Found problems: 580
1999 Abels Math Contest (Norwegian MO), 4
For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows:
If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$.
(a) Is it possible to partition $S$ into two sets having the same alternating sum?
(b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.
2014 Greece JBMO TST, 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 $i= 1,2,3, ... , (n-8)$
1989 Romania Team Selection Test, 1
Let $M$ denote the set of $m\times n$ matrices with entries in the set $\{0,1,2,3,4\}$ such that in each row and each column the sum of elements is divisible by $5$. Find the cardinality of set $M$.
2018 Austria Beginners' Competition, 3
For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that
$\bullet$ each row totals to the same sum $z$ and
$\bullet$ each column totals to the same sum $s$.
Prove:
(a) If $n$ is even, such a table does not exist.
(b) If $n = 5$, such a table does exist.
(Gerhard J. Woeginger)
1949-56 Chisinau City MO, 14
Prove that if the numbers $a, b, c$ are related by the relation $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}= \frac{1}{a+b+c}$ then the sum of some two of them is equal to zero.
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.
1963 Dutch Mathematical Olympiad, 3
Twenty numbers $a_1,a_2,..,a_{20}$ satisfy:
$$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$
$$a_1+a_2+...+a_{20}=1518$$
Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.
1984 Polish MO Finals, 4
A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.
1970 All Soviet Union Mathematical Olympiad, 137
Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.
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 $A$.
2002 Silk Road, 3
In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$ so that $|\alpha-\alpha'| <2002$ and the sum of the numbers in all rows, and in all columns will not change.
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 $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$
2021-IMOC, A9
For a given positive integer $n,$ find
$$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$
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 :
$$\frac{a^2}{b + c}
+
\frac{b^2}{c + a}
+
\frac{c^2}{a + b}.$$
2009 Postal Coaching, 3
Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions
(i) $f(0, 0) = 1$, $f(0, 1) = 1$
(ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and
(iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of
$$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$
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 :
$$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$
2013 Saudi Arabia BMO TST, 8
Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.
2011 German National Olympiad, 4
There are two points $A$ and $B$ in the plane.
a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$
b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.
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 $c_1 + c_2 + ... + c_k = n^2 - 1$.
1993 Spain Mathematical Olympiad, 2
In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above.
$0 \, 1\, 2\, 3 \,4\, ... \,1991 \,1992\, 1993$
$\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985$
$\,\,\,4 \,8 \,12\, .......... \,\,\,7968$
·······································
Prove that the bottom number is a multiple of $1993$.
2018 Dutch IMO TST, 3
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
2009 Singapore Junior Math Olympiad, 4
Let $S$ be the set of integers that can be written in the form $50m + 3n$ where $m$ and $n$ are non-negative integers. For example $3, 50, 53$ are all in $S$. Find the sum of all positive integers not in $S$.
1995 Spain Mathematical Olympiad, 1
Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such triangles. Compute the smallest possible value of $S(A)$.
1986 All Soviet Union Mathematical Olympiad, 429
A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes. Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.
2001 Switzerland Team Selection Test, 4
For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.