MathDB

4

Part of 2009 Postal Coaching

Problems(6)

ratio of areas of some 2 triangle PAB, PBC, PCD, PDA$ lies in interval [1/a, a]

Source: Indian Postal Coaching 2009 set 2 p4

5/26/2020
Determine the least real number a>1a > 1 such that for any point PP in the interior of a square ABCDABCD, the ratio of the areas of some two triangle PAB,PBC,PCD,PDAPAB, PBC, PCD, PDA lies in the interval [1/a,a][1/a, a].
areassquaregeometry
n -gonal Pythagorean triples, P(n, r) = (n - 2)r^2/2 - (n - 4) r/2

Source: Indian Postal Coaching 2009 set 1 p4

5/26/2020
For positive integers n≥3n \ge 3 and r≥1r \ge 1, define P(n,r)=(n−2)r22−(n−4)r2P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2} We call a triple (a,b,c)(a, b, c) of natural numbers, with a≤b≤ca \le b \le c, an nn-gonal Pythagorean triple if P(n,a)+P(n,b)=P(n,c)P(n, a)+P(n, b) = P(n, c). (For n=4n = 4, we get the usual Pythagorean triple.)
(a) Find an nn-gonal Pythagorean triple for each n≥3n \ge 3.
(b) Consider all triangles ABCABC whose sides are nn-gonal Pythagorean triples for some n≥3n \ge 3. Find the maximum and the minimum possible values of angle CC.
algebraanglesminmaxnumber theory
2s_1 <= s, semiperimeter inequality, incircle of 1 is circumcircle of other

Source: Indian Postal Coaching 2009 set 3 p4

5/26/2020
Let ABCABC be a triangle, and let DEFDEF be another triangle inscribed in the incircle of ABCABC. If ss and s1s_1 denote the semiperimeters of ABCABC and DEFDEF respectively, prove that 2s1≤s2s_1 \le s. When does equality hold?
geometrycircumcircleTrianglesperimeterGeometric Inequalities
4-digit number wanted, remainder by 37 related

Source: Indian Postal Coaching 2009 set 5 p4

5/26/2020
A four - digit natural number which is divisible by 77 is given. The number obtained by writing the digits in reverse order is also divisible by 77. Furthermore, both the numbers leave the same remainder when divided by 3737. Find the 4-digit number.
Digitsremaindernumber theory
8 heaps of 251 coins each, 251 heaps of 8 coins each, regular 2008-gon

Source: Indian Postal Coaching 2009 set 4 p4

5/26/2020
At each vertex of a regular 20082008-gon is placed a coin. We choose two coins and move each of them to an adjacent vertex, one in the clock-wise direction and the other in the anticlock-wise direction. Determine whether or not it is possible, by making several such pairs of moves, to move all the coins into (a) 88 heaps of 251251 coins each, (b) 251251 heaps of 88 coins each.
combinatorics
integers from 1 to 100 are arranged in a 10x10 table

Source: Indian Postal Coaching 2009 set 6 p4

5/26/2020
All the integers from 11 to 100100 are arranged in a 10×1010 \times 10 table as shown below. Prove that if some ten numbers are removed from the table, the remaining 9090 numbers contain 10 numbers in Arithmetic Progression. 1    2    3    ...  101 \,\,\,\,2\,\, \,\,3 \,\,\,\,... \,\,10 11  12  13  ...  2011 \,\,12 \,\,13 \,\,... \,\,20   .    .   .\,\,.\,\,\,\,.\,\,\,.   .    .    .\,\,.\,\,\,\,.\,\,\,\,. 91  92  93  ...  10091 \,\,92 \,\,93\,\, ... \,\,100
combinatorics