Found problems: 580
1991 All Soviet Union Mathematical Olympiad, 554
Do there exist $4$ vectors in the plane so that none is a multiple of another, but the sum of each pair is perpendicular to the sum of the other two? Do there exist $91$ non-zero vectors in the plane such that the sum of any $19$ is perpendicular to the sum of the others?
1954 Moscow Mathematical Olympiad, 286
Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.
2012 India PRMO, 16
Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions:
(a) $f(mn) =f(m)f(n)$
(b) $f(m) < f(n)$ if $m < n$
(c) $f(2) = 2$
What is the sum of $\Sigma_{k=1}^{20}f(k)$?
2010 Junior Balkan Team Selection Tests - Romania, 1
We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.
1996 Romania National Olympiad, 1
For $n ,p \in N^*$ , $ 1 \le p \le n$, we define
$$ R_n^p = \sum_{k=0}^p (p-k)^n(-1)^k C_{n+1}^k $$
Show that: $R_n^{n-p+1} =R_n^p$ .
2021 Ukraine National Mathematical Olympiad, 5
Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$ where $a\in Z$, in which one of the numbers is equal to the sum of all the others.
(Bogdan Rublev)
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.
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?
2018 Flanders Math Olympiad, 3
Write down $f(n)$ for the greatest odd divisor of $n \in N_0$.
(a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$.
(b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.
2012 Belarus Team Selection Test, 1
A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$
Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible.
(E. Barabanov)
2009 Greece JBMO TST, 4
Find positive real numbers $x,y,z$ that are solutions of the system
$x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.
2019 Peru EGMO TST, 3
For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$, not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$, not necessarily different.
For example, if $A = \{3,1,-1\}$
$\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$.
$\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$.
Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$.
2015 Silk Road, 3
Let $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$
\varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).
$$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $.
.
1991 Romania Team Selection Test, 7
Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that
$$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$
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 $n$?
2017 Purple Comet Problems, 28
Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The in finite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1939 Moscow Mathematical Olympiad, 045
Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.
2008 Dutch IMO TST, 3
Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le i\le m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$,
so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j \ge i$.
Similarly, we define, for $1\le j \le n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$.
E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$.
(a) Prove that $a_j = c_j $ for $1 \le j \le n$.
(b) Prove that for $1\le k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.
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.
2018 Danube Mathematical Competition, 3
Find all the positive integers $n$ with the property:
there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$
such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.
1985 Brazil National Olympiad, 1
$a, b, c, d$ are integers with $ad \ne bc$. Show that $1/((ax+b)(cx+d))$ can be written in the form $ r/(ax+b) + s/(cx+d)$. Find the sum $1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$.
2012 BAMO, 2
Answer the following two questions and justify your answers:
(a) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+5^{2012}$?
(b) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+...+2011^{2012}+2012^{2012}$?
1949 Moscow Mathematical Olympiad, 168
Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.
2018 Grand Duchy of Lithuania, 2
Ten distinct numbers are chosen at random from the set $\{1, 2, 3, ... , 37\}$. Show that one can select four distinct numbers out of those ten so that the sum of two of them is equal to the sum of the other two.
2005 Thailand Mathematical Olympiad, 10
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?