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-110085

Problems Algebra

15–16 4.0

The real numbers \(x\) and \(y\) are such that for any distinct prime odd \(p\) and \(q\) the number \(x^p + y^q\) is rational. Prove that \(x\) and \(y\) are rational numbers.

Problem #PRU-110122

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Calculus Functions of one variable. Continuity Monotonicity, boundedness

14–16 4.0

The functions \(f (x) - x\) and \(f (x^2) - x^6\) are defined for all positive \(x\) and increase. Prove that the function

also increases for all positive \(x\).

Problem #PRU-110153

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

13–15 3.0

The sum of the positive numbers \(a, b, c\) is \(\pi / 2\). Prove that \(\cos a + \cos b + \cos c > \sin a + \sin b + \sin c\).

Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations A homothetic transformation helps solve the problem Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots Homothetic circles Tangent lines and tangent circles Tangent lines to circles Properties and criteria of a secant

14–16 5.0

The circles \(\sigma_1\) and \(\sigma_2\) intersect at points \(A\) and \(B\). At the point \(A\) to \(\sigma_1\) and \(\sigma_2\), respectively, the tangents \(l_1\) and \(l_2\) are drawn. The points \(T_1\) and \(T_2\) are chosen respectively on the circles \(\sigma_1\) and \(\sigma_2\) so that the angular measures of the arcs \(T_1A\) and \(AT_2\) are equal (the arc value of the circle is considered in the clockwise direction). The tangent \(t_1\) at the point \(T_1\) to the circle \(\sigma_1\) intersects \(l_2\) at the point \(M_1\). Similarly, the tangent \(t_2\) at the point \(T_2\) to the circle \(\sigma_2\) intersects \(l_1\) at the point \(M_2\). Prove that the midpoints of the segments \(M_1M_2\) are on the same line, independent of the positions of the points \(T_1, T_2\).

Problem #PRU-110210

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

13–15 3.0

Prove that for each \(x\) such that \(\sin x \neq 0\), there is a positive integer \(n\) such that \(|\sin nx| \geq \sqrt{3}/2\).

Problem #PRU-110212

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

13–15 3.0

For what natural numbers \(n\) are there positive rational but not whole numbers \(a\) and \(b\), such that both \(a + b\) and \(a^n + b^n\) are integers?

Problem #PRU-111218

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

13–15 3.0

The base of the pyramid is a square. The height of the pyramid crosses the diagonal of the base. Find the largest volume of such a pyramid if the perimeter of the diagonal section containing the height of the pyramid is 5.

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 \(XAXAXA\) is divisible by 7 without a remainder.

Problem #PRU-111264

Problems Calculus Functions of one variable. Continuity Continuity and compactness Monotonicity, boundedness

15–16 4.0

A continuous function \(f(x)\) is such that for all real \(x\) the following inequality holds: \(f(x^2) - (f (x))^2 \geq 1/4\). Is it true that the function \(f(x)\) necessarily has an extreme point?