MathDB

5

Part of 2016 IFYM, Sozopol

Problems(5)

Problem 5 of First round - Factorials and powers

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

8/29/2019
Find all pairs of integers (x,y)(x,y) for which xz+zx=(x+z)!x^z+z^x=(x+z)!.
number theoryfactorial
Problem 5 of Second round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

8/31/2019
Points KK and LL are inner for ABAB for an acute ΔABC\Delta ABC, where KK is between AA and LL. Let P,QP,Q, and HH be the feet of the perpendiculars from AA to CKCK, from BB to CLCL, and from CC to ABAB, respectively. Point MM is the middle point of ABAB. If PH∩AC=XPH\cap AC=X and QH∩BC=YQH\cap BC=Y, prove that points H,P,MH,P,M, and QQ lie on one circle, if and only if the lines AY,BXAY,BX, and CHCH intersect in one point.
geometry
Problem 5 of Third round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/1/2019
We are given a ΔABC\Delta ABC with ∠BAC=39∘\angle BAC=39^\circ and ∠ABC=77∘\angle ABC=77^\circ. Points MM and NN are chosen on BCBC and CACA respectively, so that ∠MAB=34∘\angle MAB=34^\circ and ∠NBA=26∘\angle NBA=26^\circ. Find ∠BNM\angle BNM.
geometry
Problem 5 of Fourth round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/3/2019
A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides. a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle. b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic.
geometry
Problem 5 of Finals

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/19/2019
Prove that for an arbitrary ΔABC\Delta ABC the following inequality holds: lama+lbmb+lcmc>1\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1, Where la,lb,lcl_a,l_b,l_c and ma,mb,mcm_a,m_b,m_c are the lengths of the bisectors and medians through AA, BB, and CC.
geometric inequalitygeometry