MathDB

2

Part of 2022 Mexican Girls' Contest

Problems(2)

4 numbers game

Source: 1st National Women's Contest of Mexican Mathematics Olympiad 2022 , levels 1+2 p2

9/17/2022
In the training of a state, the coach proposes a game. The coach writes four real numbers on the board in order from least to greatest: a<b<c<da < b < c < d.
Each Olympian draws the figure on the right in her notebook and arranges the numbers inside the corner shapes, however she wants, putting a number on each one. Once arranged, on each segment write the square of the difference of the numbers at its ends. Then, add the 44 numbers obtained. https://cdn.artofproblemsolving.com/attachments/9/a/ea348c637ae266c908e0b97e64605808b3b1d2.png For example, if Vania arranges them as in the figure on the right, then the result would be (c−b)2+(b−a)2+(a−d)2+(d−c)2. (c - b)^2 + (b- a)^2 + (a - d)^2 + (d - c)^2. https://cdn.artofproblemsolving.com/attachments/8/b/9c5375d66a4a6344b2bce333534fa7fac2ad6c.png The Olympians with the lowest result win. In what ways can you arrange the numbers to win? Give all the possible solutions.
combinatorics
find the value of an angles

Source: 1st National Women&acute;s Contest of Mexican Mathematics Olympiad 2022, problem 2 teams

7/23/2023
Consider △ABC\triangle ABC an isosceles triangle such that AB=BCAB = BC. Let PP be a point satisfying
∠ABP=80∘,∠CBP=20∘,andAC=BP\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP
Find all possible values of ∠BCP\angle BCP.
geometryisoscelesMexicoangles