MathDB

1

Part of 2021 Iran MO (3rd Round)

Problems(4)

\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180

Source: Iran MO Third Round 2021 G1

9/25/2021
An acute triangle ABCABC is given. Let DD be the foot of altitude dropped for AA. Tangents from DD to circles with diameters ABAB and ACAC intersects with the said circles at KK and LL, in respective. Point SS in the plane is given so that ∠ABC+∠ABS=∠ACB+∠ACS=180∘\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180^\circ. Prove that A,K,LA, K, L and SS lie on a circle.
geometryIranIranMOAngle condition
\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}}, a+b+c+d = 4

Source: Iran MO Third Round 2021 A1

9/25/2021
Positive real numbers a,b,ca, b, c and dd are given such that a+b+c+d=4a+b+c+d = 4 prove that aba2−43a+43+bcb2−43b+43+cdc2−43c+43+dad2−43d+43≤4.\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.
algebrainequalities
f(n), coprime, divisor

Source: Iran MO Third Round N1

9/25/2021
For a natural number nn, f(n)f(n) is defined as the number of positive integers less than nn which are neither coprime to nn nor a divisor of it. Prove that for each positive integer kk there exist only finitely many nn satisfying f(n)=kf(n) = k.
number theory
arrange natural numbers 1 to 8 divide sum

Source: Iran MO Third Round 2021 F1

9/25/2021
Is it possible to arrange natural numbers 1 to 8 on vertices of a cube such that each number divides sum of the three numbers sharing an edge with it?
geometry3D geometry