MathDB

7

Part of 2015 IFYM, Sozopol

Problems(5)

Problem 7 of First round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

1/13/2020
Let ABCDABCD be a trapezoid, where AD∥BCAD\parallel BC, BC<ADBC<AD, and AB∩DC=TAB\cap DC=T. A circle k1k_1 is inscribed in ΔBCT\Delta BCT and a circle k2k_2 is an excircle for ΔADT\Delta ADT which is tangent to ADAD (opposite to TT). Prove that the tangent line to k1k_1 through DD, different than DCDC, is parallel to the tangent line to k2k_2 through BB, different than BABA.
geometryParallel Lines
Problem 7 of Third round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/24/2019
Determine the greatest natural number nn, such that for each set SS of 2015 different integers there exist 2 subsets of SS (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by nn.
number theoryset theorysums
Problem 7 of Second round - &quot;multicolored&quot; figures

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/19/2019
A corner with arm nn is a figure made of 2n−12n-1 unit squares, such that 2 rectangles 11 x (n−1)(n-1) are connected to two adjacent sides of a square 11 x 11, so that their unit sides coincide. The squares or a chessboard 100100 x 100100 are colored in 15 colors. We say that a corner with arm 8 is “multicolored”, if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be “mutlticolored”?
combinatoricstableChessboardColoring
Problem 7 of Fourth round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/31/2019
In a square with side 1 are placed nn equilateral triangles (without having any parts outside the square) each with side greater than 23\sqrt{\frac{2}{3}}. Prove that all of the nn equilateral triangles have a common inner point.
geometryEquilateral Triangle
Algebra(Polynomial)

Source: Problems From Previous Olympiad

10/6/2014
Determine all polynomials P(x)P(x) with real coefficients such that (x+1)P(x−1)−(x−1)P(x)(x+1)P(x-1)-(x-1)P(x) is a constant polynomial.
algebrapolynomialalgebra unsolved