For what values of \(n\) does the polynomial \((x+1)^n - x^n - 1\) divide by:
a) \(x^2 + x + 1\); b) \((x^2 + x + 1)^2\); c) \((x^2 + x + 1)^3\)?
Old calculator I.
a) Suppose that we want to find \(\sqrt[3]{x}\) (\(x> 0\)) on a calculator that can find \(\sqrt{x}\) in addition to four ordinary arithmetic operations. Consider the following algorithm. A sequence of numbers \(\{y_n\}\) is constructed, in which \(y_0\) is an arbitrary positive number, for example, \(y_0 = \sqrt{\sqrt{x}}\), and the remaining elements are defined by \(y_{n + 1} = \sqrt{\sqrt{x y_n}}\) (\(n \geq 0\)).
Prove that \(\lim\limits_{n\to\infty} y_n = \sqrt[3]{x}\).
b) Construct a similar algorithm to calculate the fifth root.
The sequence of numbers \(a_1, a_2, a_3, \dots\) is given by the following conditions \(a_1 = 1\), \(a_{n + 1} = a_n + \frac {1} {a_n^2}\) (\(n \geq 0\)).
Prove that
a) this sequence is unbounded;
b) \(a_{9000} > 30\);
c) find the limit \(\lim \limits_ {n \to \infty} \frac {a_n} {\sqrt [3] n}\).
Author: A.K. Tolpygo
12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?
Author: L.N. Vaserstein
For any natural numbers \(a_1, a_2, \dots , a_m\), no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers \(b_1, b_2, \dots , b_m\) the sum is not zero. Prove this.
Is there a line on the coordinate plane relative to which the graph of the function \(y = 2^x\) is symmetric?
Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.
There are eight points inside a circle of radius 1. Show that there are at least two points with distance between them less then 1.
Leo’s dad was making a pizza for lunch. He decided to place 7 pieces of pineapple on it. Assuming the pizza is a circle of a \(20\) cm radius, show that some two pieces of pineapple were placed closer than \(20\) cm apart.
Each point on a circle was painted red or green. Show that there is an isosceles triangle whose vertices are on the circumference of the circle, such that all three vertices are red or all three are green.