Olympiad problems

2009 ISI B.Stat Entrance Exam, 4

Tags: algebra , polynomial , calculus , derivative , arithmetic sequence

A sequence is called an [i]arithmetic progression of the first order[/i] if the differences of the successive terms are constant. It is called an [i]arithmetic progression of the second order[/i] if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k\geq 2$, a sequence is called an [i]arithmetic progression of the $k$-th order[/i] if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order. The numbers \[4,6,13,27,50,84\] are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.

2016 Dutch IMO TST, 2

Tags: arithmetic sequence , sum , algebra

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

1992 Baltic Way, 3

Tags: geometry , 3d geometry , arithmetic sequence , algebra , factorization , sum of cubes , number theory

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

2010 Romania Team Selection Test, 3

Tags: modular arithmetic , arithmetic sequence , number theory

Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.) [i]Vlad Matei[/i]

2012 USA Team Selection Test, 3

Tags: induction , arithmetic sequence , strong induction , combinatorics

Determine all positive integers $n$, $n\ge2$, such that the following statement is true: If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

1949-56 Chisinau City MO, 35

Tags: arithmetic sequence , algebra

The numbers $a^2, b^2, c^2$ form an arithmetic progression. Show that the numbers $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ also form arithmetic progression.

2020 Thailand TST, 3

Tags: algebra , functional equation , arithmetic sequence

Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

2006 Korea - Final Round, 2

Tags: search , arithmetic sequence , number theory

For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist inﬁnitely many primes that are not contained in $S_{a}$.

2014 Online Math Open Problems, 2

Tags: arithmetic sequence

Suppose $(a_n)$, $(b_n)$, $(c_n)$ are arithmetic progressions. Given that $a_1+b_1+c_1 = 0$ and $a_2+b_2+c_2 = 1$, compute $a_{2014}+b_{2014}+c_{2014}$. [i]Proposed by Evan Chen[/i]

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.

2010 Iran Team Selection Test, 1

Tags: function , prime number , arithmetic sequence , number theory , logarithm

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2015 Estonia Team Selection Test, 1

Tags: algebra , consecutive , arithmetic progression , arithmetic sequence

Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.

1973 Bulgaria National Olympiad, Problem 2

Tags: arithmetic sequence , algebra , sequence , geometric sequence

Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that: $$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$

1996 Iran MO (3rd Round), 1

Tags: arithmetic sequence , combinatorics

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

1980 Austrian-Polish Competition, 1

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.

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.

1987 IMO Longlists, 37

Tags: probability , arithmetic sequence , combinatorics

Five distinct numbers are drawn successively and at random from the set $\{1, \cdots , n\}$. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than $\frac{6}{(n-2)^3}$

2008 Tournament Of Towns, 4

Tags: arithmetic sequence , number theory

Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?

2008 National Olympiad First Round, 27

Tags: trigonometry , arithmetic sequence

The angles $\alpha, \beta, \gamma$ of a triangle are in arithmetic progression. If $\sin 20\alpha$, $\sin 20\beta$, and $\sin 20\gamma$ are in arithmetic progression, how many different values can $\alpha$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None of the above} $

2020 Spain Mathematical Olympiad, 1

Tags: algebra , polynomial , arithmetic sequence

A polynomial $p(x)$ with real coefficients is said to be [i]almeriense[/i] if it is of the form: $$ p(x) = x^3+ax^2+bx+a $$ And its three roots are positive real numbers in arithmetic progression. Find all [i]almeriense[/i] polynomials such that $p\left(\frac{7}{4}\right) = 0$

2008 Iran MO (3rd Round), 2

Tags: arithmetic sequence , number theory

Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: arithmetic progression , arithmetic sequence , ratio , combinatorics , partition

Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

2020 Estonia Team Selection Test, 1

Tags: arithmetic sequence , algebra , arithmetic progression

Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

2011 All-Russian Olympiad, 2

Tags: quadratic , invariant , vieta , induction , algebra , polynomial , arithmetic sequence

In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?

2012 Regional Competition For Advanced Students, 3

Tags: modular arithmetic , arithmetic sequence , algebra

In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?