Found problems: 580
2006 Tournament of Towns, 7
Positive numbers $x_1,..., x_k$ satisfy the following inequalities:
$$x_1^2+...+ x_k^2 <\frac{x_1+...+x_k}{2} \ \ and \ \ x_1+...+x_k < \frac{x_1^3+...+ x_k^3}{2}$$
a) Show that $k > 50$, (3)
b) Give an example of such numbers for some value of $k$ (3)
c) Find minimum $k$, for which such an example exists. (3)
1964 Vietnam National Olympiad, 1
Given an arbitrary angle $\alpha$, compute
$cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ .
Generalize this result and justify your answer.
2015 Kyiv Math Festival, P3
Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?
1986 Swedish Mathematical Competition, 5
In the arrangement of $pn$ real numbers below, the difference between the greatest and smallest numbers in each row is at most $d$, $d > 0$.
\[ \begin{array}{l} a_{11} \,\, a_{12} \,\, ... \,\, a_{1n}\\
a_{21} \,\, a_{22} \,\, ... \,\, a_{2n}\\
\,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\
\,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\
\,\, . \,\, \,\, \,\, \,\, . \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, .\\
a_{n1} \,\, a_{n2} \,\, ... \,\, a_{nn}\\
\end{array}
\]
Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.
2008 Korea Junior Math Olympiad, 6
If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we defi ne $f_s(n) = d_1^s+d_2^s+..+d_k^s$.
For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$.
Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.
2003 Junior Balkan Team Selection Tests - Moldova, 1
Let $n \ge 2003$ be a positive integer such that the number $1 + 2003n$ is a perfect square.
Prove that the number $n + 1$ is equal to the sum of $2003$ positive perfect squares.
2013 Czech-Polish-Slovak Junior Match, 2
Each positive integer should be colored red or green in such a way that the following two conditions are met:
- Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red.
- Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green.
Determine all such colorings.
2013 Korea Junior Math Olympiad, 3
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$).
For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$
Prove that $b_n$ is positive integer.
2015 Romania Team Selection Tests, 5
Given an integer $N \geq 4$, determine the largest value the sum
$$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$
may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.
2015 Czech-Polish-Slovak Junior Match, 6
The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.
2009 Chile National Olympiad, 3
Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?
2011 Indonesia TST, 1
For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$.
Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).
2016 Saudi Arabia IMO TST, 1
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
2014 Czech-Polish-Slovak Junior Match, 5
A square is given. Lines divide it into $n$ polygons.
What is he the largest possible sum of the internal angles of all polygons?
2014 Junior Balkan Team Selection Tests - Romania, 1
Let n be a positive integer and $x_1, x_2, ..., x_n > 0$ be real numbers so that $x_1 + x_2 +... + x_n =\frac{1}{x_1^2}+\frac{1}{x_2^2}+...+\frac{1}{x_n^2}$
Show that for each positive integer $k \le n$, there are $k$ numbers among $x_1, x_2, ..., x_n $ whose sum is at least $k$.
1983 Tournament Of Towns, (050) 2
Consider all nine-digit numbers, consisting of non-repeating digits from $1$ to $9$ in an arbitrary order. A pair of such numbers is called “conditional” if their sum is equal to $987654321$.
(a) Prove that there exist at least two conditional pairs (noting that ($a,b$) and ($b,a$) is considered to be one pair).
(b) Prove that the number of conditional pairs is odd.
(G Galperin, Moscow)
1937 Moscow Mathematical Olympiad, 033
* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.
2019 Gulf Math Olympiad, 3
Consider the set $S = \{1,2,3, ...,1441\}$.
1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania.
2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$.
3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.
Oliforum Contest V 2017, 8
Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$
for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$
show that at least one $a_i$ has to be equal to $\frac12$.
(Paolo Leonetti)
Estonia Open Senior - geometry, 1995.2.4
Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.
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)
2008 Indonesia TST, 2
Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$.
Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$
for all positive integers $n$.
1988 Tournament Of Towns, (167) 4
The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.
2014 Finnish National High School Mathematics, 5
Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$.
In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?
2002 Moldova Team Selection Test, 4
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.