Problems

Prove that the tangent to the graph of the function $f(x)$, constructed at coordinates $(x_0,f(x_0))$ intersects the $Ox$ axis at the coordinate: $x_0-\frac{f(x_0)}{f^{\prime }(x_0)}$.

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: $f({\alpha }_1{x}_1+{\alpha }_2{x}_2)>{\alpha }_{1}f(x_1)+{\alpha }_{2}f(x_2)$.

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.

Prove that for $n>0$ the polynomial $$P(x)=n^2{x}^{n+2}-(2n^2+2n-1)x^{n+1}+(n+1{)}^2{x}^n-x-1$$ is divisible by $(x-1{)}^3$.

Prove that for $n>0$ the polynomial $x^{2n+1}-(2n+1)x^{n+1}+(2n+1)x^n-1$ is divisible by $(x-1{)}^3$.

A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.

The height of the room is 3 meters. When it was being renovated, it turned out that more paint was needed on each wall than on the floor. Can the area of the floor of this room be more than 10 square meters?

Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?

We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?

The number $x$ is such a number that exactly one of the four numbers $a=x-\sqrt{2}$, $b=x-1/x$, $c=x+1/x$, $d=x^2+2\sqrt{2}$ is not an integer. Find all such $x$.