Let
$$
A = \{ (x, y, z) \in \mathbb{R}^3 : xyz = 1,\ x + y + z = 3 \},
$$
under the function
$$
F(x, y, z) = xy + yz + zx.
$$
Determine the image of the set $A$ under $F$.
Let $ABCD$ be a square. The line segment $AB$ is divided internally at $H$ so that $|AB| \cdot |BH| = |AH|^2$. Let $E$ be the midpoint of $AD$ and $X$ be the midpoint of $AH$. Let $Y$ be the point on $EB$ such that $XY$ is perpendicular to $BE$. Prove that $|XY| = |XH|$.
Suppose $p \ge 4$. Determine the largest constant $q$ such that, for all $a, b \gt 0$,
$$
\frac{1}{a} + \frac{1}{b} + \frac{p}{a+b} \ge \frac{q}{\sqrt{ab}}
$$
Suppose $a$, $b$, $c$ are the side lengths of a triangle. Prove that
$$
\frac{a}{1+a}, \frac{b}{1+b}, \frac{c}{1+c}
$$
are also the side lengths of a triangle.
A triangle *ABC* is inscribed in a circle, centre *O*, and $D$, $E$, $F$ are the mid-points of $BC$, $CA$, $AB$ respectively. The point $A$ is free to move on the circumference of the circle. Find the locus of $L$, the midpoint of $FE$.
Suppose $a$, $b$, $c$ are real numbers such that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Prove that
$$
a^2 b^2 c^2 \leq \frac{1}{54}
$$
and determine the cases of equality.
Let $(a_n)$ be the "iterated Fibonacci sequence": $a_n = f(f(n))$, where $f(1) = f(2) = 1$, and $f(n+2) = f(n+1) + f(n)$. Prove that $a_n$ is a multiple of 144 whenever it is a multiple of 14.
A fixed point $O$ is joined to any point $P$ on a given line which does not contain $O$. On $OP$ a point $Q$ is taken such that $|OP| \cdot |OQ|$ is constant. Show $Q$ lies on a fixed circle if $P$ varies. Examine all possible cases.
Let $p(x)$ be a polynomial with rational coefficients. Prove that there exists a positive integer $n$ such that the polynomial $q(x)$ defined by
$$
q(x) = p(x + n) - p(x)
$$
has integer coefficients.
In the triangle $ABC$ we have $|AB| \lt |AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. The lines $BD$ and $CE$ intersect at the incentre $I$ of $\triangle ABC$.
Prove that $\angle BAC = 60^\circ$ if and only if $|IE| = |ID|$.
Let $ABCD$ be a quadrilateral inscribed in a circle and let $M$ be a point on the circle. Consider the projections of the point $M$ on two opposite sides of the quadrilateral, and on its diagonals. Show that there exists a circle passing through these four points if and only if the quadrilateral is a trapezoid.
Let $ABC$ be a planar triangle. Let $\lambda \ge 0$. Define the point $D(\lambda)$ on the side $BC$ so that $D(\lambda)$ divides $BC$ in the ratio $c^\lambda : b^\lambda$. Prove that
$$
|AD(\lambda)| \le |AD(2 - \lambda)| \quad \text{iff} \quad \lambda \ge 1.
$$
Let $S$ be a set of $n$ different points in the plane so that no three lie on a line. Show that there are at least $\frac{n(n-2)}{3}$ triangles whose vertices are among the points of $S$ and such that the triangles do not contain any other points of $S$.
Suppose $r$, $R$ are the in-radius and circum-radius of triangle $ABC$. Show that
$$
9r \le a \sin B + b \sin C + c \sin A \le \frac{9R}{2}.
$$
with equality in both inequalities iff $ABC$ is equilateral.
a) For each integer $a$ not divisible by $7$, let $N(a)$ denote the least among all positive integers $n$ such that $49$ divides $a^n - 1$. For each positive integer $N$, let $A_N := \{a \in \mathbb{Z} : 0 \lt a \lt 49 \text{ and } N(a) = N\}$. Show that
$$
\begin{aligned}
|A_1| &= |A_2| = 1, & |A_3| &= |A_6| = 2, \\
|A_7| &= |A_{14}| = 6, & |A_{21}| &= |A_{42}| = 12
\end{aligned}
$$
and $A_N = \emptyset$ for all other values of $N$.
b) Find the least positive integer $n$ such that $2009$ divides $a^n - 1$ for all integers $a$ relatively prime to $2009$.