Found problems: 594
2003 Belarusian National Olympiad, 4
Positive numbers $a_1,a_2,...,a_n, b_1, b_2,...,b_n$ satisfy the condition $a_1+a_2+...+a_n=b_1+ b_2+...+b_n=1$.
Find the smallest possible value of the sum $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+...+\frac{a_n^2}{a_n+b_n}$$
(V.Kolbun)
2015 Indonesia MO Shortlist, C7
Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that :
(i) $|A| = 12$
(ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.
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 $101$?
2013 Costa Rica - Final Round, 2
Determine all even positive integers that can be written as the sum of odd composite positive integers.
2023 Germany Team Selection Test, 3
Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$.
(a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element.
(b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
(c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
2009 QEDMO 6th, 8
Given $n$ integers $a_1, a_2, ..., a_n$, which $a_1 = 1$ and $a_i \le a_{i + 1} \le 2a_i$ for each $i \in \{1,2,...,n-1\}$ .
Prove that if $a_1 + a_2 +... + a_n$ is even, you do select some of the numbers so that their sum equals $\frac{a_1 + a_2 +... + a_n}{2}$ .
1992 Czech And Slovak Olympiad IIIA, 3
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$
2006 Junior Tuymaada Olympiad, 6
[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $.
For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$.
Find the number of all [i]palindromic partitions[/i] of the number $2006$.
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.
1993 Czech And Slovak Olympiad IIIA, 4
The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?
2017 Irish Math Olympiad, 3
A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that :
$$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$
2014 Junior Balkan Team Selection Tests - Moldova, 1
Prove that $$\frac{2 }{2013 +1} +\frac{2^{2}}{2013^{2^{1}}+1} +\frac{2^{3}}{2013^{2^{2}}+1} + ...+ \frac{2^{2014}}{2013^{2^{2013}}+1} < \frac{1}{1006}$$
2014 India PRMO, 12
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?
2008 Singapore Junior Math Olympiad, 4
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
1982 Austrian-Polish Competition, 9
Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$.
Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$ holds for all $n \ge 3$.
(Note. The smaller $C$, the better the solution.)
2015 Argentina National Olympiad, 1
Express the sum of $99$ terms$$\frac{1\cdot 4}{2\cdot 5}+\frac{2\cdot 7}{5\cdot 8}+\ldots +\frac{k(3k+1 )}{(3k-1)(3k+2)}+\ldots +\frac{99\cdot 298}{296\cdot 299}$$ as an irreducible fraction.
2005 Thailand Mathematical Olympiad, 16
Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.
2013 Estonia Team Selection Test, 3
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$
(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
2014 Contests, 3
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
2001 Singapore MO Open, 4
A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer.
(As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).
2020-21 IOQM India, 3
If $\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999$ then determine the value of $N$.
2009 BAMO, 2
The Fibonacci sequence is the list of numbers that begins $1, 2, 3, 5, 8, 13$ and continues with each subsequent number being the sum of the previous two.
Prove that for every positive integer $n$ when the first $n$ elements of the Fibonacci sequence are alternately added and subtracted, the result is an element of the sequence or the negative of an element of the sequence.
For example, when $n = 4$ we have $1-2+3-5 = -3$ and $3$ is an element of the Fibonacci sequence.
2005 Thailand Mathematical Olympiad, 7
How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?
1996 Mexico National Olympiad, 4
For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$?
2010 Junior Balkan Team Selection Tests - Romania, 2
Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$.
Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.