## Search found 203 matches

- Fri May 14, 2021 11:46 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Primary Problem 5
- Replies:
**1** - Views:
**240**

### Re: BdMO National 2021 Primary Problem 5

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:
**23**

- 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:
**35**

### 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:
**23**

### 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:
**11**

### 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:
**35**

### 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 =...

- Mon May 10, 2021 9:41 am
- Forum: Combinatorics
- Topic: There are at least 2016 fixed points of the function
- Replies:
**0** - Views:
**31**

### There are at least 2016 fixed points of the function

Let $S=\{0,1,2,3,\cdots,10^{2017}+2005\}$. Let $f:S\to S$ be a function that satisfies \[f^{2017}(x)=\underbrace{f\circ f\circ f\circ\cdots\circ f(x)}_{2017}=x\]

Prove that there exists $T\subseteq S$ of $2016$ elements such that $\forall x\in T, f(x)=x$.

Prove that there exists $T\subseteq S$ of $2016$ elements such that $\forall x\in T, f(x)=x$.

- Sun May 09, 2021 8:10 pm
- Forum: National Math Camp
- Topic: National Math Camp Geometry Exam Problem-2
- Replies:
**1** - Views:
**54**

### Re: National Math Camp Geometry Exam Problem-2

$\textbf{Solution :}$ Nat Camp P2.png $D$ is the circumcenter of $BAC$. $\therefore \measuredangle BAD=\measuredangle DBA$. $M$ is the circumcenter of $XAY$. $\therefore \measuredangle XAM=\measuredangle MXA$. $M$ is the circumcenter of $DTP$. $\therefore \measuredangle MDT=\measuredangle DTM$. Now,...

- Sun May 09, 2021 4:37 pm
- Forum: National Math Camp
- Topic: Problem - 04 - National Math Camp 2021 Geometry Test - "Bonus Problem"
- Replies:
**0** - Views:
**40**

### Problem - 04 - National Math Camp 2021 Geometry Test - "Bonus Problem"

Let $ABC$ be a triangle with circumcircle $(O)$. The midpoints of $BC,CA,AB$ are $A',B',C'$ respectively. The medians $AA', BB', CC'$ cut the circumcircle $(O)$ at $A,A_1; B,B_1; C,C_1$ respectively. The line of tangency to $(O)$ at $A_1$ meets the perpendicular to $AO$ through $A'$ at $X$. Define $...