Olympiad Algebra

Final round

Ten problems from the whole studio. Nothing tells you which idea to use. 0 of 10 solved

Averaging chips

AMC

Five chips read 1, 2, 3, 4 and 15. A move replaces two chips by two copies of their average. Keep moving and the chips close in on one common value. What value?

Cubes first

AMC

$x + y = 4$ and $x^3 + y^3 = 28$ . Find $xy$ .

One equation, two unknowns

AMC

Real numbers x and y satisfy

x^2 + 2y^2 - 2xy - 6y + 9 = 0.

Find x + y.

A huge power, a small divisor

AMC

What is the remainder when $x^{2026} + x^{13} + 1$ is divided by $x^2 + 1$ ?

A function with a correction

AIME

For all real x and y,

f(x + y) = f(x) + f(y) + xy + 1,

and f(1) = 1. Find f(−5).

Reciprocal squares

AIME

r, s and t are the roots of $x^3 - 5x^2 + 6x - 1 = 0$ . Find $\tfrac{1}{r^2} + \tfrac{1}{s^2} + \tfrac{1}{t^2}$ .

Cubes over a product

AIME

For positive a and b, find the least value of

\frac{a^3 + b^3}{ab(a + b)}.

Product equals one

olympiad

Positive a, b, c have $abc = 1$ . Find the least value of $(a + b)(b + c)(c + a)$ .

The fourth power sum

olympiad

Numbers x, y, z (not necessarily real) satisfy

\begin{gathered} x + y + z = 3, \\ x^2 + y^2 + z^2 = 5, \\ x^3 + y^3 + z^3 = 7. \end{gathered}

Find $x^4 + y^4 + z^4$ .

A sum that never ends

olympiad

Let $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ . Find the infinite sum

\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \cdots

← All chapters