MathDB

1

Part of 2017 Korea Winter Program Practice Test

Problems(4)

Six tangent circles

Source: 2017 Korean Winter Program Practice Test 1 Day 1 #1

1/18/2017
Let γ1,γ2,γ3\gamma_1, \gamma_2, \gamma_3 be mutually externally tangent circles and Γ1,Γ2,Γ3\Gamma_1, \Gamma_2, \Gamma_3 also be mutually externally tangent circles. For each 1≤i≤31 \le i \le 3, γi\gamma_i and Γi+1\Gamma_{i+1} are externally tangent at AiA_i, γi\gamma_i and Γi+2\Gamma_{i+2} are externally tangent at BiB_i, and γi\gamma_i and Γi\Gamma_i do not meet. Show that the six points A1,A2,A3,B1,B2,B3A_1, A_2, A_3, B_1, B_2, B_3 lie on either a line or a circle.
geometrycircles
Inequality on a function

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #1

1/21/2017
Let f:Z→Rf : \mathbb{Z} \to \mathbb{R} be a function satisfying f(x)+f(y)+f(z)≥0f(x) + f(y) + f(z) \ge 0 for all integers x,y,zx, y, z with x+y+z=0x + y + z = 0. Prove that f(−2017)+f(−2016)+⋯+f(2016)+f(2017)≥0. f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0.
inequalitiesfunction
Find two sets of successive integers

Source: 2017 Korea Winter Program Practice Test 2 #1

8/14/2019
For every positive integers n,mn,m, show that there exist two sets A,BA,B which satisfy the following.
[*]AA is a set of nn successive positive integers, and BB is a set of mm successive positive integers. [*]A∪B=ϕA\cup B = \phi [*]For every a∈Aa\in A and b∈Bb\in B, aa and bb are not relatively prime.
number theory
Number of integer pair

Source: 2017 Korea Winter Program Practice Test 2 #5

8/14/2019
Find all prime number pp such that the number of positive integer pair (x,y)(x,y) satisfy the following is not 2929.
[*]1≤x,y≤291\le x,y\le 29 [*]29∣y2−xp−2629\mid y^2-x^p-26
number theory