MathDB

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Part of 2022 Kosovo & Albania Mathematical Olympiad

Problems(2)

Equivalence between a right angle and sum of two segments in a square

Source: KAMO 2022 Grades 7-8 P3

7/5/2022
Let ABCDABCD be a square and let MM be the midpoint of BCBC. Let XX and YY be points on the segments ABAB and CDCD, respectively. Prove that ∠XMY=90∘\angle XMY = 90^\circ if and only if BX+CY=XYBX + CY = XY.
Note: In the competition, students were only asked to prove the 'only if' direction.
geometry
Difference of sum of squares of parts of a bipartition

Source: KAMO 2022 Grade 9 P3

7/5/2022
Is it possible to partition {1,2,3,…,28}\{1, 2, 3, \ldots, 28\} into two sets AA and BB such that both of the following conditions hold simultaneously:
(i) the number of odd integers in AA is equal to the number of odd integers in BB;
(ii) the difference between the sum of squares of the integers in AA and the sum of squares of the integers in BB is 1616?
number theory