We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (other)

Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.

Problem #PRU-111245

Problems Set theory and logic

16–16 2.0

Let’s denote any two digits with the letters $A$ and $X$ . Prove that the six-digit number $X A X A X A$ is divisible by 7 without a remainder.

Problem #PRU-111345

Problems Calculus Integral Applications of integration Area by integration Algebra and arithmetic Polynomials Quadratic polynomials Quadratic equations. Vieta's rule.

16–16 3.0

The numbers $p$ and $q$ are such that the parabolas $y = - 2 x^{2}$ and $y = x^{2} + p x + q$ intersect at two points, bounding a certain figure.

Find the equation of the vertical line dividing the area of this figure in half.

Problem #PRU-111923

Problems Calculus Derivative Derivative and extrema Derivative and multiple roots

15–16 3.0

When water is drained from a pool, the water level $h$ in it varies depending on the time $t$ according to the function $h (t) = a t^{2} + b t + c$ , and at the time $t_{0}$ of when the draining is ending, the equalities $h (t_{0}) = h^{'} (t_{0}) = 0$ are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?

Problem #PRU-115449

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–16 4.0

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an $8 \times 8$ square grid, so that no more ’corners’ would fit?

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let $C_{1}$ be the point of intersection, further from the vertex $C$ , of the circles constructed from the medians $A M_{1}$ and $B M_{2}$ . Points $A_{1}$ and $B_{1}$ are defined similarly. Prove that the lines $A A_{1}$ , $B B_{1}$ and $C C_{1}$ intersect at the same point.

Problem #PRU-116433

Problems Calculus Functions of one variable. Continuity Functions. Continuity (other) Algebra and arithmetic Sequences Progressions Geometric progression

14–16 3.0

The function $f (x)$ is defined on the positive real $x$ and takes only positive values. It is known that $f (1) + f (2) = 10$ and $f (a + b) = f (a) + f (b) + 2 \sqrt{f (a) f (b)}$ for any $a$ and $b$ . Find $f (2^{2011})$ .

Problem #PRU-116435

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

14–16 3.0

On a chessboard, $n$ white and $n$ black rooks are arranged so that the rooks of different colours cannot capture one another. Find the greatest possible value of $n$ .

Problem #PRU-116563

Problems Algebra Trigonometry Identical transformations (trigonometry) Methods Proof by contradiction Calculus Real numbers Rational and irrational numbers

15–16 2.0

Does there exist a real number $α$ such that the number $\cos α$ is irrational, and all the numbers $\cos 2 α$ , $\cos 3 α$ , $\cos 4 α$ , $\cos 5 α$ are rational?

Problem #PRU-116627

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Polynomials Algebraic identities for polynomials Factoring polynomials Calculus Real numbers Integer and fractional parts. Archimedean property

14–16 3.0

Solve the inequality: $⌊ x ⌋ \times {x} < x - 1$ .