## Search found 195 matches

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,...
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

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 $... Sun May 09, 2021 4:28 pm Forum: National Math Camp Topic: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle" Replies: 1 Views: 29 ### Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle" Let$\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$\omega$. 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$\...
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.
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?
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 $N|m-a, N|n-b$.
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
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...
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 $2$.