Proofs - practice problems - last page

Instructions: Solve each problem carefully and provide a detailed solution for every item.

Number of problems found: 40

Inequality proof
Let a, b, and c be positive real numbers whose sum is 3, each of which 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.
Triangle Geometry Proof
On side AB of triangle ABC, points D and E are given such that |AD| = |DE| = |EB|. Points A and B are the midpoints of segments CF and CG. Line CD intersects line FB at point I, and line CE intersects line AG at point J. Prove that the intersection of lin
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 area 50% greater than that of the inscribed sphere.
Trapezoid proof
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 triangle
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 | < c.
Three numbers
How much do we increase the sum of three numbers when the first enlarges by 14, the second by 15, and the third by 16? Choose any three two-digit numbers and prove the results.
Diagonal in rectangle
In the ABCD rectangle is the center of BC, point E, and point F is the center of the CD. Prove that the lines AE and AF divide diagonal BD into three equal parts.
Theorem prove
We want to prove the following statement: If a natural number n is divisible by six, then n is divisible by three. What assumption do we start from?
Truncated cone
Calculate the height of the rotating truncated cone with volume V = 1471 cm³ and a base radii r₁ = 6.1 cm and r₂ = 7.9 cm.
See harmonics
Is it true that the size of the central segment of any trapezoid is the harmonic mean size of its bases? Prove it. The central segment crosses the intersection of the diagonals and is parallel to the bases.
Laws
From which law does the validity of Pythagoras' theorem in a right triangle directly follow? ...
Proof PT
Can Pythagoras' theorem be easily proved using the Euclidean theorems? If so, do it.
Triangle P2
Can a triangle have two right angles?
Triangle
Prove whether a triangle ABC with the given properties can be constructed: a=8 cm, b=6 cm, c=10 cm.
Reciprocal
Prove that if a > b > 0, then: (1)/(a)< (1)/(b)
Proof I
When the larger of two consecutive integers is added to the product of those two integers, the result is the square of the larger integer. Is this true?
Engineer bug
Comical errors, disputes and plots arise when converting physical units. The core of today's dispute was the price of natural gas. The unit was changed - from the natural unit of cubic meter of gas to the unit kWh. The goal was to be able to compare the p

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