Olympiad problems

1977 IMO Shortlist, 15

Tags: linear algebra , algebra , sequence , maximization , extremal combinatorics

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

2002 Kazakhstan National Olympiad, 4

Tags: number theory , remainder

Prove that there is a set $ A $ consisting of $2002$ different natural numbers satisfying the condition: for each $ a \in A $, the product of all numbers from $ A $, except $ a $, when divided by $ a $ gives the remainder $1$.

PEN O Problems, 44

Tags: modular arithmetic

A set $C$ of positive integers is called good if for every integer $k$ there exist distinct $a, b \in C$ such that the numbers $a+k$ and $b+k$ are not relatively prime. Prove that if the sum of the elements of a good set $C$ equals $2003$, then there exists $c \in C$ such that the set $C-\{c\}$ is good.

1957 Miklós Schweitzer, 8

Tags: number theory

[b]8.[/b] Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). [b](N. 9)[/b]

2012 Romania National Olympiad, 3

Tags: geometry , solid geometry , perpendicular

Let $ACD$ and $BCD$ be acute-angled triangles located in different planes. Let $G$ and $H$ be the centroid and the orthocenter respectively of the $BCD$ triangle; Similarly let $G'$ and $H'$ be the centroid and the orthocenter of the $ACD$ triangle. Knowing that $HH'$ is perpendicular to the plane $(ACD)$, show that $GG' $ is perpendicular to the plane $(BCD)$.

2018 Online Math Open Problems, 13

Tags:

Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers. [i]Proposed by Tristan Shin

2020 CCA Math Bonanza, L1.3

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If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $CDX$ is equilateral, compute $\angle{AXE}$ in degrees. [i]2020 CCA Math Bonanza Lightning Round #1.3[/i]

2011 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , greatest common divisor , arithmetic sequence , relatively prime , algebra , system of equations , number theory

We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality \[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\] holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i]. [b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

1974 Bulgaria National Olympiad, Problem 2

Tags: polynomial , number theory

Let $f(x)$ and $g(x)$ be non-constant polynomials with integer positive coefficients, $m$ and $n$ are given natural numbers. Prove that there exists infinitely many natural numbers $k$ for which the numbers $$f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)$$ are composite. [i]I. Tonov[/i]

1995 All-Russian Olympiad Regional Round, 11.7

Tags: geometry , equal segments

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. Ray $O_1B$ meets $S_2$ again at $F$, and ray $O_2B$ meets $ S_1$ again at $E$. The line through $B$ parallel to $ EF$ intersects $S_1$ and $S_2$ again at $M$ and $N$, respectively. Prove that $MN = AE +AF$.

2015 239 Open Mathematical Olympiad, 3

Tags: graph , coloring , combinatorics

The edges of a graph $G$ are coloured in two colours. Such that for each colour all the connected components of this graph formed by edges of this colour contains at most $n>1$ vertices. Prove there exists a proper colouring for the vertices of this graph with $n$ colours.

2002 Moldova Team Selection Test, 2

Tags: combinatorics

Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an integer. Find the smallest $k$ for which the set A can be partitioned into two subsets having the same number of elements, the same sum of elements, the same sum of the squares of elements, and the same sum of the cubes of elements.

2024 District Olympiad, P4

Tags: integral , inequalities , real analysis

Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function, with a continous derivative. Given that $f(0)=0$ and $0\leqslant f'(x)\leqslant 1$ for every $x>0$ prove that\[\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,\]for any positive integer $n{}$ and real number $a>0.$

2020 BMT Fall, 8

Tags: geometry , square , area

Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEF B] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$ , where m and $n$ are relatively prime positive integers. Compute $m + n$.

2024 USAMTS Problems, 3

Tags: geometry

$\triangle ABC$ is an equilateral triangle. $D$ is a point on $\overline{AC}$, and $E$ is a point on $\overline{BD}$. Let $P$ and $Q$ be the circumcenters of $\triangle ABD$ and $\triangle AED$, respectively. Prove that $ \triangle EPQ$ is an equilateral triangle if and only if $ \overline{AB} \perp \overline{CE}$.

2013 JBMO TST - Turkey, 1

Tags: incenter , circumcircle , geometry

Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

2010 Contests, 3

Tags: number theory

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

Kvant 2021, M2654

Tags: geometry , angle chasing

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

2004 Postal Coaching, 2

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

(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$ (b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.

2020 Final Mathematical Cup, 4

Tags: perpendicular , tangent circle , geometry

Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.

2021 Math Prize for Girls Problems, 13

Tags:

There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$. After all 1011 people have gone by, how many light bulbs are on?

2020-IMOC, N2

Tags: diophantine equation , number theory

Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with $$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$

2022 Baltic Way, 5

Tags: algebra , functional equation

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$, $$ f(xy-x)+f(x+f(y))=yf(x)+3 $$

2016 AIME Problems, 5

Tags: geometry , right triangle

Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n\geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum\limits_{n=1}^{\infty}C_{n-1}C_n = 6p$. Find $p$.

2024 HMNT, 4

Tags: team

Albert writes down all of the multiples of $9$ between $9$ and $999,$ inclusive. Compute the sum of the digits he wrote.