Problems

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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 xy is placed in correspondence. Find 19931935 if it is known that for any three numbers x,y,z, the following identities hold: xx=0 and x(yz)=(xy)+z.

For each pair of real numbers a and b, consider the sequence of numbers pn=2{an+b}. 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: c is the integer part, {c} is the fractional part of the number c.

Prove that for any natural number a1>1 there exists an increasing sequence of natural numbers a1,a2,a3,, for which a12+a22++ak2 is divisible by a1+a2++ak for all k1.

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.