Found problems: 594
Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with
$$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$
Let $S$ be a positive integer divisible by all the integers $1, 2,...,2015$ and $a_1, a_2,..., a_k$ numbers in $\{1, 2,...,2015\}$ such that $2S \le a_1 + a_2 + ... + a_k$. Prove that we can select from $a_1, a_2,..., a_k$ some numbers so that the sum of these selected numbers is equal to $S$.
Lê Anh Vinh
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 $r (A) \ge n s (A)$.
What is the sum (in base $10$) of all the natural numbers less than $64$ which have exactly three ones in their base $2$ representation?
Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$. Let $s = a_1 + a_3 +...+ a_{2n-1}$, $t = a_2 + a_4 + ... + a_{2n}$, and
$x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$). Prove that
$$\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}$$
The diameter of a circle of radius $R$ is divided into $4$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.
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.)
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
A positive integer $n$ is called [i]good [/i] if they sum up in one and only one way at least of two positive integers whose product also has the value $n$. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.
There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers.
For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?
In a set of $21$ real numbers, the sum of any $10$ numbers is less than the sum of the remaining $11$ numbers.
Prove that all the numbers are positive.
A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?
Any natural number $n, n\ge 3$ can be presented in different ways as a sum several summands (not necessarily different). Find the greatest possible value of these summands.
Folklore
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
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 $a_n$.
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 $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$
a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present.
b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.
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 $p$.
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?
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)
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 $m + n$.
Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.
Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .
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 $k > \ell\ge 0$.