MathDB

2

Part of 2022 IFYM, Sozopol

Problems(5)

Problem 2 of First round

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

9/9/2022
Does there exist a solution in integers for the equation
a2+b2+c2+d2+e2=abcde−78a^2+b^2+c^2+d^2+e^2=abcde-78
where a,b,c,d,e>2022a,b,c,d,e>2022?
algebra
Problem 2 of Second round

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

9/9/2022
Let ABCABC be a triangle with ∠BAC=40∘\angle BAC=40^\circ , OO be the center of its circumscribed circle and GG is its centroid. Point DD of line BCBC is such that CD=ACCD=AC and CC is between BB and DD. If AD∥OGAD\parallel OG, find ∠ACB\angle ACB.
geometryangles
Problem 2 of Third round

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

9/9/2022
We say that a rectangle and a triangle are similar, if they have the same area and the same perimeter. Let PP be a rectangle for which the ratio of the longer to the shorter side is at least λ−1+λ(λ−2)\lambda -1+\sqrt{\lambda (\lambda -2)} where λ=332\lambda =\frac{3\sqrt{3}}{2}. Prove that there exists a tringle that is similar to PP.
algebra
different remainders am^3 + bm^2 + cm, mod 0,1,2,..., p-1

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, Round 4 p2

11/12/2022
Finding all quads of integers (a,b,c,p)(a, b, c, p) where p≥5p \ge 5 is prime number such that the remainders of the numbers am3+bm2+cmam^3 + bm^2 + cm, m=0,1,...,p−1m = 0, 1, . . . , p - 1, upon division of pp are two by two different..
number theoryremainder
concurrent wanted, circumcircle and touchpoints of incircle related

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p2

11/13/2022
Let kk be the circumcircle of the acute triangle ABCABC. Its inscribed circle touches sides BCBC, CACA and ABAB at points D,ED, E and FF respectively. The line EDED intersects kk at the points MM and NN, so that EE lies between MM and DD. Let KK and LL be the second intersection points of the lines NFNF and MFMF respectively with kk. Let AK∩BL=QAK \cap BL = Q. Prove that the lines ALAL, BKBK and QFQF intersect at a point.
geometryincircleconcurrencyconcurrent