Olympiad problems

Russian TST 2021, P3

Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has \[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]

1986 National High School Mathematics League, 1

Tags: function , inequalities

Let $-1<a<0$, $\theta=\arcsin a$. Then the solution set to the inequality $\sin x<a$ is $\text{(A)}\{x|2n\pi+\theta<x<(2n+1)\pi-\theta,n\in\mathbb{Z}\}$ $\text{(B)}\{x|2n\pi-\theta<x<(2n+1)\pi+\theta,n\in\mathbb{Z}\}$ $\text{(C)}\{x|(2n-1)\pi+\theta<x<2n\pi-\theta,n\in\mathbb{Z}\}$ $\text{(D)}\{x|(2n-1)\pi-\theta<x<2n\pi+\theta,n\in\mathbb{Z}\}$

2003 Tuymaada Olympiad, 4

Tags: function , limit , real analysis , algebra , functional equation

Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$ \[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \] [i]Proposed by F. Petrov[/i]

2014 France Team Selection Test, 6

Tags: inequalities proposed , inequalities

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

1982 Poland - Second Round, 6

Tags: geometry , combinatorics , combinatorial geometry , 3D geometry

Given a finite set $B$ of points in space, any two distances between the points of this set are different. Each point of the set $B$ is connected by a line segment to the closest point of the set $B$. This way we will get a set of sections, one of which (any chosen one) we paint red, all the remaining sections we paint green. Prove that there are two points of the set $B$ that cannot be connected by a line composed of green segments.

2011 AMC 10, 20

Tags: geometry , rhombus , trigonometry , perpendicular bisector , AMC

Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? $ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2 $

1994 Taiwan National Olympiad, 1

Tags: geometry , cyclic quadrilateral , geometry unsolved

Let $ABCD$ be a quadrilateral with $AD=BC$ and $\widehat{A}+\widehat{B}=120^{0}$. Let us draw equilateral $ACP,DCQ,DBR$ away from $AB$ . Prove that the points $P,Q,R$ are collinear.

2016 Irish Math Olympiad, 7

Tags: combinatorics , math olympiad , counting , Array , Arrangements

A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$. Within each row, all entries are distinct. What is the maximum possible number of columns?

2006 Pre-Preparation Course Examination, 5

Tags: logarithms , floor function , algebra proposed , algebra

Powers of $2$ in base $10$ start with $3$ or $4$ more frequently? What is their state in base $3$? First write down an exact form of the question.

2008 ISI B.Stat Entrance Exam, 9

Tags: number theory , least common multiple , greatest common divisor

Suppose $S$ is the set of all positive integers. For $a,b \in S$, define \[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\] For example $8*12=6$. Show that [b]exactly two[/b] of the following three properties are satisfied: (i) If $a,b \in S$, then $a*b \in S$. (ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$. (iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.

2017 India IMO Training Camp, 3

Tags: number theory , combinatorics

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$

Ukrainian TYM Qualifying - geometry, 2019.11

Tags: geometry , incenter , excircles , excircle , Ukrainian TYM

Let $\omega_a, \omega_b, \omega_c$ be the exscribed circles tangent to the sides $a, b, c$ of a triangle $ABC$, respectively, $ I_a, I_b, I_c$ be the centers of these circles, respectively, $T_a, T_b, T_c$ be the points of contact of these circles to the line $BC$, respectively. The lines $T_bI_c$ and $T_cI_b$ intersect at the point $Q$. Prove that the center of the circle inscribed in triangle $ABC$ lies on the line $T_aQ$.

2012 National Olympiad First Round, 21

Tags: geometry , angle bisector

The angle bisector of vertex $A$ of $\triangle ABC$ cuts $[BC]$ at $D$. The circle passing through $A$ and touching to $BC$ at $D$ meets $[AB]$ and $[AC]$ at $P$ and $Q$, respectively. $AD$ and $PQ$ meet at $T$. If $|AB|=5, |BC|=6, |CA|=7$, then $\frac{|AT|}{|TD|}=?$ $ \textbf{(A)}\ \frac75 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac72 \qquad \textbf{(E)}\ 4$

2006 Peru MO (ONEM), 1

Tags: trigonometry , algebra

Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$.

1964 AMC 12/AHSME, 37

Tags: algebra , difference of squares , special factorizations , AMC

Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than: $\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$ $\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$

2003 India Regional Mathematical Olympiad, 2

Tags: induction

If $n$ is an integer greater than $7$, prove that ${n \choose 7} - \left[ \frac{n}{7} \right]$ is divisible by $7$.

2005 Miklós Schweitzer, 7

Tags: algebra , Symmetric , algebraic number

Let $t\in R$. Prove that $\exists A:R \times R \to R$ such that A is a symmetric, biadditive, nonzero function and $A(tx,x)=0 \,\forall x\in R$ iff t is transcendental or (t is algebraic and t,-t are conjugates over $\mathbb{Q}$).

2009 CentroAmerican, 5

Tags: geometry , circumcircle , projective geometry , geometric transformation , homothety , geometry proposed

Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.

2018 Stars of Mathematics, 2

Tags: number theory , prime numbers

Find the smallest natural $ k $ such that among any $ k $ distinct and pairwise coprime naturals smaller than $ 2018, $ a prime can be found. [i]Vlad Robu[/i]

2010 HMNT, 1

Tags: combinatorics

$16$ progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has $2$ players remaining?

2006 Turkey Team Selection Test, 3

Tags: inequalities , inequalities proposed , algebra

If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that \[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]

PEN H Problems, 19

Tags: Diophantine Equations

Find all $(x, y, z, n) \in {\mathbb{N}}^4$ such that $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$.

2024/2025 TOURNAMENT OF TOWNS, P4

Tags: combinatorics

Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

1966 Spain Mathematical Olympiad, 7

Tags: algebra , geometric sequence

Determine a geometric progression of seven terms, knowing the sum, $7$, of the first three, and the sum, $112$, of the last three.

2007 IMO Shortlist, 3

Tags: modular arithmetic , number theory , Divisibility , Extremal combinatorics , Additive combinatorics , IMO Shortlist

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]