MATHlympia 2021 Day 4

Systems of Equations

100

x + y = 5

2x + y = 2

What is x = -3, y = 8?

200

2x + 7y = 18

3x - 4y = -2

What is x = 2, y = 2?

300

5x - 7y = 8

10x - 14y = 17

What is inconsistent?

400

3x + 10y = 31

9x + ay = b

What values must a and b have for the system to be dependent?

What are a = 30, b = 93?

500

x + 3y - z = 2

4x + 2y + 5z = 1

3x + z = 12

What are x = 5, y = -2, z = -3?

Composite Functions

100

Using variables, turn the following into an equation: y varies directly with x.

What is y = kx?

200

Using variables, turn the following into an equation: y varies inversely with double the cube of x.

What is y = k/(2x³)?

300

Given that f(x) = 1/x² and g(x) = 6 - x, find f(g(4)).

What is 1/4?

400

Given that f(x) = 4x + 3 and g(x) = x(x + 4), find g(f(5)).

What is 621?

500

Given that f(x) = 5x² + 3 and g(x) = (3x + 1)/(x + 5), find g(g(2)) in its simplest form.

What is 2/3?

Radical and Rational Equations

100

root(x - 1) = 5

What is x = 26?

200

1/root(25x) = 75

x = 1/625

300

5/x - 3/x = 8

What is x = 1/4?

400

3/x + 8/(x-5) = 1

What is x = 15, x = 1?

500

x/(x+2) + 2/[(x+2)(x+3)] = 5/(x+3)

What is x = 4?

Logarithms

100

Find log₂512.

What is 9?

200

Find log₇343.

What is 3?

300

Find e^{ln(e) - 1}.

What is 1?

400

Find [log(100⁷)]+10(log₂4 + log₂8).

What is 64?

500

Find a if log_b[a³/b] + 1 = log_b8.

What is 2?

#mentor

100

This mentor hurt his shoulder and is currently in a sling.

Who is Andy?

200

This mentor taught you that the thick Z symbol means the set of all integers.

Who is Arjun?

300

This mentor taught the very first lesson of the camp.

Who is Bobert?

400

These mentors are twin rowers.

Who are Dylan and Heidi?

500

He's a runner, he's a cross country star.

Who is Chi?