Found problems: 594
1983 Swedish Mathematical Competition, 1
The positive integers are grouped as follows: $1, 2+3, 4+5+6, 7+8+9+10,\dots$. Find the value of the $n$-th sum.
1998 Tournament Of Towns, 4
For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result?
(G Galperin)
2016 Argentina National Olympiad, 2
For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$
.
2014 Korea Junior Math Olympiad, 2
Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$. Let $s = a_1 + a_3 +...+ a_{2n-1}$, $t = a_2 + a_4 + ... + a_{2n}$, and
$x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$). Prove that
$$\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}$$
2017 Hanoi Open Mathematics Competitions, 1
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$
The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above.
1998 Czech And Slovak Olympiad IIIA, 2
Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$
1990 Chile National Olympiad, 5
Determine a natural $n$ such that $$996 \le \sum_{k = 1}^{n}\frac{1}{k}$$
1997 Abels Math Contest (Norwegian MO), 2a
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.
2013 IFYM, Sozopol, 4
Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find
$\max S=\sum_{i\mid j}a_i a_j $.
1996 Estonia National Olympiad, 1
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.
2016 Dutch IMO TST, 2
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 di erence between consecutive terms is constant).
2010 Korea Junior Math Olympiad, 2
Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?
2019 Saudi Arabia JBMO TST, 1
Real nonzero numbers $x, y, z$ are such that $x+y +z = 0$. Moreover, it is known that $$A =\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+ 1$$Determine $A$.
1999 Poland - Second Round, 6
Suppose that $a_1,a_2,...,a_n$ are integers such that $a_1 +2^ia_2 +3^ia_3 +...+n^ia_n = 0$ for $i = 1,2,...,k -1$, where $k \ge 2$ is a given integer. Prove that $a_1+2^ka_2+3^ka_3+...+n^ka_n$ is divisible by $k!$.
1974 Poland - Second Round, 2
Prove that for every $ n = 2, 3, \ldots $ and any real numbers $ t_1, t_2, \ldots, t_n $, $ s_1, s_2, \ldots, s_n $, if
$$
\sum_{i=1}^n t_i = 0, \text{ to } \sum_{i=1}^n\sum_{j=1}^n t_it_j |s_i-s_j| \leq 0.$$
2015 Switzerland - Final Round, 7
Let $a, b, c$ be real numbers such that:
$$\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$$
Determine all values which the following expression can take :
$$\frac{a^2}{b + c}
+
\frac{b^2}{c + a}
+
\frac{c^2}{a + b}.$$
2017 India PRMO, 6
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$.
2008 Greece JBMO TST, 4
Product of two integers is $1$ less than three times of their sum. Find those integers.
1999 Tournament Of Towns, 1
In a row are written $1999$ numbers such that except the first and the last , each is equal to the sum of its neighbours. If the first number is $1$, find the last number.
(V Senderov)
1973 Chisinau City MO, 67
The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?
2015 Dutch IMO TST, 3
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
1992 Chile National Olympiad, 7
$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$.
$\bullet$ Construct or describe the construction of this magic square.
1999 Tournament Of Towns, 3
Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards.
(H. Lebesgue - A Kanel)
2007 Estonia Math Open Junior Contests, 10
Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.
1945 Moscow Mathematical Olympiad, 099
Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).