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.

Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.

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 $8$.

Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.

Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$

Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that (a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$. (b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$. ( V . V . Proizvolov , Moscow)

Prove that for all natural numbers $n$ greater than $1$ : $$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$ (VV Kisil)

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.

There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$, but the sum of any $n$ ticket numbers is at most $1165$. What is the maximum value of $n$?

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?

Let $a_1,...,a_n$ be positive real numbers such that $\sum_{i=1}^n a_i =\prod_{i=1}^n a_i $ , and let $b_1,...,b_n$ be positive real numbers such that $a_i \le b_i$ for all $i$. Prove that $\sum_{i=1}^n b_i \le\prod_{i=1}^n b_i $

Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$? Recall that Pascal’s triangle is the pattern of numbers that begins as follows [img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img] where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$. Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.

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 $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.

Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.

Calculate $$\sum_{k=5}^{k=49}\frac{11_(k}{2\sqrt[3]{1331_(k}}$$ knowing that the numbers $11$ and $1331$ are written in base $k \ge 4$.

Let $n$ be a nonnegative integer and $M =\{n^3, n^3+1, n^3+2, ..., n^3+n\}$. Consider $A$ and $B$ two nonempty, disjoint subsets of $M$ such that the sum of elements of the set $A$ divides the sum of elements of the set $B$. Prove that the number of elements of the set $A$ divides the number of elements of the set $B$.