Math IOQM

Q1. Let n be a positive integer such that 1 ≤ n ≤ 1000 . Let M n be the number of integers in the set X n = { 4 n + 1 , 4 n + 2 , … , 4 n + 1000 } . Let a = max ⁡ { M n : 1 ≤ n ≤ 1000 } , and b = min ⁡ { M n : 1 ≤ n ≤ 1000 } . a = \max\{M_n : 1 \leq n \leq 1000\}, \quad \text{and} \quad b = \min\{M_n : 1 \leq n \leq 1000\}. Find a − b a - b . Solution: Quick tip: If n^2 = k then (n+1)^2 = k + (2n + 1) We will use this here. Also as the numbers grow larger the gap between 2 perfect squares becomes less and less. For. e.g. there are 10 perfect squares between 1 and 100 but only 4 perfect squares between 101 and 200. So X1 = {sqrt(5) ... sqrt(1004)} will have the most number of perfect squares. While X1000 = {sqrt(4001)...sqrt(5000)} will have the least. In X1 the first integer square root is 3 and we know that 1024 is the square of 32 so the last integer will be 31. Total: 31 - 3 + 1 = 29 integers in X1. In X1000, We know that 64^2 = 4096 so 64 is the first integer. 70^2 = 4900 and ...

IOQM 2023 Q5: In a triangle ABC , let E be the midpoint of AC and F be the midpoint of AB . The medians BE and CF intersect at G . Let Y and Z be the midpoints of BE and CF respectively. If the area of triangle ABC is 480 , find the area of triangle GYZ . Answer 10:  Solution: 1. 3 medians intersecting at centroid divide the triangle into 6 triangles of equal area. 2. GBC has 2 of those triangles so its area is 1/3 of the total area. 3. Look at GYZ. GB is 2/3 of the median BE. BY is 1/2 of the median BE. So GY = 2/3 - 1/2 = 1/6BE. GYZ and GBC are similar with side ratio of 1:4. So [GYZ] = 1/16[GBC] = 1/16 * 1/3 [ABC] = 480/16*3 = 10. Prerequisites: Centroid and median properties in triangle . Similarity criterion for triangles.