Olympiad problems

2021 Saudi Arabia Training Tests, 26

Given an infinite sequence of numbers $a_1, a_2, a_3, ...$ such that for each positive integer $k$, there exists positive integer $t$ for which $a_k = a_{k+t} = a_{k+2t} = ....$ Does this sequences must be periodic?

1976 Chisinau City MO, 119

Tags: algebra , number theory

The Serpent Gorynych has $1976$ heads. The fabulous hero can cut down $33, 21, 17$ or $1$ head with one blow of the sword, but at the same time, the Serpent grows, respectively, $48, 0, 14$ or $349$ heads. If all the heads are cut off, then no new heads will grow. Will the hero be able to defeat the Serpent?

2019 AMC 10, 5

Tags:

What is the greatest number of consecutive integers whose sum is $45 ?$ $\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$

2005 Silk Road, 4

Tags: algebra , induction , inequalities

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: \[ a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3\] and $a(1)=a(2)=1$. Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.

2008 iTest Tournament of Champions, 3

Tags:

The $260$ volumes of the [i]Encyclopedia Galactica[/i] are out of order in the library. Fortunately for the librarian, the books are numbered. Due to his religion, which holds both encyclopedias and the concept of parity in high esteem, the librarian can only rearrange the books two at a time, and then only by switching the position of an even numbered volume with that of an odd numbered volume. Find the minimum number of such transpositions sufficient to get the books back into ordinary sequential order, regardless of the starting positions of the books. (Find the minimum number of transpositions in the worst-case scenario.)

2017 Hong Kong TST, 2

Tags: algebra

Suppose all of the 200 integers lying in between (and including) 1 and 200 are written on a blackboard. Suppose we choose exactly 100 of these numbers and circle each one of them. By the [i]score[/i] of such a choice, we mean the square of the difference between the sum of the circled numbers and the sum of the non-circled numbers. What is the average scores over all possible choices for 100 numbers?

2015 Singapore MO Open, 4

Tags: fibonacci sequence , number theory

Let $f_0, f_1,...$ be the Fibonacci sequence: $f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2}$ if $n \geq 2$. Determine all possible positive integers $n$ so that there is a positive integer $a$ such that $f_n \leq a \leq f_{n+1}$ and that $a( \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n} )$ is an integer.

2013 Harvard-MIT Mathematics Tournament, 3

Tags: circumcircle , rectangle , geometry

Let $ABC$ be a triangle with circumcenter $O$ such that $AC = 7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.

2014 Hanoi Open Mathematics Competitions, 13

Tags: algebra , inequalities

Let $a,b, c > 0$ and $abc = 1$. Prove that $\frac{a - 1}{c}+\frac{c - 1}{b}+\frac{b - 1}{a} \ge 0$

2021 Caucasus Mathematical Olympiad, 3

Tags: combinatorics , combinatorial geometry

Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)

MOAA Team Rounds, 2018.4

Tags: team , combinatorics

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

2016 AMC 12/AHSME, 24

Tags: algebra , polynomial

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$? $\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

2004 USAMTS Problems, 3

Tags:

Define the recursive sequence $1, 4, 13, \ldots$ by $s_1 = 1$ and $s_{n+1} = 3s_n + 1$ for all positive integers $n$. The element $s_{18} = 193710244$ ends in two identical digits. Prove that all the elements in the sequence that end in two or more identical digits come in groups of three consecutive elements that have the same number of identical digits at the end.

1987 IMO Shortlist, 18

Tags: partition , coloring , ramsey theory , set systems , combinatorics

For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class. [i]Proposed by Romania[/i]

JBMO Geometry Collection, 1999

Tags: geometry , perpendicular bisector , trigonometry , circumcircle

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

2024 India Regional Mathematical Olympiad, 4

Tags: number theory

Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ [i]nice[/i] if for every $k = 2 ,3, \cdots , n$, we have that $a_1^2 + a_2^2 + \cdots + a_k^2$ is [b]not[/b] divisible by $k$. Determine which positive integers $n>1$ have a [i]nice[/i] arrangement.

2013 IFYM, Sozopol, 6

Tags: number theory

Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

1997 Nordic, 3

Tags: ratio , distance , point set , geometry

Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$, and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$. .

2003 Peru Cono Sur TST, P1

Tags: algebra

Determine all real numbers $a$ such that the equation: $$x^8+ax^4+1=0$$ have four real roots that form an arithmetic progression.

2022 AMC 10, 8

Tags: arithmetic mean

A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$? $\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

2016 239 Open Mathematical Olympiad, 5

Tags: geometry

Through point $ P $ inside triangle $ ABC $, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point $ P $, are drawn. Points $ A_1 $, $ B_1 $ and $ C_1 $ are the intersection points of the lines containing these diagonals such that $A_1$ and $A$ are in different sides of line $BC$ and $B_1$ and $C_1$ are similar. Prove that if hexagon $ AC_1BA_1CB_1 $ is inscribed and convex, then point $ P $ is the orthocenter of triangle $ A_1B_1C_1 $.

1967 IMO Longlists, 5

Tags: linear algebra , polynomial , system of equations , matrix , algebra

Solve the system of equations: $ \begin{matrix} x^2 + x - 1 = y \\ y^2 + y - 1 = z \\ z^2 + z - 1 = x. \end{matrix} $

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: divisibility , fbh , perfect square , number theory

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

2012 AMC 12/AHSME, 5

Tags:

Two integers have a sum of $26$. When two more integers are added to the first two integers the sum is $41$. Finally when two more integers are added to the sum of the previous four integers the sum is $57$. What is the minimum number of even integers among the $6$ integers? ${{ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4}\qquad\textbf{(E)}\ 5} $

2020 MBMT, 21

Tags:

Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with \$256, find the probability that after 8 bets, he will have at least \$50. [i]Proposed by Jeffrey Tong[/i]