Found problems: 594
Find all natural numbers that have exactly six divisors whose sum is $3500$.
For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$
An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum?
(V. Lebed)
[img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]
Let $a, b$ be positive real numbers satisfying $a + b = 1$. Show that if $x_1,x_2,...,x_5$ are positive real numbers such that $x_1x_2...x_5 = 1$, then $(ax_1+b)(ax_2+b)...(ax_5+b)>1$
Given a sequence of $19$ positive integers not exceeding $88$ and another sequence of $88$ positive integers not exceeding $19$. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.
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.
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.
A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers
$b_1=a_1$
$b_2= a_1 + a_2$
$b_3=a_1 + a_2 + a_3$
...
$b_{100}=a_1 + a_2 + ...+a_{100}$
Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different .
( L . D . Kurlyandchik , Leningrad)
We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?
Is it true that every integer is a sum of finite number of cubes of distinct integers?
Evaluate the sum: $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$.
Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$
For which positive integers $m$ is $f(m)$ rational?
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.
Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.
a) $100$ numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive, 3) every number whose sum with the two numbers following it is positive.
Can the sum of all underlined numbers be
(i) negative?
(ii) equal to zero?
b) $n$ numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.
For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the di erence between consecutive terms is constant).
For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.
Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.
Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$.
Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$, denote $r_k$ as the remainder of the division $k^p$ by $p^2$. Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$
Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.
For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows:
If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$.
(a) Is it possible to partition $S$ into two sets having the same alternating sum?
(b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.
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 $8$.
For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the di erence between consecutive terms is constant).
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.