MathDB

2

Part of 2023 India IMO Training Camp

Problems(5)

Yet another configy

Source: India TST 2023 Day 2 P2

7/9/2023
In triangle ABCABC, let DD be the foot of the perpendicular from AA to line BCBC. Point KK lies inside triangle ABCABC such that ∠KAB=∠KCA\angle KAB = \angle KCA and ∠KAC=∠KBA\angle KAC = \angle KBA. The line through KK perpendicular to like DKDK meets the circle with diameter BCBC at points X,YX,Y. Prove that AX⋅DY=DX⋅AYAX \cdot DY = DX \cdot AY
indiageometryTSTanant mudgal geo
Ouroboros functions

Source: India TST 2023 Day 1 P2

7/9/2023
Let g:N→Ng:\mathbb{N}\to \mathbb{N} be a bijective function and suppose that f:N→Nf:\mathbb{N}\to \mathbb{N} is a function such that:
[*] For all naturals xx, f(⋯(f⏟x2023  f’s(x)))=x.\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. [*] For all naturals x,yx,y such that x∣yx|y, we have f(x)∣g(y)f(x)|g(y).
Prove that f(x)=xf(x)=x.
Proposed by Pulkit Sinha
algebrafunctional equationfunction
Dr. Supercali and The Multiverse of Mischief

Source: India TST 2023 Day 3 P2

7/9/2023
In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students A1,A2,…,A2023A_1, A_2, \dots, A_{2023} is called mischievous if
∙\bullet Total number of friends of A1A_1 is odd. ∙\bullet AiA_i and Ai+1A_{i+1} are friends for i=1,2,…,2022i=1, 2, \dots, 2022. ∙\bullet Total number of friends of A2023A_{2023} is even.
Prove that the total number of mischievous sequences is even.
combinatoricsgraph theory
FE with too many squares

Source: India TST 2023 Practice Test 2 P2

7/9/2023
Let R+\mathbb R^+ be the set of all positive real numbers. Find all functions f:R+→R+f:\mathbb{R}^+ \rightarrow \mathbb{R}^+ satisfying f(x+y2f(x2))=f(xy)2+f(x)f(x+y^2f(x^2))=f(xy)^2+f(x) for all x,y∈R+x,y \in \mathbb{R}^+.
Proposed by Shantanu Nene
algebrafunctional equation
s(2^n) > s(2^{n+1})

Source:

1/17/2015
For a positive integer kk, let s(k)s(k) denote the sum of the digits of kk. Show that there are infinitely many natural numbers nn such that s(2n)>s(2n+1)s(2^n) > s(2^{n+1}).
logarithmsmodular arithmeticnumber theory unsolvednumber theory