Problems

A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be exactly seven times their product. Determine these numbers.

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.

To each pair of numbers $x$ and $y$ some number $x\ast y$ is placed in correspondence. Find $1993\ast 1935$ if it is known that for any three numbers $x,y,z$, the following identities hold: $x\ast x=0$ and $x\ast (y\ast z)=(x\ast y)+z$.

For each pair of real numbers $a$ and $b$, consider the sequence of numbers $p_n=\lfloor 2\{an+b\}\rfloor$. Any $k$ consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length $k$ is a word of the sequence given by some $a$ and $b$ for $k=4$; when $k=5$?

Note: $\lfloor c\rfloor$ is the integer part, $\{c\}$ is the fractional part of the number $c$.

The function $f$ is such that for any positive $x$ and $y$ the equality $f(xy)=f(x)+f(y)$ holds. Find $f(2007)$ if $f(1/2007)=1$.

Replace $a,b$ and $c$ with integers not equal to $1$ in the equality $(a{y}^b{)}^c=-64y^6$, so it would become an identity.

Prove that for any natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_1,a_2,a_3,\dots$, for which $a_1^2+a_2^2+\cdots +a_k^2$ is divisible by $a_1+a_2+\cdots +a_k$ for all $k\ge 1$.

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Prove that for any positive integer $n$ the inequality

is true.

Find the sum $1/3+2/3+2^2/3+2^3/3+\cdots +2^{1000}/3$.