We solve problems

Problem #PRU-109997

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Continuous functions (general properties)

15–16 4.0

On a function \(f (x)\) defined on the whole line of real numbers, it is known that for any \(a > 1\) the function \(f (x)\) + \(f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.

Problem #PRU-110035

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Absolute value Inequalities containing absolute values Trigonometry Trigonometrical inequalities

15–16 4.0

Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?

Problem #PRU-110061

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

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-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-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?

Problem #PRU-111345

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

16–16 3.0

The numbers \(p\) and \(q\) are such that the parabolas \(y = - 2x^2\) and \(y = x^2 + px + q\) 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-111763

Problems Calculus Derivative Calculating a derivative Algebra Absolute value Inequalities containing absolute values Polynomials Quadratic polynomials Quadratic polynomilal (other)

15–16 4.0

The quadratic trinomials \(f (x)\) and \(g (x)\) are such that \(f' (x) g' (x) \geq | f (x) | + | g (x) |\) for all real \(x\). Prove that the product \(f (x) g (x)\) is equal to the square of some trinomial.