2025 AIME II Problems/Problem 12

Problem

Let $A_1A_2\dots A_{11}$ be a non-convex $11$ -gon such that

• The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$ ,

• $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$ ,

• The perimeter of $A_1A_2\dots A_{11}$ is $20$ .

If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$ , find $m+n+p+q$ .

Solution 1

Set $A_1A_2 = x$ and $A_1A_3 = y$ . By the first condition, we have $\frac{1}{2}xy\sin\theta = 1$ , where $\theta = \angle A_2 A_1 A_3$ . Since $\cos\theta = \frac{12}{13}$ , we have $\sin\theta = \frac{5}{13}$ , so $xy = \frac{26}{5}$ . Repeating this process for $\triangle A_i A_1 A_{i+1}$ , we get $A_1A_2 = A_1A_4 = \ldots A_1A_{10} = x$ and $A_1A_3 = A_1A_5 = \ldots A_1A_{11} = y$ . Since the included angle of these $9$ triangles is $\theta$ , the square of the third side is $x^2 + y^2 - 2xy\cos\theta = x^2 + y^2 - \frac{52}{5}\cdot \frac{12}{13} = x^2 + y^2 - \frac{48}{5} = (x+y)^2 - 20.$ Thus the third side has length $\sqrt{(x+y)^2 - 20}.$ The perimeter is constructed from $9$ of these lengths, plus $A_{11}A_1 + A_1A_2 = x + y$ , so $9\sqrt{(x+y)^2 - 20} + x + y = 20$ . We seek the value of $x + y,$ so let $x + y = a$ so $\begin{align*} 9\sqrt{a^2 - 20} + a &= 20\\ 81(a^2 - 20) &= 400 - 40a + a^2\\ 4a^2 + 2a - 101 &= 0 \\ a &= \frac{-2 \pm \sqrt{1620}}{8} = \frac{-1 \pm \sqrt{405}}{4} = \frac{-1 \pm 9\sqrt{5}}{4}. \end{align*}$ Taking the positive solution gives $m + n + p + q = 1 + 9 + 5 + 4 = \boxed{\textbf{(019)}}.$

-Benedict T (countmath1)

Solution 2(Very similar to Solution 1)

Let $A_1 A_2 = a_1, A_1 A_3 = a_2, ... A_1 A_{10} = a_9, A_1 A_11 a_{10}$ . Now the area of each triangle are build up of these lengths multiplied by the sine of the angle in between. This angle's cosine is given to us to be $\frac{12}{13}$ . We recognize this as a $5-12-13$ triangle hence the sine of the angle will be simply $\frac{5}{13}$ . Hence each area will be $\frac{1}{2} \cdot a_n a_{n + 1} \cdot \frac{5}{13} = 1 \implies a_n a_{n+ 1} = \frac{26}{5}$ . Therefore, we have:

$a_1 a_2 = \frac{26}{5}$

$a_2 a_3 = \frac{26}{5}$ . . .

$a_9 a_{10} = \frac{26}{5}$

$a_{10} a_1 = \frac{26}{5}$

Now note that the perimeter will be build up of all the other lengths. For instance, $A_2 A_3$ will be part of this perimeter which happens to be the only unknown side of $\triangle A_1 A_2 A_3$ . By Law of Cosines

$(A_2 A_3)^2 = a_1^{2} + a_2^{2} - 2a_1 a_2 \frac{12}{13}$ after substituting the cosine value. We know $a_1 a_2 = \frac{26}{5}$ so substituting this in we get

$(A_2 A_3)^2 = a_1^{2} + a_2^{2} - \frac{48}{5} \implies A_2 A_3 = \sqrt{a_1^{2} + a_2^{2} - \frac{48}{5}}$ . By symmetry, we conclude

$\sqrt{a_1^{2} + a_2^{2} - \frac{48}{5}} + ... + \sqrt{a_{10}^{2} + a_1^{2} - \frac{48}{5}} + a_1 + a_2 = 20$ because we need to account for $A_1 A_2 + A_1 A_11$ as they are actually known sides that are included in the perimeter.

Note that from the $a_1 a_2 = \frac{26}{5}$ system of equations, we can see clearly that

$a_1 = a_3$

$a_2 = a_4$ . . .

$a_8 = a_{10}$

$a_9 = a_1$

$a_{10} = a_2$

So we see that $a_{2n + 1} = a_1$ and $a_{2n} = a_2$ so we can substitute this in to get

$\sqrt{a_1^{2} + a_2^{2} - \frac{48}{5}} + ... + \sqrt{a_1^{2} + a_2^{2} - \frac{48}{5}} + a_1 + a_2 = 20$

Now the square root terms happen $9$ times as there are $11$ sides and two of them are $a_1, a_2$ given at the end of the $LHS$ . So we conclude:

$9\sqrt{a_1^{2} + a_2^{2} - \frac{48}{5}} + a_1 + a_2 = 20$

Note that $a_1^{2} + a_2^{2} - \frac{48}{5} = (a_1 + a_2)^2 - 2a_1 a_2 - \frac{48}{5} = (a_1 + a_2)^2 - \frac{52 + 48}{5} = (a_1 + a_2)^2 - 20$ . Note that we want to find $A_1 A_2 + A_1 A_11 = a_1 + a_{10} = a_1 + a_2$ . So we let $a_1 + a_2 = x$ to get

$9\sqrt{x^2 - 20} + x = 20 \implies 81(x^2 - 20) = (20 - x)^2 \implies 81(x^2 - 20) = 400 - 40x + x^2 \implies 81x^2 - 1620 = 400 - 40x + x^2 \implies 80x^2 + 40x - 2020 = 0 \implies 4x^2 + 2x - 101 = 0 \implies x = \frac{-1 \pm 9\sqrt{5}}{4}$ . Now take the positive value to get

$x = \frac{9\sqrt{5} - 1}{4} \implies 9 + 5 + 1 + 4 = \boxed{19}$ .

~ilikemath24736 ~Sylesh (Latex errors)

2025 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2025 AIME II Problems/Problem 12

Contents

Problem

Solution 1

Solution 2(Very similar to Solution 1)

See also