Found problems: 594
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)
Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$
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.
The sum $$\sum_{k=84}^{8000}{k \choose 84}{{8084 - k} \choose 84}$$
can be written as a binomial coefficient $a \choose b$ for integers $a, b$. Find a possible pair $(a, b)$
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?
We build a modified version of Pascal’s triangle as follows: in the first row we write a $2$ and a $3$, and in the further rows, every number is the sum of the two numbers directly above it (and rows always begin with a $2$ and end with a $3$). In the $13$th row, what is the $5$th number from the left?
[img]https://cdn.artofproblemsolving.com/attachments/7/2/58e1a9f43fa7c304bfd285fc1b73bed883e9a6.png[/img]
Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?
Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.
Calculate if $n\in N$ with $n>2$ the value of
$$B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}} $$
Let $n\geq1$ be an integer. We have two sequences, each of $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ and $b_1,b_2,\ldots ,b_n$ such that $a_1+a_2+\ldots +a_n=1$ and $ b_1+b_2+\ldots +b_n=1$. Find the smallest possible value that the sum can take $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.$$
$64$ non-negative numbers whose sum equals $1956$ are arranged in a square table, eight numbers in each row and each column. The sum of the numbers on the two longest diagonals is equal to $112$. The numbers situated symmetrically with respect to any of the longest diagonals are equal.
(a) Prove that the sum of numbers in any column is less than $1035$.
(b) Prove that the sum of numbers in any row is less than $518$.
Let $a_i$, $i=1,2,…2016$, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples $x_1,x_2…x_{2016}$ of natural numbers, for which the sum
$\sum_{i=1}^{2016}{a_i x_i^i}$
is a 2017-th power of a natural number.
Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$.
If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.
The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?
For which natural $ n \geq 3 $ numbers from 1 to $ n $ can be arranged by a circle so that each number does not exceed $60$ % of the sum of its two neighbors?
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$
A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.
For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?
Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$.
(a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element.
(b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
(c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $10$ (E): None of the above.
A $x>2$ real number is given. Bob has got $1997$ labels and writes one of the numbers $"x^0, x^1, x^2 ,\dotsm x^{1995}, x^{1996}"$ each labels such that all labels has distinct numbers. Bob puts some labels to right pocket, some labels to left pocket. Prove that sum of numbers of the right pocket never equal to sum of numbers of the left pocket.
Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.
The numbers $x_1,x_2,x_3, ..., x_n$ satisfy the two conditions
$$\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1$$
Prove that there are two numbers among them whose product is no greater than $- 1/n$.
(Stolov, Kharkov)
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 $n_1 + \ldots + n_k = N$.
Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.