Olympiad problems

Estonia Open Senior - geometry, 2017.1.5

On the sides $BC, CA$ and $AB$ of triangle $ABC$, respectively, points $D, E$ and $F$ are chosen. Prove that $\frac12 (BC + CA + AB)<AD + BE + CF<\frac 32 (BC + CA + AB)$.

2020 Belarusian National Olympiad, 11.6

Tags: algebra , functional equation

Functions $f(x)$ and $g(x)$ are defined on the set of real numbers and take real values. It is known that $g(x)$ takes all real values, $g(0)=0$, and for all $x,y \in \mathbb{R}$ the following equality holds $$f(x+f(y))=f(x)+g(y)$$ Prove that $g(x+y)=g(x)+g(y)$ for all $x,y \in \mathbb{R}$.

2024 LMT Fall, 10

Tags: team

Find the sum of all positive integers $n\le 2024$ such that all pairs of distinct positive integers $(a,b)$ that satisfy $ab=n$ have a sum that is a perfect square.

2020 Iranian Our MO, 4

Tags: graph theory , combinatorics

In a school there are $n$ classes and $k$ student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than $k$ students. If $k-1$ is not a perfect square, prove that there exist a student that has attended in at least $\sqrt k$ classes. [i]Proposed by Mohammad Moshtaghi Far, Kian Shamsaie[/i] [b]Rated 4[/b]

1967 IMO Longlists, 31

Tags: combinatorics , counting

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

2007 France Team Selection Test, 3

Tags: geometry

Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel. Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.

2021 Azerbaijan Senior NMO, 2

Tags: number theory , perfect square

Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

2011 Romania National Olympiad, 2

Tags: number theory

Prove that any natural number smaller or equal than the factorial of a natural number $ n $ is the sum of at most $ n $ distinct divisors of the factorial of $ n. $

2015 EGMO, 1

Tags: geometry , triangle , tangent

Let $\triangle ABC$ be an acute-angled triangle, and let $D$ be the foot of the altitude from $C.$ The angle bisector of $\angle ABC$ intersects $CD$ at $E$ and meets the circumcircle $\omega$ of triangle $\triangle ADE$ again at $F.$ If $\angle ADF = 45^{\circ}$, show that $CF$ is tangent to $\omega .$

2022 Singapore MO Open, Q3

Tags: functional equation , number theory , factorial , function

Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

2008 Princeton University Math Competition, B1

Tags: combinatorics

Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange?

2010 Postal Coaching, 3

Tags: algebra

Determine the smallest odd integer $n \ge 3$, for which there exist $n$ rational numbers $x_1 , x_2 , . . . , x_n$ with the following properties: $(a)$ \[\sum_{i=1}^{n} x_i =0 , \ \sum_{i=1}^{n} x_i^2 = 1.\] $(b)$ \[x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.\]

2013 Purple Comet Problems, 13

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Find $n$ so that $4^{4^{4^2}}=2^{8^n}$.

2008 District Olympiad, 2

Tags: real analysis , periodic , analysis , integral , primitive , function

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that: [b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic. [b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $

2018 IMAR Test, 3

Tags: combinatorics , set

Let $S$ be a finite set and let $\mathcal{P}(S)$ be its power set, i.e., the set of all subsets of $S$, the empty set and $S$, inclusive. If $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S),$ let \[\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.\] Given a non-negative integer $n\leqslant |S|,$ determine the minimal size $\mathcal{A}\vee \mathcal{B}$ may have, where $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S)$ such that $|\mathcal{A}|+|\mathcal{B}|>2^n$. [i]Amer. Math. Monthly[/i]

2017 Hanoi Open Mathematics Competitions, 12

Tags: fixed , geometry , segment

Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?

2007 IMO Shortlist, 8

Tags: geometry , quadrilateral , incircle , triangle

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]

1986 AIME Problems, 11

Tags: algebra , polynomial , ratio , binomial theorem

The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and thet $a_i$'s are constants. Find the value of $a_2$.

1986 Tournament Of Towns, (108) 2

Tags: number theory , maximum , digit , divide

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

2006 Argentina National Olympiad, 6

Tags: number theory

We will say that a natural number $n$ is [i]adequate[/i] if there exist $n$ integers $a_1,a_2,\ldots ,a_n$ (which are not necessarily positive and can be repeated) such that$$a_1+a_2+\cdots +a_n=a_1a_2 \cdots a_n=n.$$Determine all [i]adequate[/i] numbers.

2017 Balkan MO Shortlist, C4

Tags: combinatorics , minimum , radius , combinatorial geometry , circles

For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$

2002 German National Olympiad, 4

Tags: inequalities , convergence , sequence , recursive , algebra

Given a positive real number $a_1$, we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let $$b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }.$$ Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound.

2008 APMO, 2

Tags: pigeonhole principle , combinatorics , algorithm

Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is $ 46$, there is a set of $ 10$ students in which no group is properly contained.

2021 AMC 10 Fall, 11

Tags:

A regular hexagon of side length $1{ }$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? $(\textbf{A})\: \frac{5\sqrt{3}}{2} - \pi\qquad(\textbf{B}) \: 3\sqrt{3}-\pi\qquad(\textbf{C}) \: 4\sqrt{3}-\frac{3\pi}{2}\qquad(\textbf{D}) \: \pi - \frac{\sqrt{3}}{2}\qquad(\textbf{E}) \: \frac{\pi + \sqrt{3}}{2}$

1997 Federal Competition For Advanced Students, Part 2, 1

Tags: number theory

Let $a$ be a fixed integer. Find all integer solutions $x, y, z$ of the system \[5x + (a + 2)y + (a + 2)z = a,\]\[(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,\]\[(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.\]