MathDB

5

Part of 2018 IFYM, Sozopol

Problems(5)

Problem 5 of First round

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

9/21/2018
Find the solutions in prime numbers of the following equation
p4+q4+r4+119=s2.p^4 + q^4 + r^4 + 119 = s^2 .
number theoryprime numbers
Problem 5 of Third round

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

9/22/2018
On the sides ABAB,BCBC, and CACA of △ABC\triangle ABC are chosen points C1C_1, A1A_1, and B1B_1 respectively, in such way that AA1AA_1, BB1BB_1, and CC1CC_1 intersect in one point XX. If ∠A1C1B=∠B1C1A\angle A_1C_1B = \angle B_1C_1A, prove that CC1CC_1 is perpendicular to ABAB.
geometry
Problem 5 of Second round

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

9/22/2018
Point XX lies in a right-angled isosceles △ABC\triangle ABC (∠ABC=90∘\angle ABC = 90^\circ). Prove that
AX+BX+2CX≥5ABAX+BX+\sqrt{2}CX \geq \sqrt{5}AB
and find for which points XX the equality is met.
geometry
Problem 5 of Fourth round

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

9/22/2018
Find all functions f:[0,+∞)→[0,+∞)f :[0, +\infty) \rightarrow [0, +\infty) for which
f(f(x)+f(y))=xyf(x+y)f(f(x)+f(y)) = xy f (x+y)
for every two non-negative real numbers xx and yy.
functional equationalgebra
Problem 5 of Finals

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

9/22/2018
On the extension of the heights AH1AH_1 and BH2BH_2 of an acute △ABC\triangle ABC, after points H1H_1 and H2H_2, are chosen points MM and NN in such way that
∠MCB=∠NCA=30∘\angle MCB = \angle NCA = 30^\circ.
We denote with C1C_1 the intersection point of the lines MBMB and NANA. Analogously we define A1A_1 and B1B_1. Prove that the straight lines AA1AA_1, BB1BB_1, and CC1CC_1 intersect in one point.
geometrygeometry unsolved