We solve problems

Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots

14–16 5.0

The circles $σ_{1}$ and $σ_{2}$ intersect at points $A$ and $B$ . At the point $A$ to $σ_{1}$ and $σ_{2}$ , respectively, the tangents $l_{1}$ and $l_{2}$ are drawn. The points $T_{1}$ and $T_{2}$ are chosen respectively on the circles $σ_{1}$ and $σ_{2}$ so that the angular measures of the arcs $T_{1} A$ and $A T_{2}$ 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 $σ_{1}$ intersects $l_{2}$ at the point $M_{1}$ . Similarly, the tangent $t_{2}$ at the point $T_{2}$ to the circle $σ_{2}$ intersects $l_{1}$ at the point $M_{2}$ . Prove that the midpoints of the segments $M_{1} M_{2}$ 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 $| \sin n x | \geq \sqrt{3} / 2$ .

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-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-111415

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

13–15 3.0

The volume of the regular quadrangular pyramid $S A B C D$ is equal to $V$ . The height $S P$ of the pyramid is the edge of the regular tetrahedron $S P Q R$ , the plane of the face $P Q R$ which is perpendicular to the edge $S C$ . Find the volume of the common part of these pyramids.

Problem #PRU-111417

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

13–15 3.0

The height $S O$ of a regular quadrilateral pyramid $S A B C D$ forms an angle $α$ with a side edge and the volume of this pyramid is equal to $V$ . The vertex of the second regular quadrangular pyramid is at the point $S$ , the centre of the base is at the point $C$ , and one of the vertices of the base lies on the line $S O$ . Find the volume of the common part of these pyramids.

Problem #PRU-111763

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

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.

Problem #PRU-111805

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

13–15 3.0

The sequence $(a_{n})$ is given by the conditions $a_{1} = 1000000$ , $a_{n + 1} = n ⌊ a_{n} / n ⌋ + n$ . Prove that an infinite subsequence can be found within it, which is an arithmetic progression.

Problem #PRU-111815

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

Given a square trinomial $f (x) = x^{2} + a x + b$ . It is known that for any real $x$ there exists a real number $y$ such that $f (y) = f (x) + y$ . Find the greatest possible value of $a$ .