Found problems: 594
2019 Silk Road, 4
The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $
Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers.
(Here, $ [x] $ is the largest integer not exceeding $ x $.)
2011 Saudi Arabia Pre-TST, 2.2
Consider the sequence $x_n = 2^n-n$, $n = 0,1 ,2 ,...$.
Find all integers $m \ge 0$ such that $s_m = x_0 + x_1 + x_2 + ... + x_m$ is a power of $2$.
2013 BAMO, 4
Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$.
Example:
[img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]
VMEO IV 2015, 12.4
We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$
Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.
1988 Swedish Mathematical Competition, 3
Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$.
Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$.
1977 All Soviet Union Mathematical Olympiad, 245
Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.
2011 Tournament of Towns, 4
The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?
1995 Swedish Mathematical Competition, 4
The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.
2013 India PRMO, 10
Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?
2000 Nordic, 1
In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.)
1991 All Soviet Union Mathematical Olympiad, 557
The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$
2002 Mexico National Olympiad, 5
A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.
2003 Junior Balkan Team Selection Tests - Moldova, 8
In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.
2000 Tournament Of Towns, 4
Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon.
(G Galperin)
1981 Tournament Of Towns, (013) 3
Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$.
(E Turkevich)
2020 Argentina National Olympiad, 5
Determine the highest possible value of:
$$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$
where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$.
Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.
2013 Danube Mathematical Competition, 1
Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$
2000 Tournament Of Towns, 1
Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table.
(R Zhenodarov)
2004 Thailand Mathematical Olympiad, 10
Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.
1986 Tournament Of Towns, (122) 4
Consider subsets of the set $1 , 2,..., N$.
For each such subset we can compute the product of the reciprocals of each member.
Find the sum of all such products.
2016 Hanoi Open Mathematics Competitions, 8
Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.
1947 Moscow Mathematical Olympiad, 139
In the numerical triangle
$................1..............$
$...........1 ...1 ...1.........$
$......1... 2... 3 ... 2 ... 1....$
$.1...3...6...7...6...3...1$
$...............................$
each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.
2008 Indonesia TST, 2
Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$.
Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$
for all positive integers $n$.
2003 Estonia National Olympiad, 4
Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that
a) all primes are lonely,
b) there exist infinitely many non-lonely positive integers.
2005 Thailand Mathematical Olympiad, 16
Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.