MathDB

3

Part of 2023 Saint Petersburg Mathematical Olympiad

Problems(3)

Blocks in fractions

Source: St. Petersburg 2023 11.3

8/12/2023
The infinite periodic fractions ab\frac{a} {b} and cd\frac{c} {d} with (a,b)=(c,d)=1(a, b)=(c, d)=1 are such that every finite block of digits in the first fraction after the decimal point appears in the second fraction as well (again after the decimal point). Show that b=db=d.
number theory
NT equation

Source: St. Petersburg 2023 10.3

8/12/2023
Find all positive integers x,yx, y and primes pp, such that x5+y4=pxyx^5+y^4=pxy.
number theory
Median geo

Source: St. Petersburg 2023 9.3

8/12/2023
Let MM be the midpoint of ACAC in an acute triangle ABCABC. Let KK be a point on the minor arc ACAC, such that ∠AKM=90o\angle AKM=90^{o}. Let BK∩AM=XBK \cap AM=X and the AA-altitude meets BMBM at YY. Show that XY∥ABXY \parallel AB.
geometry