AIME 2026 I · 第 5 题

Solution 1

The points $A$, $B$, $A'$, $B'$, make a parallelogram (because we have a pair of equal parallel sides) with one pair of sides of length $1$, diagonals of $\frac{4}{3}$ and $1$. The diagonals split the parallelogram into four triangles. Because parallelogram diagonals bisect each other, we know that one of the triangles containing $\theta$ will have side lengths of $\frac{2}{3}$ opposite of $\frac{1}{2}$, and $1$. Using law of cosines, we can find that $cos(\theta)=29/36$, and the answer is $\boxed{065}$.

~ Logibyte

Note: We have diagonals of length $1$ because a rotated point sweeps out a with radius $1$. Here, the wedge is the circle with center $A$ and arc $BB'$. ~math660

Solution 2 (Straight to the point)

Notice that $\angle ABA ' = \theta$, $\angle A ' B B ' = 90 - \frac{\theta}{2}$, so $\angle ABB ' = 90 + \frac{\theta}{2}$. Law of Cosines on $\triangle ABB '$ gives $\sin ^2 (\frac{\theta}{2}) = \frac{7}{72}$, so double angle formula shows $\cos \theta = \frac{29}{36}$ and the sum is $29 + 36 = \boxed {065}$.

~ Ddk001

Solution 3 (Outline how to solve)

Drawing the diagram we get parallel lines (due to congruent angles). We can then use congruent triangles to realize that the side lengths of the triangle are $1, 1/2,$ and $2/3$. (1/2 is half of 1 and 2/3 is half of 4/3).We then use law of cosines on this triangle and get cos theta =$\frac{29}{36}$ we then get $\boxed{65}$ by answer extraction

~Aarav22 (I did not make AIME).

Solution 4 (Pythagorean Theorem, no fancy LoC)

Drawing the diagram out and placing an altitude from $A'$ perpendicular to $AB$, we can see that $AB'$ is simply a diagonal of the parallelogram.

Therefore, we can create an equation in terms of $\theta$. The total horizontal distance is equal to $1 + 1 - (overlap)$, and we can see that overlap is simply just $\cos \theta$ (by simple right-triangle trigonometry definitions). Thus, the total horizontal distance is $2 - \cos \theta$.

Next, we can see that the total vertical distance is just $\sin \theta$, so now we can create our equation:

$(2 - \cos \theta) ^2 + (\sin \theta) ^2 = (4/3)^2$.

Now, we let $\cos \theta = a$, and $\sin \theta$ as $\sqrt{1-a^2}$ (by the Pythagorean Identity).

This eventually turns into a linear equation and we can see that the answer is $29/36$ thus $\boxed{065}$.

~notvalid (I got 3 on AIME and 70.5/99 on amc10)

~ $\LaTeX$ by Logibyte

~shaodavin (minor edit)

Video Solution (Fast and Easy 🔥🚀)

2026 AIME I #5

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