MathDB

3

Part of 2022 IFYM, Sozopol

Problems(5)

Problem 3 of First round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Let p1,p2,…,pnp_1,p_2,\dots ,p_n be all prime numbers lesser than 21002^{100}. Prove that
1p1+1p2+⋯+1pn<10\frac{1}{p_1} +\frac{1}{p_2} +\dots +\frac{1}{p_n} <10.
number theoryprime numbers
Problem 3 of Second round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
The set of quadruples (a,b,c,d)(a,b,c,d) where each of a,b,c,da,b,c,d is either 00 or 11 is called vertices of the four dimensional unit cube or 4-cube for short. Two vertices are called adjacent, if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from (0,0,0,0)(0,0,0,0) and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to (0,0,0,0)(0,0,0,0) after 40424042 turns? Note that it is NOT forbidden for the robot to pass through (0,0,0,0)(0,0,0,0) before the 40424042-nd turn.
combinatorics
min (AB + BC + CD + DA)/ (AC + BD) for tangential ABCD

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p3

11/13/2022
Quadrilateral ABCDABCD is circumscribed around circle kk. Gind the smallest possible value of AB+BC+CD+DAAC+BD\frac{AB + BC + CD + DA}{AC + BD}, as well as all quadrilaterals with the above property where it is reached.
geometrytangential quadrilateralgeometric inequalitytangential
Problem 3 of Third round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
The positive integers pp, qq are such that for each real number xx (x+1)p(x−3)q=xn+a1xn−1+a2xn−2+⋯+an−1x+an(x+1)^p (x-3)^q=x^n+a_1 x^{n-1}+a_2 x^{n-2}+\dots +a_{n-1} x+a_n where n=p+qn=p+q and a1,…,ana_1,\dots ,a_n are real numbers. Prove that there exists infinitely many pairs (p,q)(p,q) for which a1=a2a_1=a_2.
algebra
AK = KP if H'P_|_ AB

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, 4th round p3

11/12/2022
Given an acute-angled △AB\vartriangle ABC with altitude AHAH ( ∠BAC>45o>∠AB\angle BAC > 45^o > \angle ABC). The perpendicular bisector of ABAB intersects BCBC at point DD. Let KK be the midpoint of BFBF, where FF is the foot of the perpendicular from CC on ADAD. Point H′H' is the symmetric to HH wrt KK. Point PP lies on the line ADAD, such that H′P⊥ABH'P \perp AB. Prove that AK=KPAK = KP.
equal segmentsgeometry