Olympiad problems

2000 Moldova National Olympiad, Problem 2

Solve the system \begin{align*} 36x^2y-27y^3&~=~8,\\ 4x^3-27xy^2&~=~4.\end{align*}

2006 France Team Selection Test, 1

Tags: circumcircle , projective geometry , cyclic quadrilateral , angle bisector , geometry

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

1987 AIME Problems, 3

Tags: calculus , integration , geometry , 3d geometry , tetrahedron

By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

2019 Greece Team Selection Test, 4

Tags: functional equation , functional , algebra

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

2024 LMT Fall, 24

Tags: guts

Let $ABC$ be a triangle with $AB=13, BC=15, AC=14$. Let $P$ be the point such that $AP$ $=$ $CP$ $=$ $\tfrac12 BP$. Find $AP^2$.

2001 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory

Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!

2019-IMOC, N4

Tags: number theory

Given a sequence of prime numbers $p_1, p_2,\cdots$ , with the following property: $p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$ Show that the set $\{p_i\}_{i\in \mathbb{N}}$ is finite.

2015 NIMO Problems, 4

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Find the sum of all positive integers $1\leq k\leq 99$ such that there exist positive integers $a$ and $b$ with the property that \[x^{100}-ax^k+b=(x^2-2x+1)P(x)\] for some polynomial $P$ with integer coefficients. [i]Proposed by David Altizio[/i]

VMEO IV 2015, 10.1

Tags: algebra , sequence , recurrence , recurrence relation , limit of sequence

Where $n$ is a positive integer, the sequence $a_n$ is determined by the formula $$a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1.$$ Find the limit of the sequence $S_n$ defined by $S_n=a_1 + a_2 +... + a_n$.

2013 National Chemistry Olympiad, 59

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All of the following atoms comprise part of a peptide functional group except: $ \textbf{(A)}\ \text{Hydrogen} \qquad\textbf{(B)}\ \text{Nitrogen}\qquad$ ${\textbf{(C)}\ \text{Oxygen} \qquad\textbf{(D)}}\ \text{Phosphorous} \qquad$

1991 Polish MO Finals, 3

Tags: inequalities , three variable inequality , algebra

If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$, prove the inequality \[ x + y + z \leq 2 + xyz \] When does equality occur?

2023 Kyiv City MO Round 1, Problem 4

Tags: number theory

Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer. [i]Proposed by Oleksiy Masalitin[/i]

2007 Today's Calculation Of Integral, 232

Tags: calculus , integration , trigonometry , inequalities , calculus computations

For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$ Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$

2015 Israel National Olympiad, 3

Tags: algebra , root , cube roots

Prove that the number $\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3$ is an integer.

2020 MIG, 23

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There exists a positive integer $b$ such that the base-$10$ fraction $\tfrac{59}{48}$ can be expressed as $1.\overline{14}_b$ (or $1.141414\ldots_b$), a value in base $b$. Find $b$. $\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

2009 Tournament Of Towns, 3

Tags: combinatorics

In each square of a $101\times 101$ board, except the central one, is placed either a sign " turn" or a sign " straight". The chess piece " car" can enter any square on the boundary of the board from outside (perpendicularly to the boundary). If the car enters a square with the sign " straight" then it moves to the next square in the same direction, otherwise (in case it enters a square with the sign " turn") it turns either to the right or to the left ( its choice). Can one place the signs in such a way that the car never enter the central square?

2023 MOAA, 9

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Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. [color=#00f]Note that this problem is null because a diagram is impossible.[/color] [i]Proposed by Andy Xu[/i]

1973 USAMO, 1

Tags: geometry , 3d geometry , tetrahedron

Two points $ P$ and $ Q$ lie in the interior of a regular tetrahedron $ ABCD$. Prove that angle $ PAQ < 60^\circ$.

2015 Regional Olympiad of Mexico Southeast, 2

Tags: altitude , geometry

In a acutangle triangle $ABC, \angle B>\angle C$. Let $D$ the foot of the altitude from $A$ to $BC$ and $E$ the foot of the perpendicular from $D$ to $AC$. Let $F$ a point in $DE$. Prove that $AF$ and $BF$ are perpendiculars if and only if $EF\cdot DC=BD\cdot DE$.

2024 All-Russian Olympiad, 3

Tags: game , combinatorics

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

1998 Tournament Of Towns, 6

Tags: combinatorics

A gang of robbers took away a bag of coins from a merchant . Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies) . Prove that after one coin is removed, the number of the remaining coins is divisible by the number of robbers. (Folklore, modified by A Shapovalov)

Kvant 2021, M2677

Tags: combinatorics

There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob? [i]Alexandr Gribalko[/i]

2019 ASDAN Math Tournament, 1

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A kite is a quadrilateral with $2$ pairs of equal adjacent sides. Given a cyclic kite with side lengths $3$ and $4$, compute the distance between the intersection of its diagonals and the center of the circle circumscribing it.

2004 All-Russian Olympiad Regional Round, 8.3

Tags: geometry , equilateral , median

In an acute triangle, the distance from the midpoint of any side to the opposite vertex is equal to the sum of the distances from it to sides of the triangle. Prove that this triangle is equilateral.

2021 China Second Round Olympiad, Problem 1

Tags: vector

Given two vectors $\overrightarrow a$, $\overrightarrow b$, find the range of possible values of $\|\overrightarrow a - 2 \overrightarrow b\|$ where $\|\overrightarrow v\|$ denotes the magnitude of a vector $\overrightarrow v$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 1)[/i]