2022 AIME I Problems/Problem 14
Contents
Problem
Given and a point on one of its sides, call line the splitting line of through if passes through and divides into two polygons of equal perimeter. Let be a triangle where and and are positive integers. Let and be the midpoints of and , respectively, and suppose that the splitting lines of through and intersect at . Find the perimeter of .
The Geometry Part - Solution 1
Consider the splitting line through . Extend on ray such that . Then the splitting line bisects segment , so in particular it is the midline of triangle and thus it is parallel to . But since triangle is isosceles, we can easily see is parallel to the angle bisector of , so the splitting line is also parallel to this bisector, and similar for the splitting line through . Some simple angle chasing reveals the condition is now equivalent to .
- MortemEtInteritum
The Geometry Part - Solution 2
Let and be the splitting lines. Reflect across to be and across to be . Take and , which are spiral similarity centers on the other side of as such that and . This gets that because and , then and are on 's circumcircle. Now, we know that and so because and , then and and and .
We also notice that because and correspond on and , and because and correspond on and , then the angle formed by and is equal to the angle formed by and which is equal to . Thus, . Similarly, and so and .
- kevinmathz
The NT Part
We now need to solve . A quick check gives that and . Thus, it's equivalent to solve .
Let be one root of . Then, recall that is the ring of integers of and is a unique factorization domain. Notice that . Therefore, it suffices to find an element of with the norm .
To do so, we factor in . Since it's , it must split. A quick inspection gives . Thus, , so \begin{align*} 73^2 &= N((8-\omega)^2) \\ &= N(64 - 16\omega + \omega^2) \\ &= N(64 - 16\omega + (-1-\omega)) \\ &= N(63 - 17\omega), \end{align*}giving the solution and , yielding and , so the sum is . Since and are primes in , the solution must divide . One can then easily check that this is the unique solution.
- MarkBcc168
Solution (Geometry + Number Theory)
Denote , , .
Let the splitting line of through (resp. ) crosses at another point (resp. ).
WLOG, we assume .
: .
We extend segment to , such that . We extend segment to , such that .
In this case, is the midpoint of , and is the midpoint of .
Because and are the midpoints of and , respectively, . Because and are the midpoints of and , respectively, .
Because , . Because , .
Let and intersect at . Because and , the angle formed between lines and is congruent to . Hence, or .
We have
Hence, we must have , not . Hence, .
This implies and . This contradicts the condition specified for this case.
Therefore, this case is infeasible.
: .
We extend segment to , such that . We extend segment to , such that .
In this case, is the midpoint of , and is the midpoint of .
Because and are the midpoints of and , respectively, . Because and are the midpoints of and , respectively, .
Because , . Because , .
Let be a point of , such that . Hence, .
Because and and , the angle formed between lines and is congruent to . Hence, or .
We have
Hence, we must have , not . Hence, .
This implies and . This contradicts the condition specified for this case.
Therefore, this case is infeasible.
: .
We extend segment to , such that . We extend segment to , such that .
In this case, is the midpoint of , and is the midpoint of .
Because and are the midpoints of and , respectively, . Because and are the midpoints of and , respectively, .
Because , . Because , .
Because and , the angle formed between lines and is congruent to . Hence, or .
We have
Hence, we must have , not . Hence, .
In , by applying the law of cosines, we have
Because , we have
Now, we find integer solution(s) of this equation with .
Multiplying this equation by 4, we get
Denote . Because , .
Because , . Thus, . This implies .
We also have . Hence, . This implies .
Denote and . Hence, . Hence, Equation (1) can be written as
Now, we solve this equation.
First, we find an upper bound of .
We have . Hence, . Hence, . Because is an integer, we must have .
Second, we find a lower bound of .
We have . Hence, . Hence, . Because is an integer, we must have .
Now, we find the integer solutions of and that satisfy Equation (2) with .
First, modulo 9,
Hence .
Second, modulo 5,
Because , we must have . Hence, .
Third, modulo 7,
Because , we must have . Hence, .
Given all conditions above, the possible are 74, 83, 88, 92, 97, 101, 106, 109, 116, 118, 127.
By testing all these numbers, we find that the only solution is . This implies .
Hence, and . Hence, .
Therefore, the perimeter of is .
~Steven Chen (www.professorchenedu.com)
Video Solution
https://www.youtube.com/watch?v=kkous52vPps&t=3023s
~Steven Chen (wwww.professorchenedu.com)
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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