Olympiad problems

2021 Tuymaada Olympiad, 5

Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.

1980 IMO Shortlist, 13

Tags: algebra , arithmetic sequence , arithmetic progression , sequence

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

2017 District Olympiad, 3

Tags: arithmetic sequence , number theory

Denote $ S_n $ as being the sum of the squares of the first $ n\in\mathbb{N} $ terms of a given arithmetic sequence of natural numbers. [b]a)[/b] If $ p\ge 5 $ is a prime, then $ p\big| S_p. $ [b]b)[/b] $ S_5 $ is not a perfect square.

1966 AMC 12/AHSME, 19

Tags: 3d geometry , sphere , arithmetic sequence , algebra , linear equation , geometry

Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $8,12,\cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $17,19\cdots$. Assume $n\ne 0$. Then $s_1=s_2$ for: $\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}$ $\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}$

2013 China Team Selection Test, 2

Tags: arithmetic sequence , number theory , algebra

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

2019 Malaysia National Olympiad, B3

Tags: arithmetic sequence , arithmetic progression , algebra

An arithmetic sequence of five terms is considered $good$ if it contains 19 and 20. For example, $18.5,19.0,19.5,20.0,20.5$ is a $good$ sequence. For every $good$ sequence, the sum of its terms is totalled. What is the total sum of all $good$ sequences?

PEN O Problems, 38

Tags: induction , strong induction , arithmetic sequence

Prove that for every real number $M$ there exists an infinite arithmetical progression of positive integers such that [list] [*] the common difference is not divisible by $10$, [*] the sum of digits of each term exceeds $M$. [/list]

2004 Austrian-Polish Competition, 6

Tags: arithmetic sequence , permutations with restrictions , combinatorics

For $n=2^m$ (m is a positive integer) consider the set $M(n) = \{ 1,2,...,n\}$ of natural numbers. Prove that there exists an order $a_1, a_2, ..., a_n$ of the elements of M(n), so that for all $1\leq i < j < k \leq n$ holds: $a_j - a_i \neq a_k - a_j$.

2022-23 IOQM India, 13

Tags: geometry , trig bash , arithmetic sequence

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC$ such that $AD=BC$. \\ Suppose $\angle{CAD}=x^{\circ}, \angle{ABC}=y^{\circ}$ and $\angle{ACB}=z^{\circ}$ and $x,y,z$ are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of $\angle{ABC}$ in degrees.

1996 All-Russian Olympiad, 5

Tags: ratio , euler , arithmetic sequence , number theory

Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10. [i]L. Kuptsov[/i]

2020 CCA Math Bonanza, I13

Tags: number theory , arithmetic sequence

Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$. [i]2020 CCA Math Bonanza Individual Round #13[/i]

2001 India National Olympiad, 2

Tags: modular arithmetic , algebra , polynomial , arithmetic sequence , number theory

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

1982 Tournament Of Towns, (019) 5

Tags: arithmetic sequence

Consider the sequence $1, \frac12, \frac13, \frac14 ,...$ Does there exist an arithmetic progression composed of terms of this sequence (a) of length $5$, (b) of length greater than $5$ (if so, what possible length)? (G Galperin, Moscow)

2013 AMC 12/AHSME, 13

Tags: geometry , trapezoid , arithmetic sequence

The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA=\angle DCB$ and $\angle ADB=\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$? ${\textbf{(A)}\ 210\qquad\textbf{(B)}\ 220\qquad\textbf{(C)}\ 230\qquad\textbf{(D}}\ 240\qquad\textbf{(E)}\ 250$

1994 China Team Selection Test, 2

Tags: function , ratio , arithmetic sequence , algebra , system of equations , combinatorics

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

PEN J Problems, 4

Tags: algebra , polynomial , arithmetic sequence , divisor function

Let $m$, $n$ be positive integers. Prove that, for some positive integer $a$, each of $\phi(a)$, $\phi(a+1)$, $\cdots$, $\phi(a+n)$ is a multiple of $m$.

2012 Math Prize For Girls Problems, 11

Tags: geometric sequence , arithmetic sequence

Alison has an analog clock whose hands have the following lengths: $a$ inches (the hour hand), $b$ inches (the minute hand), and $c$ inches (the second hand), with $a < b < c$. The numbers $a$, $b$, and $c$ are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day: $A$ inches (the hour hand), $B$ inches (the minute hand), and $C$ inches (the second hand). The numbers $A$, $B$, and $C$ (in this order) are consecutive terms of a geometric sequence. What is the value of $\frac{B}{A}$?

2021 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , arithmetic sequence , induction

Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a [i]ghost[/i] if it doesn’t appear in $S$. What is the smallest ghost that is not a multiple of $p$? [i]Proposed by Guerrero[/i]

2004 Italy TST, 2

Tags: induction , least common multiple , arithmetic sequence , number theory

A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$. $(\text{a})$ Find $2004$ perfect powers in arithmetic progression. $(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.

2002 India National Olympiad, 5

Tags: arithmetic sequence , algebra

Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).

2014 India Regional Mathematical Olympiad, 2

Tags: arithmetic sequence

Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let $(i)$ $\sum_{i=1}^{n}a_{2i-1}^2 =x$ $(ii)$ $\sum _{i=1}^{n}a_{2i}^2=y$ $(iii)$ $a_n+a_{n+1}=z$ Express $d$ in terms of $x,y,z,n$

2014 Contests, 2

Tags: calculus , integration , arithmetic sequence , algebra

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2018 Moldova Team Selection Test, 4

Tags: algorithm , arithmetic sequence , combinatorics

A pupil is writing on a board positive integers $x_0,x_1,x_2,x_3...$ after the following algorithm which implies arithmetic progression $3,5,7,9...$.Each term of rank $k\ge2$ is a difference between the product of the last number on the board and the term of arithmetic progression of rank $k$ and the last but one term on the bord with the sum of the terms of the arithemtic progression with ranks less than $k$.If $x_0=0 $ and $x_1=1$ find $x_n$ according to n.

2009 Romania National Olympiad, 4

Tags: arithmetic sequence , ratio , number theory

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

2008 China Team Selection Test, 3

Tags: search , arithmetic sequence , combinatorics , ramsey theory

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.