Daily Olympiad: Math - Expansions [20260521]

Solution

Correct: B

Using the identity (a+b)^2 = a^2 + 2ab + b^2, here a = 3x and b = 4y. So, (3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2.

Solution

Correct: C

Using the identity (a-b)^2 = a^2 - 2ab + b^2, here a = 5p and b = 2q. So, (5p - 2q)^2 = (5p)^2 - 2(5p)(2q) + (2q)^2 = 25p^2 - 20pq + 4q^2.

Solution

Correct: B

Using the identity (a+b)(a-b) = a^2 - b^2, here a = 7m and b = 3n. So, (7m + 3n)(7m - 3n) = (7m)^2 - (3n)^2 = 49m^2 - 9n^2.

Solution

Correct: B

Using the identity (a+b)^2 = a^2 + 2ab + b^2, here a = x/2 and b = 2/x. So, (x/2 + 2/x)^2 = (x/2)^2 + 2(x/2)(2/x) + (2/x)^2 = x^2/4 + 2(1) + 4/x^2 = x^2/4 + 2 + 4/x^2.

Solution

Correct: C

Method 1: Expand directly. (x + 1/x)^2 = x^2 + 2(x)(1/x) + (1/x)^2 = x^2 + 2 + 1/x^2 (x - 1/x)^2 = x^2 - 2(x)(1/x) + (1/x)^2 = x^2 - 2 + 1/x^2 So, (x^2 + 2 + 1/x^2) - (x^2 - 2 + 1/x^2) = x^2 + 2 + 1/x^2 - x^2 + 2 - 1/x^2 = 2 + 2 = 4. Method 2: Use the difference of squares identity, A^2 - B^2 = (A+B)(A-B), where A = (x + 1/x) and B = (x - 1/x). [(x + 1/x) + (x - 1/x)][(x + 1/x) - (x - 1/x)] = [x + 1/x + x - 1/x][x + 1/x - x + 1/x] = [2x][2/x] = 4.

Solution

Correct: B

Using the identity (x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx, here x = 2a, y = -3b, z = c. So, (2a - 3b + c)^2 = (2a)^2 + (-3b)^2 + (c)^2 + 2(2a)(-3b) + 2(-3b)(c) + 2(c)(2a) = 4a^2 + 9b^2 + c^2 - 12ab - 6bc + 4ca.

Solution

Correct: A

Given x + 1/x = 5. Square both sides: (x + 1/x)^2 = 5^2 Using (a+b)^2 = a^2 + 2ab + b^2, with a = x and b = 1/x. x^2 + 2(x)(1/x) + (1/x)^2 = 25 x^2 + 2 + 1/x^2 = 25 x^2 + 1/x^2 = 25 - 2 x^2 + 1/x^2 = 23.

Solution

Correct: B

Using the identity (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, here a = 2x and b = 3. (2x + 3)^3 = (2x)^3 + 3(2x)^2(3) + 3(2x)(3)^2 + (3)^3 = 8x^3 + 3(4x^2)(3) + 3(2x)(9) + 27 = 8x^3 + 36x^2 + 54x + 27.

Solution

Correct: B

Using the identity (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3, here a = m and b = 2n. (m - 2n)^3 = (m)^3 - 3(m)^2(2n) + 3(m)(2n)^2 - (2n)^3 = m^3 - 3(m^2)(2n) + 3(m)(4n^2) - 8n^3 = m^3 - 6m^2n + 12mn^2 - 8n^3.

Solution

Correct: B

Given x - 1/x = 3. Cube both sides: (x - 1/x)^3 = 3^3 Using (a-b)^3 = a^3 - b^3 - 3ab(a-b), with a = x and b = 1/x. x^3 - (1/x)^3 - 3(x)(1/x)(x - 1/x) = 27 x^3 - 1/x^3 - 3(1)(3) = 27 x^3 - 1/x^3 - 9 = 27 x^3 - 1/x^3 = 27 + 9 x^3 - 1/x^3 = 36.

Solution

Correct: B

This is a direct application of the identity (X+Y)^2 - (X-Y)^2 = 4XY. Here, X = a and Y = b. So, (a + b)^2 - (a - b)^2 = 4ab. Alternatively, expanding directly: (a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2 Subtracting the two: (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = a^2 + 2ab + b^2 - a^2 + 2ab - b^2 = 4ab.

Solution

Correct: A

First, expand (x + 1)(x + 2): (x + 1)(x + 2) = x(x+2) + 1(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2. Now multiply this result by (x + 3): (x^2 + 3x + 2)(x + 3) = x(x^2 + 3x + 2) + 3(x^2 + 3x + 2) = x^3 + 3x^2 + 2x + 3x^2 + 9x + 6 = x^3 + (3x^2 + 3x^2) + (2x + 9x) + 6 = x^3 + 6x^2 + 11x + 6.

Solution

Correct: C

This expression is of the form (X+Y)^2 - (X-Y)^2, which simplifies to 4XY. Here, X = a/b and Y = b/a. So, 4 * (a/b) * (b/a) = 4 * 1 = 4.

Solution

Correct: A

We know the identity (a+b)^2 = a^2 + b^2 + 2ab. We are given a + b = 7 and ab = 10. Substitute these values into the identity: (7)^2 = a^2 + b^2 + 2(10) 49 = a^2 + b^2 + 20 a^2 + b^2 = 49 - 20 a^2 + b^2 = 29.

Solution

Correct: C

We know the identity (x-y)^2 = x^2 + y^2 - 2xy. We are given x - y = 4 and xy = 21. Substitute these values into the identity: (4)^2 = x^2 + y^2 - 2(21) 16 = x^2 + y^2 - 42 x^2 + y^2 = 16 + 42 x^2 + y^2 = 58.

Solution

Correct: A

Using the identity (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca, here a = -x, b = 2y, c = -3z. (-x + 2y - 3z)^2 = (-x)^2 + (2y)^2 + (-3z)^2 + 2(-x)(2y) + 2(2y)(-3z) + 2(-3z)(-x) = x^2 + 4y^2 + 9z^2 - 4xy - 12yz + 6zx.

Solution

Correct: A

We know the identity x^3 + y^3 = (x + y)^3 - 3xy(x + y). Given x + y = 12 and xy = 32. Substitute these values into the identity: x^3 + y^3 = (12)^3 - 3(32)(12) = 1728 - (96)(12) = 1728 - 1152 = 576.

Solution

Correct: B

This expression matches the identity (a - b)(a^2 + ab + b^2) = a^3 - b^3. Here, a = 2x and b = y. So, (2x - y)((2x)^2 + (2x)(y) + (y)^2) = (2x)^3 - (y)^3 = 8x^3 - y^3.

Solution

Correct: C

This expression matches the identity (a + b)(a^2 - ab + b^2) = a^3 + b^3. Here, a = 3p and b = 2q. So, (3p + 2q)((3p)^2 - (3p)(2q) + (2q)^2) = (3p)^3 + (2q)^3 = 27p^3 + 8q^3.

Solution

Correct: C

Given a + b + c = 0. This implies: a + b = -c b + c = -a c + a = -b Substitute these into the expression: (a + b)^2 / ab + (b + c)^2 / bc + (c + a)^2 / ca = (-c)^2 / ab + (-a)^2 / bc + (-b)^2 / ca = c^2 / ab + a^2 / bc + b^2 / ca To add these fractions, find a common denominator, which is abc. Multiply the first term by c/c, second by a/a, third by b/b: = (c^2 * c) / (ab * c) + (a^2 * a) / (bc * a) + (b^2 * b) / (ca * b) = c^3 / abc + a^3 / abc + b^3 / abc = (a^3 + b^3 + c^3) / abc From the identity for cubes, if a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. Since a + b + c = 0 is given, we can apply this identity. Therefore, (a^3 + b^3 + c^3) / abc = 3abc / abc = 3.