Problems

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f(f(x))=x$, $g(g(x))=x$, $f(g(x))>x$, $g(f(x))>x$?

b) $f(f(x))<x$, $g(g(x))<x$, $f(g(x))>x$, $g(f(x))>x$?

Find all functions $f(x)$ such that $f(2x+1)=4x^2+14x+7$.

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 $c$.

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 $k\ge 1$.

The quadratic trinomials $f(x)$ and $g(x)$ are such that $f^{\prime }(x)g^{\prime }(x)\ge |f(x)|+|g(x)|$ for all real $x$. Prove that the product $f(x)g(x)$ is equal to the square of some trinomial.

How many rational terms are contained in the expansion of

a) $(\sqrt{2}+\sqrt[4]{3}{)}^{100}$;

b) $(\sqrt{2}+\sqrt[3]{3}{)}^{300}$?

Suppose that in each issue of our journal in the “Quantum” problem book there are five mathematics problems. We denote by $f(x,y)$ the number of the first of the problems of the $x$-th issue for the $y$-th year. Write a general formula for $f(x,y)$, where $1\ge x\ge 12$ and $1970\ge y\ge 1989$. Solve the equation $f(x,y)=y$. For example, $f(6,1970)=26$. Since $1989$, the number of tasks has become less predictable. For example, in recent years, half the issues have 5 tasks, and in other issues there are 10. Even the number of magazine issues has changed, no longer being 12 but now 6.

Author: V.A. Popov

On the interval $[0;1]$ a function $f$ is given. This function is non-negative at all points, $f(1)=1$ and, finally, for any two non-negative numbers $x_1$ and $x_2$ whose sum does not exceed 1, the quantity $f(x_1+x_2)$ does not exceed the sum of $f(x_1)$ and $f(x_2)$.

a) Prove that for any number $x$ on the interval $[0;1]$, the inequality $f(x_2)\le 2x$ holds.

b) Prove that for any number $x$ on the interval $[0;1]$, the $f(x_2)\le 1.9x$ must be true?

The function $f(x)$ for each real value of $x\in (-\mathrm{\infty },+\mathrm{\infty })$ satisfies the equality $f(x)+(x+1/2)\times f(1-x)=1$.

a) Find $f(0)$ and $f(1)$. b) Find all such functions $f(x)$.