MathDB

B1

Part of 2018 Malaysia National Olympiad

Problems(3)

OMK 2018 Sulong, Section B Problem 1

Source: OMK 2018 Sulong, Section B Problem 1

6/21/2021
Let ABCABC be an acute triangle. Let DD be the reflection of point BB with respect to the line ACAC. Let EE be the reflection of point CC with respect to the line ABAB. Let Γ1\Gamma_1 be the circle that passes through A,BA, B, and DD. Let Γ2\Gamma_2 be the circle that passes through A,CA, C, and EE. Let PP be the intersection of Γ1\Gamma_1 and Γ2\Gamma_2 , other than AA. Let Γ\Gamma be the circle that passes through A,BA, B, and CC. Show that the center of Γ\Gamma lies on line APAP.
geometryProofcircle
square by n sticks of lengths 1,2, 3,..., n 2018 Malaysia OMK Intermediate B1

Source:

9/19/2021
Let nn be an integer. Dayang are given nn sticks of lengths 1,2,3,...,n1,2, 3,..., n. She may connect the sticks at their ends to form longer sticks, but cannot cut them. She wants to use all these sticks to form a square. For example, for n=8n = 8, she can make a square of side length 99 using these connected sticks: 1+81 + 8, 2+72 + 7, 3+63 + 6, and 4+54 + 5. How many values of nn, with 1≤n≤20181 \le n \le 2018, that allow her to do this?
combinatorics
OMK 2018 Bongsu, Section B Problem 1

Source: OMK 2018 Bongsu, Section B Problem 1

6/20/2021
Given two triangles with the same perimeter. Both triangles have integer side lengths. The first triangle is an equilateral triangle. The second triangle has a side with length 1 and a side with length dd. Prove that when dd is divided by 3, the remainder is 1.
Proofnumber theory