Olympiad problems

2015 Tournament of Towns, 3

Tags: algebra , polynomial

Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$. Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$. [i]($6$ points)[/i]

2010 Albania National Olympiad, 4

Tags: algorithm , algebra , system of equations , number theory unsolved , number theory

The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. [b](a)[/b] Prove that $f_{2010} $ is divisible by $10$. [b](b)[/b] Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

2023 Baltic Way, 16

Tags: algebra , polynomial

Prove that there exist nonconstant polynomials $f, g$ with integer coefficients, such that for infinitely many primes $p$, $p \nmid f(x)-g(y)$ for any integers $x, y$.

Ukrainian TYM Qualifying - geometry, VIII.3

Tags: geometric inequality , geometry , Ukrainian TYM

Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.

2000 Moldova National Olympiad, Problem 2

Tags: limit , real analysis

For $n\in\mathbb N$, define $$a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.$$ (a) Prove that the sequence $b_n=a_n^n$ is convergent and determine the limit. (b) Show that $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$.

2006 Italy TST, 3

Tags: algebra , polynomial , complex numbers , algebra unsolved

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

2000 National High School Mathematics League, 2

Tags: trigonometry

If $\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}$, then the range value of $\frac{\alpha}{3}$ is $\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$ $\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$

2018 Greece Team Selection Test, 2

Tags: geometry proposed , Power of Point , homothety , geometry

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

2022 Kyiv City MO Round 1, Problem 1

Tags: number theory , algebra , greatest common divisor

Consider $5$ distinct positive integers. Can their mean be a)Exactly $3$ times larger than their largest common divisor? b)Exactly $2$ times larger than their largest common divisor?

2025 Polish MO Finals, 6

Tags: algebra , functions , function

A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy $$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\;\;b_{n+1}=b_n+f(a_n)$$ for every $n\geq 1$. Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$.

2002 National Olympiad First Round, 1

Tags: geometry

Let $C', A', B'$ be the midpoints of sides $[AB]$, $[BC]$, $[CA]$ of $\triangle ABC$, respectively. Let $H$ be the foot of perpendicular from $A$ to $BC$. If $|A'C'| = 6$, what is $|B'H|$? $ \textbf{a)}\ 5 \qquad\textbf{b)}\ 6 \qquad\textbf{c)}\ 5\sqrt 2 \qquad\textbf{d)}\ 6\sqrt 2 \qquad\textbf{e)}\ 7 $

2008 239 Open Mathematical Olympiad, 7

Tags: number theory

Find all natural numbers $n, k$ such that $$ 2^n – 5^k = 7. $$

2023 CMI B.Sc. Entrance Exam, 6

Tags: function , number theory , strong induction , CMI , IMO

Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite. The following part problems may attract partial credit. $\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$. $\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$. $\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$. $\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.

2022 Saudi Arabia JBMO TST, 4

Tags: number theory , combinatorics

You plan to organize your birthday party, which will be attended either by exactly $m$ persons or by exactly $n$ persons (you are not sure at the moment). You have a big birthday cake and you want to divide it into several parts (not necessarily equal), so that you are able to distribute the whole cake among the people attending the party with everybody getting cake of equal mass (however, one may get one big slice, while others several small slices - the sizes of slices may differ). What is the minimal number of parts you need to divide the cake, so that it is possible, regardless of the number of guests.

2014 Contests, 2

Tags: number theory proposed , number theory

Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

2004 May Olympiad, 2

Tags: combinatorics

Pepito's mother wants to prepare $n$ packages of $3$ candies to give away at the birthday party, and for this she will buy assorted candies of $3$ different flavors. You can buy any number of candies but you can't choose how many of each taste. She wants to put one candy of each flavor in each package, and if this is not possible she will use only candy of one flavor and all the packages will have $3$ candies of that flavor. Determine the least number of candies that must be purchased in order to assemble the n packages. He explains why if he buys fewer candies, he is not sure that he will be able to assemble the packages the way he wants.

1987 IMO Shortlist, 22

Tags: function , algebra , functional equation , IMO Shortlist , IMO , IMO 1987

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

2011 Mongolia Team Selection Test, 2

Tags: Putnam , function , number theory unsolved , number theory

Mongolia TST 2011 Test 1 #2 Let $p$ be a prime number. Prove that: $\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem) Note: I believe they meant to say $p>2$ as well.

1968 IMO Shortlist, 26

Tags: function , algebra , periodic function , functional equation , IMO , IMO 1968

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2014 Dutch Mathematical Olympiad, 5

Tags: geometry , rectangle , combinatorics , combinatorial geometry , circumradius

We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape. a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$? b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?

2022 Princeton University Math Competition, A6 / B8

Tags: combinatorics

Fine Hall has a broken elevator. Every second, it goes up a floor, goes down a floor, or stays still. You enter the elevator on the lowest floor, and after $8$ seconds, you are again on the lowest floor. If every possible such path is equally likely to occur, the probability you experience no stops is $\tfrac{a}{b},$ where $a,b$ are relatively prime positive integers. Find $a + b.$

2010 Bosnia Herzegovina Team Selection Test, 4

Tags: geometry proposed , geometry

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

2020 Silk Road, 2

Tags: circumcircle , circles

The triangle $ ABC $ is inscribed in the circle $ \omega $. Points $ K, L, M $ are marked on the sides $ AB, BC, CA $, respectively, and $ CM \cdot CL = AM \cdot BL $. Ray $ LK $ intersects line $ AC $ at point $ P $. The common chord of the circle $ \omega $ and the circumscribed circle of the triangle $ KMP $ meets the segment $ AM $ at the point $ S $. Prove that $ SK \parallel BC $.

2009 Regional Olympiad of Mexico Northeast, 3

Tags: geometry , incircle , tangent , circumcircle

The incircle of triangle $\vartriangle ABC$ is tangent to side $AB$ at point $P$ and to side $BC$ at point $Q$. The circle passing through points $A,P,Q$ intersects line $BC$ a second time at $M$ and the circle passes through the points $C,P,Q$ and cuts the line $AB$ a second time at point$ N$. Prove that $NM$ is tangent to the incircle of $ABC$.

1974 Chisinau City MO, 71

Tags: geometry , Centroid , Triangle construction

The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.