Russian Math Olympiad Problems And Solutions Pdf Verified <2025>

Let $f(x) = x^2 + 4x + 2$. Find all $x$ such that $f(f(x)) = 2$.

Note that $2007 = 3 \cdot 669 = 3 \cdot 3 \cdot 223$. We can write $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$. Since $x^2 - xy + y^2 > 0$, we must have $x + y > 0$. Also, $x + y$ must divide $2007$, so $x + y \in {1, 3, 669, 2007}$. If $x + y = 1$, then $x^2 - xy + y^2 = 2007$, which has no integer solutions. If $x + y = 3$, then $x^2 - xy + y^2 = 669$, which also has no integer solutions. If $x + y = 669$, then $x^2 - xy + y^2 = 3$, which gives $(x, y) = (1, 668)$ or $(668, 1)$. If $x + y = 2007$, then $x^2 - xy + y^2 = 1$, which gives $(x, y) = (1, 2006)$ or $(2006, 1)$. russian math olympiad problems and solutions pdf verified

Find all pairs of integers $(x, y)$ such that $x^3 + y^3 = 2007$. Let $f(x) = x^2 + 4x + 2$

(From the 2010 Russian Math Olympiad, Grade 10) We can write $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

(From the 2001 Russian Math Olympiad, Grade 11)

(From the 2007 Russian Math Olympiad, Grade 8)

By Cauchy-Schwarz, we have $\left(\frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x}\right)(y + z + x) \geq (x + y + z)^2 = 1$. Since $x + y + z = 1$, we have $\frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \geq 1$, as desired.