# The proof - practice problems

The proof is a convincing demonstration in mathematics that a statement is true under certain conditions.#### Number of problems found: 26

- Bisector 2

ABC is an isosceles triangle. While AB=AC, AX is the bisector of the angle ∢BAC meeting side BC at X. Prove that X is the midpoint of BC. - Fraction and ratios

Fraction and ratios are different names for the same thing? - Simplify logarithm expr

Given that logxU + logxV =p and logxU - logxV =q Prove that U=x^½(p+q) - Directly 55591

If n is a natural number that gives a division of 2 or 3 when divided by 5, then n gives a residue of 4 when divided by 5. Prove directly - Indirectly: 55581

Prove indirectly: No odd natural number is divisible by four. - Decreasing 36183

Prove that the sequence {3 - 4. n} from n = 1 to ∞ is decreasing. - Justification 8468

The natural number n has at least 73 two-digit divisors. Prove that one of them is the number 60. Also, give an example of the number n, which has exactly 73 double-digit divisors, including a proper justification. - Workers

Three factory operators produced 480 units in 50 minutes. How many hours worked? I'm trying to prove or disapprove the idea that the company tell me that it's 2.5 hours. So what is right? Thank you, Petra - Inequality 7320

Let a, b, c be positive real numbers whose sum is 3, and each of them is at most 2. Prove that the inequality holds: a2 + b2 + c2 + 3abc - Odd/even number

Pick any number. If that number is even, divide it by 2. If it's odd, multiply it by three and add one. Now repeat the process with your new number. If you keep going, you'll eventually end up at one every time. Prove. - Prove

Prove that k1 and k2 are the equations of two circles. Find the equation of the line that passes through the centers of these circles. k1: x²+y²+2x+4y+1=0 k2: x²+y²-8x+6y+9=0 - Sum of inner angles

Prove that the sum of all inner angles of any convex n-angle equals (n-2) . 180 degrees. - Equilateral cylinder

A sphere is inserted into the rotating equilateral cylinder (touching the bases and the shell). Prove that the cylinder has both a volume and a surface half larger than an inscribed sphere. - Trapezoid 4908

Trapezoid ABCD with bases AB = a, CD = c has height v. The point S is the center of the arm BC. Prove that the area of the ASD triangle is equal to half the area of the ABCD trapezoid. - Inequality 4434

The heel of height from the vertex C in the triangle ABC divides the side AB in the ratio 1: 2. Prove that in the usual notation of the lengths of the sides of the triangle ABC, the inequality 3 | a-b | holds - Three numbers

How much we increases the sum of three numbers when the first enlarge by 14, second by 15 and third by 16? Choose any three two-digit numbers and prove results. - Diagonal in rectangle

In that rectangle ABCD is the center of BC point E and point F is center of CD. Prove that the lines AE and AF divide diagonal BD into three equal parts. - Theorem prove

We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started? - Truncated cone

Calculate the height of the rotating truncated cone with volume V = 1354 cm³ and a base radii r_{1}= 9.1 cm and r_{2}= 5.4 cm. - See harmonics

It is true that the size of the central segment of any trapezoid is the harmonic mean size of its bases? Prove it. Central segment crosses the intersection of the diagonals and is parallel to the bases.

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