We solve problems

Problems Geometry Plane geometry Area Area of a circle, sector, segment

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?

Problem #PRU-7653

Problems Geometry Plane geometry Area Area ratio Finding the area by rearrangment

12–14 4.0

In a pentagon $A B C D E$ , diagonal $A D$ is parallel to the side $B C$ and the diagonal $C E$ is parallel to the side $A B$ . Show that the areas of the triangles $△ A B E$ and $△ B C D$ are the same.

Problem #PRU-76543

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–15 3.0

Prove that, for any integer $n$ , among the numbers $n, n + 1, n + 2, \dots, n + 9$ there is at least one number that is mutually prime with the other nine numbers.

Problem #PRU-7710

Problems Algebra Fun Problems Arithmetic Puzzles

12–14 2.0

How can you arrange the numbers $5 / 177$ , $51 / 19$ and $95 / 9$ and the arithmetical operators “ $+$ ”, “ $-$ ”, “ $\times$ ” and “ $\div$ ” such that the result is equal to 2006? Note: you can use the given numbers and operators more than once.

Problem #PRU-77892

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

There are 13 weights, each weighing an integer number of grams. It is known that any 12 of them can be divided into two cups of weights, six weights on each one, which will come to equilibrium. Prove that all the weights have the same weight.

Problem #PRU-77894

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

If we are given any 100 whole numbers then amongst them it is always possible to choose one, or several of them, so that their sum gives a number divisible by 100. Prove that this is the case.

Problem #PRU-77915

Problems Methods Pigeonhole principle

13–15 3.0

Numbers $1, 2, 3, \dots, 101$ are written out in a row in some order. Prove that one can cross out 90 of them so that the remaining 11 will be arranged in their magnitude (either increasing or decreasing).

Problem #PRU-77973

Problems Algebra Graphs and locus of points on the Cartesian plane Trigonometry Trigonometrical equations

14–16 2.0

Find the locus of points whose coordinates $(x, y)$ satisfy the relation $\sin (x + y) = 0$ .

Problem #PRU-77989

Problems Methods Examples and counterexamples. Constructive proofs Algebra Polynomials Quadratic polynomials Quadratic equations and systems of equations

13–16 4.0

The equations $\begin{matrix} (1) & a x^{2} + b x + c = 0 \end{matrix}$ and $\begin{matrix} (2) & - a x^{2} + b x + c \end{matrix}$ are given. Prove that if $x_{1}$ and $x_{2}$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_{3}$ of the equation $\frac{1}{2} a x^{2} + b x + c$ such that either $x_{1} \leq x_{3} \leq x_{2}$ or $x_{1} \geq x_{3} \geq x_{2}$ .

Problem #PRU-78003

Problems Number Theory Divisibility Arithmetic functions Algebra Polynomials Polynomials(other)

15–16 3.0

Prove that if $x_{0}^{4} + a_{1} x_{0}^{3} + a_{2} x_{0}^{2} + a_{3} x_{0} + a_{4}$ and $4 x_{0}^{3} + 3 a_{1} x_{0}^{2} + 2 a_{2} x_{0} + a_{3} = 0$ then $x^{4} + a_{1} x^{3} + a_{2} x^{2} + a_{3} x + a_{4}$ is divisible by $(x - x_{0})^{2}$ .