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Found: 86

Problem #PRU-109573

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

Prove that for all \(x\), \(0 < x < \pi /3\), we have the inequality \(\sin 2x + \cos x > 1\).

Problem #PRU-109704

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Methods Algebraic methods Counting in two ways Calculus Real numbers Integer and fractional parts. Archimedean property

Prove that for any positive integer \(n\) the inequality

is true.

Problem #PRU-109715

Problems Algebra Sequences Progressions Geometric progression Calculus Real numbers Integer and fractional parts. Archimedean property Sums of number sequences and difference series

Find the sum \(1/3 + 2/3 + 2^2/3 + 2^3/3 + \dots + 2^{1000}/3\).

Problem #PRU-109881

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

The polynomial \(P (x)\) of degree \(n\) has \(n\) distinct real roots.

What is the largest number of its coefficients that can be equal to zero?

Problem #PRU-109946

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations Calculus Real numbers Integer and fractional parts. Archimedean property Polynomials Quadratic polynomials Investigating the quadratic polynomial

Solve the equation \((x + 1)^3 = x^3\).

Problem #PRU-110153

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

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-110210

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

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)

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)

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-111415

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

The volume of the regular quadrangular pyramid \(SABCD\) is equal to \(V\). The height \(SP\) of the pyramid is the edge of the regular tetrahedron \(SPQR\), the plane of the face \(PQR\) which is perpendicular to the edge \(SC\). Find the volume of the common part of these pyramids.

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