Notes to a video lecture on http://www.unizor.com Trigonometry+ 05 Problem A Given ∠α, ∠β and ∠γ are acute angles of a triangle. Evaluate the expression tan(α)·tan(β) + tan(β)·tan(γ) + + tan(γ)·tan(α) Hint A α + β + γ = π/2 Solution A γ = π/2 − (α+β) tan(γ) = sin(γ)/cos(γ) = = cos(π/2−γ)/sin(π/2−γ) = = cos(α+β)/sin(α+β) = = [cos(α)·cos(β)−sin(α)·sin(β)]/ /[sin(α)·cos(β)+cos(α)·sin(β)] = divide both numerator and denominator by cos(α)·cos(β) = [1−tan(α)·tan(β)]/ /[tan(α)+tan(β)] Therefore, tan(α)·tan(β) + tan(β)·tan(γ) + + tan(γ)·tan(α) = = tan(α)·tan(β) + + [tan(α)+tan(β)]·tan(γ) Substi ..read more

Notes to a video lecture on http://www.unizor.com Algebra+ 06 Problem A Prove that sum of square roots of 2, 3 and 5 is an irrational number. Hint A Assume, this sum is rational, that is √2 + √3 + √5 = p/q where p and q are integer numbers without common divisors (if they do, we can reduce the fraction by dividing a numerator p and denominator q by a common divisor without changing the value of a fraction). Then simplify the above expression by getting rid of square roots and prove that p must be an even number and, therefore, can be represented as p=2r. Then prove that q must be even as ..read more

Notes to a video lecture on http://www.unizor.com Algebra+ 05 Problem A Given a system of two equations with three unknown variables x, y and z: x + y + z = A x−1 + y−1 + z−1 = A−1 Prove that one of the unknown variables equals to A. Hint A System of equations x + y = p x · y = q fully defines a pair of numbers (generally speaking, complex numbers) as solutions to a quadratic equation X² − p·X + q. Indeed, if X1 and X2 are the solution of the equation, then, according to the Vieta's Theorem, X1 + X2 = −(−p) = p and X1 · X2 = q (See a lecture Math 4 Teens - Algebra - Quadratic Equations ..read more

Notes to a video lecture on http://www.unizor.com Geometry+ 07 Problem A Given any circle with a center at point O, its diameter MN and any point P on this circle not coinciding with the ends M, N of a given diameter. Let point Q be a projection of point P on a diameter MN. This point Q divides diameter MN into two parts: MQ = a and QN= b Prove that (1) Radius of a OP circle is an arithmetic average of a and b. (2) Projection segment PQ is a geometric average of a and b. (3) Based on these proofs, conclude that geometric average of two non-negative real numbers is less or equal to their a ..read more

Notes to a video lecture on http://www.unizor.com Logic+ 06 Problem A There are 5 towns. Some of them are connected by direct roads, that is by roads not going through other towns. It's known that among any group of 4 towns out of these 5 there is always one town connected by direct roads with each of the other 3 towns of this group. Prove that there is at least one town connected with all 4 others by direct roads. Proof A Choose any 4 towns from given 5 as the first group towns. One of these towns is connected to 3 others, as the problem states. Let's call this town A and the others wi ..read more

Notes to a video lecture on http://www.unizor.com Geometry+ 06 Problem A Given an isosceles triangle ΔABC with AB=BC and ∠ABC=20°. Point D on side BC is chosen such that ∠CAD=60°. Point E on side AB is chosen such that ∠ACE=50°. Find angle ∠ADE. Solution A Problem B Given two triangles ΔA1B1C1 and ΔA2B2C2 with the following properties: (a) side A1B1 of the first triangle equals to side A2B2 of the second; (b) angles opposite to these sides, ∠A1C1B1 and ∠A2C2B2, are equal to each other; (c) bisectors of these angles, C1X1 and C2X2, are also equal to each other. Prove that these triang ..read more

Notes to a video lecture on http://www.unizor.com Trigonometry+ 04 Problem A Find the sums Σk∈[0,n−1]sin²(x+k·π/n) Solution A First, convert sin²(...) into cos(...) by using the identity cos(2φ) = cos²(φ) − sin²(φ) = = 1 − 2sin²(φ) from which follows sin²(φ) = ½(1 − cos(2φ)) Now our sum looks like this Σk∈[0,n−1] ½(1−cos(2x+2k·π/n)) = = n/2 − − ½Σk∈[0,n−1]cos(2x+2k·π/n) To calculate Σk∈[0,n−1] above, let's use the result of the previous lecture Trigonometry 03 that proved the following Σk∈[0,n]cos(x+k·y) = = [sin(x+(2n+1)·y/2) − − sin(x−y/2)] / / [2·sin(y/2 ..read more

Notes to a video lecture on http://www.unizor.com Algebra+ 04 Problem A Prove that if X+Y+Z=1 then X²+Y²+Z² ≥ 1/3. Hint A1 X² + Y² ≥ 2·X·Y. Hint A2 X²+Y²+Z² = = (X−a)²+(Y−a)²+(Z−a)²+ +2·a·(X+Y+Z)−3a² ≥ ≥ 2a −3a² Quadratic polynomial 2a−3a² has roots a=0 and a=2/3 and maximum at a=1/3 with value 2·(1/3)−3·(1/3)²=1/3. Problem B Solve an equation X8 + X4 + 1 = 0 Hint B1 Substitute y=X4. Hint B2 Represent the left part as a product of 4 polynomials of a second degree by adding and subtracting X4. Answer B X1,2,3,4 = ±1/2 ± i·√3/2 X5,6,7,8 = ±√3/2 ± i/2 Problem C Find a number from 1 ..read more

Notes to a video lecture on http://www.unizor.com Arithmetic+ 05 Problem A Prove that a remainder of division of any prime number P by 24 is a prime number, assuming P is greater than 24. Hint A If P is prime and R is a remainder of division of P by 24 then P=24·n+R where R≤23. If R is not prime, it should be divisible by a product of, at least, two prime numbers. Also, 24=2·2·2·3. Problem B Prove that for any natural number N the number 10N+18·N−1 is divisible by 27. Hint B First, prove a theorem that any natural number n divided by 3 (or 9) has the same remainder as a sum of its digit ..read more

Notes to a video lecture on http://www.unizor.com Trigonometry+ 03 Problem A Find the sums Ssin(x,y,n)= Σk∈[0,n]sin(x+k·y) Scos(x,y,n)= Σk∈[0,n]cos(x+k·y) as algebraic expression with arguments x, y and n. Solution A1 Let's start with a sum of sines. Recall the formula for a cosine of a sum of two angles cos(a+b) = = cos(a)·cos(b) − sin(a)·sin(b) From this formula it's easy to derive another trigonometric identity formula 2·sin(a)·sin(b) = = cos(a−b)−cos(a+b) Let's use this formula to transform our series into only a couple of members. Multiply Ssin (that is, each member of this serie ..read more