MathDB

Problem 1

Part of 1999 Mongolian Mathematical Olympiad

Problems(5)

3^k|m^3+10, existence of infinite m for any k

Source: Mongolia 1999 Grade 8 P1

5/4/2021
Prove that for any positive integer kk there exist infinitely many positive integers mm such that 3k∣m3+103^k\mid m^3+10.
number theoryDivisibility
painting unit cells in the plane

Source: Mongolia 1999 Grade 9 P1

5/4/2021
The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the 1616 possible colored 2×22\times2 squares.
geometrycombinatoricscombinatorial geometry
exists k:k*2^n+1 is composite for any n in N

Source: Mongolia 1999 Grade 10 P1

5/5/2021
Prove that for any nn there exists a positive integer kk such that all the numbers k⋅2s+1 (s=1,…,n)k\cdot2^s+1~(s=1,\ldots,n) are composite.
number theory
convex quadrilateral

Source:

7/20/2013
In a convex quadrilateral ABCDABCD, ∠ABD=65∘{\angle}ABD=65^\circ,∠CBD=35∘{\angle}CBD=35^\circ, ∠ADC=130∘{\angle}ADC=130^\circ and BC=ABBC=AB.Find the angles of ABCDABCD.
geometry unsolvedgeometry
weird FE, f(x)≠f(x+h), f is semiconstant

Source: Mongolia 1999 Teachers secondary level P1

5/6/2021
Suppose that a function f:R→Rf:\mathbb R\to\mathbb R is such that for any real hh there exist at most 1999050919990509 different values of xx for which f(x)≠f(x+h)f(x)\ne f(x+h). Prove that there is a set of at most 99952569995256 real numbers such that ff is constant outside of this set.
fefunctional equationalgebra