## Search found 203 matches

Fri May 14, 2021 11:46 pm
Topic: BdMO National 2021 Primary Problem 5
Replies: 1
Views: 247

In decimal system, we can write a number $n$ as $n=a_0+a_1\cdot10+a_2\cdot10^2+\cdots+a_k\cdot10^k$ where $a_0,a_1,a_2,\cdots a_k\in\{0,1,2,\cdots,9\}$ and $a_k\neq0$. Then $\begin{equation} \begin{split} g(n)&=a_0+a_1+a_2+\cdots+a_k \\ \therefore n-g(n)&=(10^0-1)a_0+(10^1-1)a_1+(10^2-1)a_2+\cdo... Thu May 13, 2021 3:38 pm Forum: National Math Camp Topic: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!" Replies: 3 Views: 30 ### Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!" Dustan wrote: Thu May 13, 2021 2:36 pm Easy! Ans Thu May 13, 2021 1:22 pm Forum: National Math Camp Topic: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks" Replies: 4 Views: 45 ### Re: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks" Let a_1 \leq a_2 \leq a_3 \cdots \leq a_n be a sequence of positive integers. For 1 \leq i \leq a_n, let b_i be the number of terms in the sequence that are not smaller than i. It is given that, b_1 > b_2 > \cdots > b_{a_n}, for each 1 \leq i \leq a_n, b_i is a power of three, and b_1 ... Thu May 13, 2021 12:25 am Forum: National Math Camp Topic: Problem - 04 - National Math Camp 2021 Mock Exam - "Angle bisector, spiral similarity etc." Replies: 0 Views: 12 ### Problem - 04 - National Math Camp 2021 Mock Exam - "Angle bisector, spiral similarity etc." Let ABC be a triangle such that AB < AC. Let D be a point on AC such that CD = BD. The line parallel to BC through D meet the minor arc AB of \odot ABC at E. Let I, J be the incenters of \triangle ADE and \triangle BDE respectively. Prove that the internal angle bisector of... Thu May 13, 2021 12:12 am Forum: National Math Camp Topic: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!" Replies: 3 Views: 30 ### Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!" Find all functions f:\mathbb{R}\to\mathbb{R} such that for all x,y\in\mathbb{R}, \[f(f(f(x)+y))=f(x+y)+f(x)+y$
Thu May 13, 2021 12:08 am
Forum: National Math Camp
Topic: Problem - 02 - National Math Camp 2021 Mock Exam - "Modular equation involving power tower of 2"
Replies: 0
Views: 12

### Problem - 02 - National Math Camp 2021 Mock Exam - "Modular equation involving power tower of 2"

Let $n = 2^{2^x}$ for some $x > 0$. If $d\mid n^2+1$, then show that $d\equiv1\pmod{2^{x+3}}$
Thu May 13, 2021 12:03 am
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Mock Exam - "Not as bad as it looks"
Replies: 4
Views: 45