Search found 195 matches

by Anindya Biswas
Sun May 09, 2021 8:10 pm
Forum: National Math Camp
Topic: National Math Camp Geometry Exam Problem-2
Replies: 1
Views: 23

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,...
by Anindya Biswas
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: 12

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 $...
by Anindya Biswas
Sun May 09, 2021 4:28 pm
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
Replies: 1
Views: 29

Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"

Let $\triangle ABC$ be a triangle inscribed in a circle $\omega$. $D,E$ are two points on the arc $BC$ of $\omega$ not containing $A$. Points $F,G$ lie on $BC$ such that \[\angle BAF = \angle CAD, \angle BAG = \angle CAE\] Prove that the two lines $DG$ and $EF$ meet on $\omega$.
by Anindya Biswas
Thu May 06, 2021 5:03 pm
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Replies: 1
Views: 279

Re: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"

Let $g=\text{gcd}(a,b)$ $\therefore\exists x,y\in\mathbb{Z}$ such that $\text{gcd}(x,y)=1$ and $a=gx, b=gy$. By Bézout's identity , $\exists k_1,k_2\in\mathbb{Z}$ such that $k_2x-k_1y=1$. Claim : Choosing $m=a+Nk_1,n=b+Nk_2$ satisfies the necessary condition. Proof : It's sufficient to prove that $\...
by Anindya Biswas
Thu May 06, 2021 4:32 pm
Forum: National Math Camp
Topic: Problem - 03 - National Math Camp 2021 Number Theory Exam - "Infinitely many prime divisors"
Replies: 0
Views: 163

Problem - 03 - National Math Camp 2021 Number Theory Exam - "Infinitely many prime divisors"

Let $P(x)$ be a nonzero integer polynomial, that is, the coefficients are all integers. We call a prime $q$ "interesting" if there exists some natural number $n$ for which $q|2^n+P(n)$. Prove that there exist infinitely many “interesting” primes.
by Anindya Biswas
Thu May 06, 2021 4:27 pm
Forum: National Math Camp
Topic: Problem - 02 - National Math Camp 2021 Number Theory Exam - "Group theory"
Replies: 0
Views: 170

Problem - 02 - National Math Camp 2021 Number Theory Exam - "Group theory"

Let $p$ be a prime number. We call a subset $S$ of $\{1,2,\cdots,p-1\}$ "good" if it satisfies the property that for every $x,y\in S, xy\text{ mod }{p}$ is also in $S$. How many "good" sets are there?
by Anindya Biswas
Thu May 06, 2021 4:19 pm
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Replies: 1
Views: 279

Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"

Let $N$ be a nonzero integer. Given any $a,b$ such that $\text{gcd}(a,b,N)=1$. Prove that you can find $m,n$ such that $\text{gcd}(m,n)=1$ and $N|m-a, N|n-b$.
by Anindya Biswas
Tue May 04, 2021 11:22 pm
Forum: National Math Camp
Topic: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
Replies: 3
Views: 313

Re: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"

Mehrab4226 wrote:
Tue May 04, 2021 10:11 pm
No one?? :'(
We have to show if condition 1,2 satisfies, then the conclusion is, but not necessarily the 2nd condition must be true if the inequality is true
by Anindya Biswas
Fri Apr 30, 2021 5:40 pm
Forum: National Math Camp
Topic: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
Replies: 3
Views: 313

Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"

We place some checkers on an $n\times n$ checkerboard so that they follow the conditions : Every square that does not contain a checker shares a side with one that does; Given any pair of squares that contain checkers, we can find a sequence of squares occupied by checkers that start and end with th...
by Anindya Biswas
Fri Apr 30, 2021 5:33 pm
Forum: National Math Camp
Topic: Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"
Replies: 2
Views: 271

Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"

Let $n\geq1$ be an integer. A non-empty set is called “good” if the arithmetic mean of its elements is an integer. Let $T_n$ be the number of good subsets of $\{1,2,3,\cdots,n\}$. Prove that for all integers $n$, $T_n$ and $n$ leave the same remainder when divided by $2$.