, 11.3

Tags: algebra , hyperbola , conic , geometry

Four points $A$, $B$, $C$, $D$ lie on the hyperbola $y=\frac{1}{x}$. In triangle $BCD$ the point $A_1$ is the circumcenter of the triangle, which vertices are the midpoints of sides of $BCD$. In triangles $ACD$, $ABD$ and $ABC$ points $B_1$, $C_1$ and $D_1$ are chosen similarly. It turned out that points $A_1$, $B_1$, $C_1$ and $D_1$ are pairwise different and concyclic. Prove that the center of that circle coincides with the $(0,0)$ point.

2003 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: inequalities , inradius , geometry

Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.

2010 Iran MO (3rd Round), 1

Tags: polynomial , limit , algebra

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

2009 Argentina Team Selection Test, 3

Tags: geometry , circumcircle , geometric transformation , homothety , reflection , power of a point , radical axis

Let $ ABC$ be a triangle, $ B_1$ the midpoint of side $ AB$ and $ C_1$ the midpoint of side $ AC$. Let $ P$ be the point of intersection ($ \neq A$) of the circumcircles of triangles $ ABC_1$ and $ AB_1C$. Let $ Q$ be the point of intersection ($ \neq A$) of the line $ AP$ and the circumcircle of triangle $ AB_1C_1$. Prove that $ \frac{AP}{AQ} \equal{} \frac{3}{2}$.

1990 Tournament Of Towns, (249) 3

Tags: weighing , combinatorics

Fifteen elephants stand in a row. Their weights are expressed by integer numbers of kilograms. The sum of the weight of each elephant (except the one on the extreme right) and the doubled weight of its right neighbour is exactly $15$ tonnes. Determine the weight of each elephant. (F.L. Nazarov)

LMT Guts Rounds, 5

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Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?

2020 HK IMO Preliminary Selection Contest, 11

Tags: algebra , polynomial , vieta

Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.

2023 MIG, 9

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Which answer choice correctly fills the blank in the statement below? "The probability of flipping heads on a fair coin is the equal to the probability of rolling a $\underline{~~~~~~~~~~~}$ on a fair dice." $\textbf{(A) }\text{prime number}\qquad\textbf{(B) }\text{number divisible by 3}\qquad\textbf{(C) }\text{number with four factors}\qquad\textbf{(D) }2~\text{or}~3\qquad\textbf{(E) }4$

2014 Greece Junior Math Olympiad, 2

Tags: number theory

Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $ p(p+m)+p=(m+1)^3$