Found problems: 580
1996 All-Russian Olympiad Regional Round, 9.5
Find all natural numbers that have exactly six divisors whose sum is $3500$.
1992 Romania Team Selection Test, 4
Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$.
If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.
2022 New Zealand MO, 6
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.
2020 Durer Math Competition Finals, 4
Endre wrote $n$ (not necessarily distinct) integers on a paper. Then for each of the $2^n$ subsets, Kelemen wrote their sum on the blackboard.
a) For which values of $n$ is it possible that two different $n$-tuples give the same numbers on the blackboard?
b) Prove that if Endre only wrote positive integers on the paper and Ferenc only sees the numbers on the blackboard, then he can determine which integers are on the paper.
1981 Spain Mathematical Olympiad, 1
Calculate the sum of $n$ addends
$$7 + 77 + 777 +...+ 7... 7.$$
2015 Caucasus Mathematical Olympiad, 1
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?
2004 BAMO, 4
Suppose one is given $n$ real numbers, not all zero, but such that their sum is zero.
Prove that one can label these numbers $a_1, a_2, ..., a_n$ in such a manner that $a_1a_2 + a_2a_3 +...+a_{n-1}a_n + a_na_1 < 0$.
2006 Korea Junior Math Olympiad, 8
De ne the set $F$ as the following: $F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\}$
Prove that there exists a subset of $F$, called $S$ which satis es the following:
$|S| = 2006$
and for all $(a_1,a_2,... , a_{2006})\in F$ there exists $(b_1,b_2,... , b_{2006}) \in S$, such that $\Sigma_{i=1} ^{2006}a_ib_i = 0$.
1975 Dutch Mathematical Olympiad, 3
Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$.
Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$
2004 Abels Math Contest (Norwegian MO), 1a
If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.
2020-21 IOQM India, 3
If $\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999$ then determine the value of $N$.
1988 All Soviet Union Mathematical Olympiad, 482
Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.
2020 Paraguay Mathematical Olympiad, 5
The general term of a sequence of numbers is defined as $a_n =\frac{1}{n^2 - n}$, for every integer $n \ge 3$.
That is, $a_3 =\frac16$, $a_4 =\frac{1}{12}$, $a_5 =\frac{1}{20}$, and so on.
Find a general expression for the sum $S_n$, which is the sum of all terms from $a_3$ until $a_n$.
2002 Mexico National Olympiad, 4
A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?
1998 ITAMO, 1
Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .
You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.
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.)
1977 Bundeswettbewerb Mathematik, 2
A beetle crawls along the edges of an $n$-lateral pyramid, starting and ending at the midpoint $A$ of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)? Show that the sum of the numbers of passed vertices (over all these ways) equals $1^2 +2^2 +\ldots +n^2. $
2017 Singapore MO Open, 3
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
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]
1980 Bundeswettbewerb Mathematik, 4
Consider the sequence $a_1, a_2, a_3, \ldots$ with
$$ a_n = \frac{1}{n(n+1)}.$$
In how many ways can the number $\frac{1}{1980}$ be represented as the sum of finitely many consecutive terms of
this sequence?
1947 Putnam, A5
Let $a_1 , b_1 , c_1$ be positive real numbers whose sum is $1,$ and for $n=1, 2, \ldots$ we define
$$a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.$$
Show that $a_n , b_n ,c_n$ approach limits as $n\to \infty$ and find those limits.
1984 All Soviet Union Mathematical Olympiad, 380
$n$ real numbers are written in increasing order in a line. The same numbers are written in the second line below in unknown order. The third line contains the sums of the pairs of numbers above from two previous lines. It comes out, that the third line is arranged in increasing order. Prove that the second line coincides with the first one.
1988 Dutch Mathematical Olympiad, 3
For certain $a,b,c$ holds: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$
Prove that for all odd $n$ holds, $$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}.$$
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)
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.