MathDB

2

Part of 2021 Iran MO (3rd Round)

Problems(5)

\angle KMC = 90, \angle KAC = 180 - \angle ABC

Source: Iran MO Third Round G2

9/25/2021
Given an acute triangle ABCABC let MM be the midpoint of ABAB. Point KK is given on the other side of line ACAC from that of point BB such that ∠KMC=90∘\angle KMC = 90 ^ \circ and ∠KAC=180∘−∠ABC\angle KAC = 180^\circ - \angle ABC. The tangent to circumcircle of triangle ABCABC at AA intersects line CKCK at EE. Prove that the reflection of line BCBC with respect to CMCM passes through the midpoint of line segment MEME.
geometrycircumcircle
Permuting Integers on Integer Lattice to Make Increasing

Source: Iran MO Third Round C2

9/25/2021
Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?
combinatorics
2|a-b|\leq |b|, \frac{7}{2} <\left| \frac{b}{a}

Source: Iran MO Third Round 2021 A2

9/25/2021
If a,b,ca, b, c and dd are complex non-zero numbers such that 2∣a−b∣≤∣b∣,2∣b−c∣≤∣c∣,2∣c−d∣≤∣d∣,2∣d−a∣≤∣a∣.2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|. Prove that 72<∣ba+cb+dc+ad∣.\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .
algebra
f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)

Source: Iran MO Third Round 2021 N2

9/25/2021
Find all functions f:N→Nf:\mathbb{N} \to \mathbb{N} such that for any two positive integers aa and bb we have fa(b)+fb(a)∣2(f(ab)+b2−1) f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1) Where fn(m)f^n(m) is defined in the standard iterative manner.
functionnumber theory
AC\| DX DY\| AB reflection of H

Source: Iran MO Third Round 2021 F2

9/25/2021
Given an acute triangle ABCABC, let ADAD be an altitude and HH the orthocenter. Let EE denote the reflection of HH with respect to AA. Point XX is chosen on the circumcircle of triangle BDEBDE such that AC∥DXAC\| DX and point YY is chosen on the circumcircle of triangle CDECDE such that DY∥ABDY\| AB. Prove that the circumcircle of triangle AXYAXY is tangent to that of ABCABC.
geometrygeometric transformationreflectioncircumcircleIranIranMOtangent circles