We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

Prove that any convex polygon contains not more than $35$ vertices with an angle of less than $170^{\circ}$ .

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Ceva's theorem and Menelaus's theorem Calculus Integral Existence of a definite integral Vectors Law of polygon of vectors Similar triangles

13–14 3.0

On the sides $A B$ , $B C$ and $A C$ of the triangle $A B C$ points $P$ , $M$ and $K$ are chosen so that the segments $A M$ , $B K$ and $C P$ intersect at one point and $\vec{A M} + \vec{B K} + \vec{C P} = 0$ Prove that $P$ , $M$ and $K$ are the midpoints of the sides of the triangle $A B C$ .

Problem #PRU-108731

Problems Fun Problems Word Problems Percentages and ratios

12–14 2.0

A field that will be used to grow wheat has a rectangular shape. This year, the farmer responsible for this field decided to increase the length of one of the sides by $20 %$ and decrease the length of another side by $20 %$ . The field remains rectangular. Will the harvest of wheat change this year and, if so, then by how much?

Problem #PRU-108995

Problems Discrete Mathematics Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (area and volume) Systems of points and line segments Systems of points

13–15 3.0

Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.

Problem #PRU-109018

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than $720^{\circ}$ .

Problem #PRU-109041

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Algorithm Theory Theory of algorithms (other)

13–16 4.0

$x_{1}$ is the real root of the equation $x^{2} + a x + b = 0$ , $x_{2}$ is the real root of the equation $x^{2} - a x - b = 0$ .

Prove that the equation $x^{2} + 2 a x + 2 b = 0$ has a real root, enclosed between $x_{1}$ and $x_{2}$ . ( $a$ and $b$ are real numbers).

Problem #PRU-109196

Problems Calculus Real numbers Rational and irrational numbers

14–16 4.0

In the number $a = 0.12457 \dots$ the $n$ th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that $α$ is an irrational number.

Problem #PRU-109438

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Exponential functions and logarithms Exponential functions and logarithms (other)

13–16 3.0

The function $f$ is such that for any positive $x$ and $y$ the equality $f (x y) = f (x) + f (y)$ holds. Find $f (2007)$ if $f (1 / 2007) = 1$ .

Problem #PRU-109440

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Solve the equation: $(x^{3} - 2) (2^{\sin x} - 1) + (2^{x^{3}} - 4) \sin x = 0$ .

Problem #PRU-109441

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

We are given a $100 \times 100$ square grid and $N$ counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.

What is the largest value of $N$ for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.