## IMO 2016

This year’s International Mathematical Olympiad (IMO) took place in Hong Kong from 6-16 July. The problems can be downloaded from this page or viewed at the Art of Problem Solving (AoPS) forum page for IMO 2016 (here).

Congratulations to my country, Singapore’s team, for coming in 4th overall! And for winning the country’s:

• Best medal haul (4G 2S, previous best was 4G 1S 1B in 2011)
• Highest absolute score (196 out of a possible 252, previous best was 179 in 2011)
• 2nd best overall team ranking (4th, best was 3rd in 2011)

Some factoids on the results:

• The Gold, Silver and Bronze medal score cutoffs were 29, 22 and 16 respectively. Nothing too out of the ordinary. (Interestingly, IMO 2014 had the exact same cutoffs.)
• The top 3 countries were the United States, South Korea and China. (These 3 countries were the top 3 in 2015 as well, just a different order: USA, China, South Korea.)
• Team USA won a best possible 6 Gold medals. (Next best was 4G 2S by South Korea, China and Singapore.)
• There were 6 contestants with perfect scores: 3 from South Korea, 2 from USA and 1 from China.
• 3 of them were first-time participants of the competition. 2 of them had won Gold in 2015, while the remaining contestant (Allen Liu, USA) had won Gold in 2014 and 2015.
• Question 3 was the most difficult question, with a mean score of 0.251, 91% of students scoring 0 points and only 10 contestants scoring a perfect 7 points. The easiest was Question 1 with a mean score of 5.272. (The questions from easiest to hardest: 1, 4, 2, 5, 6, 3.)

## [Soln] 2013 AIME I Problem 15

2013 AIME I 15. Let $N$ be the number of ordered triples $(A, B, C)$ of integers satisfying the conditions

(a) $0 \leq A < B < C \leq 99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0 \leq b < a < c < p$,
(c) $p$ divides $A - a$, $B - b$, and $C - c$, and
(d) each ordered triple $(A, B, C)$ and each ordered triple $(b, a, c)$ form arithmetic sequences.

Find $N$.

## 2013 AIME I Problem 15

2013 AIME I 15. Let $N$ be the number of ordered triples $(A, B, C)$ of integers satisfying the conditions

(a) $0 \leq A < B < C \leq 99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0 \leq b < a < c < p$,
(c) $p$ divides $A - a$, $B - b$, and $C - c$, and
(d) each ordered triple $(A, B, C)$ and each ordered triple $(b, a, c)$ form arithmetic sequences.

Find $N$.

## [Soln] 2013 AIME I Problem 14

2013 AIME I 14. For $\pi \leq \theta < 2\pi$, let

\begin{aligned} P &= \displaystyle\frac{1}{2}\cos \theta - \frac{1}{4} \sin 2\theta - \frac{1}{8} \cos 3 \theta + \frac{1}{16} \sin 4\theta + \frac{1}{32}\cos 5\theta \\ &\hspace{2em} - \frac{1}{64}\sin 6\theta - \frac{1}{128} \cos 7\theta + \dots \end{aligned}

and

\begin{aligned} Q &= 1 - \displaystyle\frac{1}{2}\sin \theta - \frac{1}{4} \cos 2\theta + \frac{1}{8} \sin 3 \theta + \frac{1}{16} \cos 4\theta - \frac{1}{32}\sin 5\theta \\ &\hspace{2em} - \frac{1}{64}\cos 6\theta + \frac{1}{128} \sin 7\theta + \dots \end{aligned}

so that $\frac{P}{Q} = \frac{2\sqrt{2}}{7}$. Then $\sin \theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

## 2013 AIME I Problem 14

2013 AIME I 14. For $\pi \leq \theta < 2\pi$, let

\begin{aligned} P &= \displaystyle\frac{1}{2}\cos \theta - \frac{1}{4} \sin 2\theta - \frac{1}{8} \cos 3 \theta + \frac{1}{16} \sin 4\theta + \frac{1}{32}\cos 5\theta \\ &\hspace{2em} - \frac{1}{64}\sin 6\theta - \frac{1}{128} \cos 7\theta + \dots \end{aligned}

and

\begin{aligned} Q &= 1 - \displaystyle\frac{1}{2}\sin \theta - \frac{1}{4} \cos 2\theta + \frac{1}{8} \sin 3 \theta + \frac{1}{16} \cos 4\theta - \frac{1}{32}\sin 5\theta \\ &\hspace{2em} - \frac{1}{64}\cos 6\theta + \frac{1}{128} \sin 7\theta + \dots \end{aligned}

so that $\frac{P}{Q} = \frac{2\sqrt{2}}{7}$. Then $\sin \theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2013 AIME I 13. Triangle $AB_0C_0$ has side lengths $AB_0 = 12$$B_0C_0 = 17$$C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$ respectively, creating three similar triangles $\Delta AB_nC_n \sim \Delta B_{n-1}C_nC_{n-1} \sim \Delta AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n \geq 1$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
2013 AIME I 13. Triangle $AB_0C_0$ has side lengths $AB_0 = 12$$B_0C_0 = 17$$C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$ respectively, creating three similar triangles $\Delta AB_nC_n \sim \Delta B_{n-1}C_nC_{n-1} \sim \Delta AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n \geq 1$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $q$.