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.

We are given rational positive numbers $p,q$ where $1/p+1/q=1$. Prove that for positive $a$ and $b$, the following inequality holds: $ab\le \frac{a^p}{p}+\frac{b^q}{q}$.

Prove that the equation $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=1$ is unsolvable using positive integers.

Prove the inequality: $$\frac{(b_1+\dots b_n{)}^{b_1+\dots b_n}}{(a_1+\dots a_n{)}^{b_1+\cdots +b_n}}\le {\left(\frac{b_1}{a_1}\right)}^{b_1}\dots {\left(\frac{b_n}{a_n}\right)}^{b_n}$$ where all variables are considered positive.

Prove that if the function $f(x)$ is convex upwards on the line $[a,b]$, then for any distinct points $x_1,x_2$ in $[a;b]$ and for any positive ${\alpha }_1,{\alpha }_2$ such that ${\alpha }_1+{\alpha }_2=1$ the following inequality holds: $f({\alpha }_1{x}_1+{\alpha }_2{x}_2)>{\alpha }_{1}f(x_1)+{\alpha }_{2}f(x_2)$.

Inequality of Jensen. Prove that if the function $f(x)$ is convex upward on $[a,b]$, then for any distinct points $x_1,x_2,\dots ,x_n$ ($n\ge 2$) from $[a;b]$ and any positive ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ such that ${\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n=1$, the following inequality holds: $f({\alpha }_1{x}_1+\cdots +{\alpha }_n{x}_n)>{\alpha }_{1}f(x_1)+\cdots +{\alpha }_{n}f(x_n)$.

Let $p$ and $q$ be positive numbers where $1/p+1/q=1$. Prove that $$a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n\le (a_1^p+\dots a_n^p{)}^{1/p}(b_1^q+\cdots +b_n^q{)}^{1/q}$$ The values of the variables are considered positive.

Let the sequences of numbers $\{a_n\}$ and $\{b_n\}$, that are associated with the relation $\mathrm{\Delta }{b}_n=a_n$ ($n=1,2,\dots$), be given. How are the partial sums $S_n$ of the sequence $\{a_n\}$ $S_n=a_1+a_2+\cdots +a_n$ linked to the sequence $\{b_n\}$?

Definition. Let the function $f(x,y)$ be valid at all points of a plane with integer coordinates. We call a function $f(x,y)$ harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: $$f(x,y)=1/4(f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)).$$ Let $f(x,y)$ and $g(x,y)$ be harmonic functions. Prove that for any $a$ and $b$ the function $af(x,y)+bg(x,y)$ is also harmonic.

Let $f(x,y)$ be a harmonic function. Prove that the functions ${\mathrm{\Delta }}_{x}f(x,y)=f(x+1,y)-f(x,y)$ and ${\mathrm{\Delta }}_{y}f(x,y)=f(x,y+1)-f(x,y)$ will also be harmonic.