Found problems: 580
2013 Dutch Mathematical Olympiad, 5
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result?
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.
1980 All Soviet Union Mathematical Olympiad, 285
The vertical side of a square is divided onto $n$ segments. The sum of the segments with even numbers lengths equals to the sum of the segments with odd numbers lengths. $n-1$ lines parallel to the horizontal sides are drawn from the segments ends, and, thus, $n$ strips are obtained. The diagonal is drawn from the lower left corner to the upper right one. This diagonal divides every strip onto left and right parts. Prove that the sum of the left parts of odd strips areas equals to the sum of the right parts of even strips areas.
2011 QEDMO 10th, 2
Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$
2018 India PRMO, 25
Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?
2005 Thailand Mathematical Olympiad, 8
For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.
2004 Bosnia and Herzegovina Junior BMO TST, 3
Let $a, b, c, d$ be reals such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7$ and $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12$.
Find the value of $w =\frac{a}{b}+\frac{c}{d}$
.
1971 Spain Mathematical Olympiad, 1
Calculate $$\sum_{k=5}^{k=49}\frac{11_(k}{2\sqrt[3]{1331_(k}}$$ knowing that the numbers $11$ and $1331$ are written in base $k \ge 4$.
2015 Bundeswettbewerb Mathematik Germany, 2
A sum of $335$ pairwise distinct positive integers equals $100000$.
a) What is the least number of uneven integers in that sum?
b) What is the greatest number of uneven integers in that sum?
2018 India PRMO, 6
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?
2018 Hanoi Open Mathematics Competitions, 13
For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively.
1) Find all values of n such that $n = P(n)$:
2) Determine all values of n such that $n = S(n) + P(n)$.
2001 Estonia National Olympiad, 5
A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column.
(a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table.
(b) Show that there is such a table not all of whose entries are $0$.
1967 Swedish Mathematical Competition, 4
The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges.
Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.
1973 Czech and Slovak Olympiad III A, 4
For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]
2018 Saudi Arabia BMO TST, 2
Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.
2009 Postal Coaching, 5
Let $P$ be an interior point of a circle and $A_1,A_2...,A_{10}$ be points on the circle such that $\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o$. Prove that $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.
1999 Bosnia and Herzegovina Team Selection Test, 5
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that
$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$
$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$
where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$
2004 Junior Tuymaada Olympiad, 6
We call a positive integer [i] good[/i] if the sum of the reciprocals of all its natural divisors are integers. Prove that if
$ m $ is a [i]good [/i] number, and $ p> m $ is a prime number, then $ pm $ is not [i]good[/i].
2011 Singapore Junior Math Olympiad, 5
Initially, the number $10$ is written on the board. In each subsequent moves, you can either
(i) erase the number $1$ and replace it with a $10$, or
(ii) erase the number $10$ and replace it with a $1$ and a $25$ or
(iii) erase a $25$ and replace it with two $10$.
After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?
1990 Mexico National Olympiad, 4
Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$
2016 India PRMO, 12
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.
2011 Ukraine Team Selection Test, 4
Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount.
For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.
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 Czech And Slovak Olympiad III A, 1
Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.
1983 Tournament Of Towns, (045) 2
Find all natural numbers $k$ which can be represented as the sum of two relatively prime numbers not equal to $1$.