Call a real-valued function $f$ *very convex* if
\[ \frac{f(x)+f(y)}{2}
\ge f\left( \frac{x+y}{2} \right) + \left\lvert x-y \right\rvert \]
holds for all real numbers $x$ and $y$.
Prove that no very convex function exists.
Let $S$ be the set of all triangles $ABC$ for which
\[ 5 \left( \frac{1}{AP} + \frac{1}{BQ} + \frac{1}{CR} \right)
- \frac{3}{\min\{ AP, BQ, CR \}} = \frac{6}{r}, \]
where $r$ is the inradius and $P$, $Q$, $R$ are the points of tangency
of the incircle with sides $AB$, $BC$, $CA$ respectively.
Prove that all triangles in $S$ are isosceles and similar to one another.
A game of solitaire is played with $R$ red cards,
$W$ white cards, and $B$ blue cards.
A player plays all the cards one at a time.
With each play he accumulates a penalty.
If he plays a blue card, then he is charged a penalty
which is the number of white cards still in his hand.
If he plays a white card, then he is charged a penalty
which is twice the number of red cards still in his hand.
If he plays a red card, then he is charged a penalty
which is three times the number of blue cards still in his hand.
Find, as a function of $R$, $W$, and $B$, the minimal total penalty
a player can amass and the number of ways in which this minimum can be achieved.
Find the smallest positive integer $n$ such
that if $n$ squares of a $1000 \times 1000$ chessboard are colored,
then there will exist three colored squares
whose centers form a right triangle with sides
parallel to the edges of the board.
Let $A_1 A_2 A_3$ be a triangle,
and let $\omega_1$ be a circle in its plane
passing through $A_1$ and $A_2$.
Suppose there exists circles $\omega_2$, $\omega_3$, …, $\omega_7$
such that for $k = 2, 3, …, 7,$
circle $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes
through $A_k$ and $A_{k+1}$ (indices mod $3$).
Prove that $\omega_7 = \omega_1$.