Problems

A sequence of natural numbers $a_1<a_2<a_3<\cdots <a_n<\dots$ is such that each natural number is either a term in the sequence, can be expressed as the sum of two terms in the sequence, or perhaps the same term twice. Prove that $a_n\le n^2$ for any $n=1,2,3,\dots$

Out of the given numbers 1, 2, 3, ..., 1000, find the largest number $m$ that has this property: no matter which $m$ of these numbers you delete, among the remaining $1000-m$ numbers there are two, of which one is divisible by the other.

Author: A.A. Egorov

Calculate the square root of the number $0.111\dots 111$ (100 ones) to within a) 100; b) 101; c)* 200 decimal places.

The triangle $C_1{C}_{2}O$ is given. Within it the bisector $C_2{C}_3$ is drawn, then in the triangle $C_2{C}_{3}O$ – bisector $C_3{C}_4$ and so on. Prove that the sequence of angles ${\gamma }_n=C_{n+1}{C}_{n}O$ tends to a limit, and find this limit if $C_{1}O{C}_2=\alpha$.

A rectangular chocolate bar size $5\times 10$ is divided by vertical and horizontal division lines into 50 square pieces. Two players are playing the following game. The one who starts breaks the chocolate bar along some division line into two rectangular pieces and puts the resulting pieces on the table. Then players take turns doing the same operation: each time the player whose turn it is at the moment breaks one of the parts into two parts. The one who is the first to break off a square slice $1\times 1$ (without division lines) a) loses; b) wins. Which of the players can secure a win: the one who starts or the other one?

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is $18$, find the area of the small blue triangle.

What has a greater value: $300!$ or ${100}^{300}$?

A quadrilateral is given; $A$, $B$, $C$, $D$ are the successive midpoints of its sides, $P$ and $Q$ are the midpoints of its diagonals. Prove that the triangle $BCP$ is equal to the triangle $ADQ$.

On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?

In a pentagon $ABCDE$, diagonal $AD$ is parallel to the side $BC$ and the diagonal $CE$ is parallel to the side $AB$. Show that the areas of the triangles $\mathrm{△}ABE$ and $\mathrm{△}BCD$ are the same.