MathDB

3

Part of 2023 Bulgaria JBMO TST

Problems(3)

Diophantine Equation with prime numbers and bonus conditions

Source: 2023 Bulgaria JBMO TST Problem 3

5/7/2023
Find all natural numbers aa, bb, cc and prime numbers pp and qq, such that:
■\blacksquare 4∤c4\nmid c ■\blacksquare p≢11(mod16)p\not\equiv 11\pmod{16} ■\blacksquare paqb−1=(p+4)cp^aq^b-1=(p+4)^c
number theoryprime numbersDiophantine equation
Playing with balls in boxes

Source: Bulgaria JBMO TST 2023, Day 2, Problem 3

6/17/2023
There are infinitely many boxes - initially one of them contains nn balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became 20232023. What is the smallest possible value of nn?
combinatoricsinvariant
Hidden devil properties of incircle and excircle

Source: Bulgaria JBMO TST 2023, Day 3, Problem 3

6/17/2023
Let ABCABC be a non-isosceles triangle with circumcircle kk, incenter II and CC-excenter ICI_C. Let MM be the midpoint of ABAB and NN be the midpoint of arc ACB^\widehat{ACB} on kk. Prove that ∠IMIC+∠INIC=180∘\angle IMI_C + \angle INI_C = 180^{\circ}.
geometryincircleexcirclesimilaritycircumcircleincenter