Olympiad problems

2016 AIME Problems, 5

Tags:

Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.

2018 Kyiv Mathematical Festival, 3

Tags: inequalities

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge2\sqrt{2xy}.$

1967 IMO Shortlist, 5

Tags: function , geometry , algebra , inequality , linear function

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

2006 Mexico National Olympiad, 1

Tags: number theory

Let $ab$ be a two digit number. A positive integer $n$ is a [i]relative[/i] of $ab$ if: [list] [*] The units digit of $n$ is $b$. [*] The remaining digits of $n$ are nonzero and add up to $a$.[/list] Find all two digit numbers which divide all of their relatives.

2021 Iranian Geometry Olympiad, 3

Tags: geometry , intermediate p3

Given a convex quadrilateral $ABCD$ with $AB = BC $and $\angle ABD = \angle BCD = 90$.Let point $E$ be the intersection of diagonals $AC$ and $BD$. Point $F$ lies on the side $AD$ such that $\frac{AF}{F D}=\frac{CE}{EA}$.. Circle $\omega$ with diameter $DF$ and the circumcircle of triangle $ABF$ intersect for the second time at point $K$. Point $L$ is the second intersection of $EF$ and $\omega$. Prove that the line $KL$ passes through the midpoint of $CE$. [i]Proposed by Mahdi Etesamifard and Amir Parsa Hosseini - Iran[/i]

2016 India PRMO, 8

Tags: floor function , algebra

Find the number of integer solutions of $\left[\frac{x}{100} \left[\frac{x}{100}\right]\right]= 5$

2022 German National Olympiad, 1

Tags: algebra , system of equations , quadratic equation , parameterization

Determine all real numbers $a$ for which the system of equations \begin{align*} 3x^2+2y^2+2z^2&=a\\ 4x^2+4y^2+5z^2&=1-a \end{align*} has at least one solution $(x,y,z)$ in the real numbers.

2019 Estonia Team Selection Test, 5

Tags: combinatorics

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country

2002 Hong kong National Olympiad, 1

Tags: trigonometry , geometry

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

2012 Czech And Slovak Olympiad IIIA, 5

Tags: combinatorics

In a group of $90$ children each has at least $30$ friends (friendship is mutual). Prove that they can be divided into three $30$-member groups so that each child has its own a group of at least one friend.

2022 JHMT HS, 3

Tags: expected value

Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?

1979 Austrian-Polish Competition, 1

Tags: square , right angle , equal segments , geometry

On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.

2012 District Olympiad, 2

Tags: function , algebra , polynomial , linear algebra

Let $ A,B\in\mathcal{M} \left( \mathbb{R} \right) $ that satisfy $ AB=O_3. $ Prove that: [b]a)[/b] The function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ defined as $ f(x)=\det \left( A^2+B^2+xBA \right) $ is a polynomial one, of degree at most $ 2. $ [b]b)[/b] $ \det\left( A^2+B^2 \right)\ge 0. $

2007 Sharygin Geometry Olympiad, 11

Tags: ratio , geometry , distance

A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases? (Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)

1965 Dutch Mathematical Olympiad, 2

Tags: number theory , perfect square , divide , divisible

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

2010 Harvard-MIT Mathematics Tournament, 8

Tags: integration , calculus , factorial

Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.

2021 Final Mathematical Cup, 3

Tags: inequalities , number theory , algebra , sum of digits

For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$

2022 Durer Math Competition Finals, 16

Tags: number theory , divisor

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

2009 239 Open Mathematical Olympiad, 3

Tags:

$200$ sticks are given whose lengths are $1, 2, 4, \ldots , 2^{199}$. What is the smallest number of sticks needed to be broken so that out of all the resulting sticks, several triangles could be created, if each stick could be broken only once, and each triangle can be created out of only three sticks?

2015 Sharygin Geometry Olympiad, 8

Tags: geometry , rectangle

Does there exist a rectangle which can be divided into a regular hexagon with sidelength $1$ and several congruent right-angled triangles with legs $1$ and $\sqrt{3}$?

2008 Middle European Mathematical Olympiad, 4

Tags: number theory

Determine that all $ k \in \mathbb{Z}$ such that $ \forall n$ the numbers $ 4n\plus{}1$ and $ kn\plus{}1$ have no common divisor.

2008 China Girls Math Olympiad, 8

Tags: number theory , search

For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.

1986 Traian Lălescu, 2.3

Tags: calculus , limit , real analysis , continuity

Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $

2020 Costa Rica - Final Round, 4

Tags: algebra , functional equation , function , functional

Consider the function $ h$, defined for all positive real numbers, such that: $$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$ for all $x > 0$. Find $h(x)$ and the value of $h(4)$.

2001 Greece Junior Math Olympiad, 4

Tags: geometry , equal segments , angle chasing

Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.