2004 USAMO Problems/Problem 1

Problem

(Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that

$$\frac {1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.$$

When does equality hold?

Solution 1

It suffices to show that $$|BC^3-CD^3| \leq 3|AB^3-AD^3|,$$ because the other side of the inequality follows by symmetry. By Pitot's Theorem, we have $AB-AD=BC-CD$. WLOG, let $AB \geq AD$. Then we have that $BC \geq CD$, so we must show $BC^3-CD^3 \leq 3(AB^3-AD^3)$. If $AB=AD$, we are done. Therefore, we will assume for the rest of the proof that $AB>AD$, $BC>CD$.

Then we would like to show that $$BC^2+BC\cdot CD+CD^2 \leq 3(AB^2+AB\cdot AD+AD^2).$$ By the Law of Cosines, we have that $$BD^2=BC^2+k_1(BC\cdot CD)+CD^2=AB^2+k_2(AB\cdot AD)+AD^2,$$ where $-1 \leq k_1, k_2 \leq 1$ (this is because $60^{\circ} \leq \angle BAD,\angle BCD \leq 120^{\circ}$). Therefore, $$3(BC^2-BC\cdot CD+CD^2) \leq 3(BC^2+k_1(BC\cdot CD)+CD^2)=3(AB^2+k_2(AB\cdot AD)+AD^2)\leq 3(AB^2+AB\cdot AD+AD^2).$$ Then we wish to show that $$3(BC^2-BC\cdot CD+CD^2)-(BC^2+BC\cdot CD+CD^2) \geq 0 \rightarrow 2(BC-CD)^2 \geq 0,$$ and we are done. $\blacksquare$

Solution 2

By a well-known property of tangential quadrilaterals, the sum of the two pairs of opposite sides are equal; hence $a + c = b + d \Rightarrow a - b = d - c \Rightarrow |a - b| = |d - c|$ Now we factor the desired expression into $\frac {|d - c|(c^2 + d^2 + cd)}{3} \le|a - b|(a^2 + b^2 + ab)\le 3|d - c|(c^2 + d^2 + cd)$. Temporarily discarding the case where $a = b$ and $c = d$, we can divide through by the $|a - b| = |d - c|$ to get the simplified expression $(c^2 + d^2 + cd)/3\le a^2 + b^2 + ab\le 3(c^2 + d^2 + cd)$.

Now, draw diagonal $BD$. By the law of cosines, $c^2 + d^2 + 2cd\cos A = BD$. Since each of the interior and exterior angles of the quadrilateral is at least 60 degrees, we have that $A\in [60^{\circ},120^{\circ}]$. Cosine is monotonically decreasing on this interval, so by setting $A$ at the extreme values, we see that $c^2 + d^2 - cd\le BD^2 \le c^2 + d^2 + cd$. Applying the law of cosines analogously to $a$ and $b$, we see that $a^2 + b^2 - ab\le BD^2 \le a^2 + b^2 + ab$; we hence have $c^2 + d^2 - cd\le BD^2 \le a^2 + b^2 + ab$ and $a^2 + b^2 - ab\le BD^2 \le c^2 + d^2 + cd$.

We wrap up first by considering the second inequality. Because $c^2 + d^2 - cd\le BD^2 \le a^2 + b^2 + ab$, $\text{RHS}\ge 3(a^2 + b^2 - ab)$. This latter expression is of course greater than or equal to $a^2 + b^2 + ab$ because the inequality can be rearranged to $2(a - b)^2\ge 0$, which is always true. Multiply the first inequality by $3$ and we see that it is simply the second inequality with the variables swapped; hence by symmetry it is true as well.

Equality occurs when $a = b$ and $c = d$, or when $ABCD$ is a kite.

Resources

 2004 USAMO (Problems • Resources) Preceded byFirst problem Followed byProblem 2 1 • 2 • 3 • 4 • 5 • 6 All USAMO Problems and Solutions