Problems

We took several positive numbers and constructed the following sequence: $a_1$ is the sum of the initial numbers, $a_2$ is the sum of the squares of the original numbers, $a_3$ is the sum of the cubes of the original numbers, and so on.

a) Could it happen that up to $a_5$ the sequence decreases ($a_1>a_2>a_3>a_4>a_5$), and starting with $a_5$ – it increases ($a_5<a_6<a_7<\dots$)?

b) Could it be the other way around: before $a_5$ the sequence increases, and starting with $a_5$ – decreases?

A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of $n$ cells. We will call the natural number $n$ jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one $n>50$ that is not jumpable.

One hundred gnomes weighing each 1, 2, 3, ..., 100 pounds, gathered on the left bank of a river. They cannot swim, but on the same shore is a rowing boat with a carrying capacity of 100 pounds. Because of the current, it’s hard to swim back, so each gnome has enough power to row from the right bank to the left one no more than once (it’s enough for any one of the gnomes to row in the boat, the rower does not change during one voyage). Will all gnomes cross to the right bank?

The number $x$ is such that both the sums $S=\sin 64x+\sin 65x$ and $C=\cos 64x+\cos 65x$ are rational numbers.

Prove that in both of these sums, both terms are rational.

Author: A.K. Tolpygo

An irrational number $\alpha$, where $0<\alpha <\frac{1}{2}$, is given. It defines a new number ${\alpha }_1$ as the smaller of the two numbers $2\alpha$ and $1-2\alpha$. For this number, ${\alpha }_2$ is determined similarly, and so on.

a) Prove that for some $n$ the inequality ${\alpha }_n<3/16$ holds.

b) Can it be that ${\alpha }_n>7/40$ for all positive integers $n$?

There was a football match of 10 versus 10 players between a team of liars (who always lie) and a team of truth-tellers (who always tell the truth). After the match, each player was asked: “How many goals did you score?” Some participants answered “one”, Callum said “two”, some answered “three”, and the rest said “five”. Is Callum lying if it is known that the truth-tellers won with a score of 20:17?

For all real $x$ and $y$, the equality $f(x^2+y)=f(x)+f(y^2)$ holds. Find $f(-1)$.

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.

Prove that for any odd natural number, $a$, there exists a natural number, $b$, such that $2^b-1$ is divisible by $a$.

On a lottery ticket, it is necessary for Mary to mark 8 cells from 64. What is the probability that after the draw, in which 8 cells from 64 will also be selected (all such possibilities are equally probable), it turns out that Mary guessed

a) exactly 4 cells? b) exactly 5 cells? c) all 8 cells?