Search found 16 matches
- Wed Jul 14, 2021 11:18 am
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Combinatorics Test - "Points not maintaining social distance"
- Replies: 2
- Views: 6368
Re: Problem - 01 - National Math Camp 2021 Combinatorics Test - "Points not maintaining social distance"
Let $G$ be a graph with $19$ vertices and by an edge between any two vertices we mean that they are less than. $1 $ unit apart. Let $G'$ be the complement of the graph $G$. Then that means in $ G' $,an edge between any two points means that they are At least $1$ unit apart. Now ATQ there cannot be a...
- Thu Jun 17, 2021 8:46 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
- Replies: 6
- Views: 8137
Re: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
SOL 2 (Moving points) : Fix $\triangle ABC, E,G$ Let $D$ be a moving point on $ \omega$ $ . AF$ is just the reflection of $AD$ over the angle bisector of $ \angle BAC$ $\therefore D\rightarrow AD \rightarrow AF \rightarrow F$ is projective . Let $ EF \cap \omega = P_2$ then $D \rightarrow F \rightar...
- Thu Jun 17, 2021 8:31 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
- Replies: 6
- Views: 8137
Re: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
SOL 1 ( Pascal): Let $ AG \cap \omega =G_1 , AF \cap \omega = F_1 $ $ \angle CBD =\angle CAD = \angle BAF = \angle BAF_1 =\angle BDF_1 \Longrightarrow F_1D \parallel BC$ Similarly $ EG_1 \parallel BC \parallel DF_1 \Longrightarrow EG_1 \parallel DF_1$ Let $ P_{\infty}$ be the point at infinity along...
- Thu Jun 17, 2021 8:11 pm
- Forum: National Math Camp
- Topic: Problem - 02 - National Math Camp 2021 Number Theory Exam - "Group theory"
- Replies: 3
- Views: 10296
Re: Problem - 02 - National Math Camp 2021 Number Theory Exam - "Group theory"
ig it may have something to do with primitive roots
- Thu Jun 17, 2021 9:45 am
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P1
- Replies: 2
- Views: 10504
Re: APMO 2021 P1
Sol by Kazi Nadid Let $a=$ $\lfloor x \rfloor$ $a\le x< a+1$ $a^2\le x^2< (a+1)^{2}$ $a$ can't be $0$ Because then $L.H.S>0$ But $R.H.S=0$ which is a contradiction So, $a\ge 1$ $\implies a^2+3a\ge a^2+2a+1$ $\implies a^2+3a \ge (a+1)^{2}$ So, $a^2\le x^2< a^2+3a$ $ \implies a^2 \le ar<a^2+3a$ $ \imp...
- Thu Jun 17, 2021 9:42 am
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P3 - Geometry
- Replies: 2
- Views: 5606
Re: APMO 2021 P3 - Geometry
Here is a another cool sol by an aops user Let $I_1$ and $I_2$ be the incircle of $\triangle ABC$ and $\triangle DBC$(Note that $L$ must be the intersection of line $BI_1$ and $CI_2$), $F$ the excircle of $EBC$, and $X = AI_1 \cap CF$, $Y = DI_2 \cap BF$. Claim: $I_1I_2 // XY$ Proof. By incenter - e...
- Thu Jun 17, 2021 9:40 am
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P3 - Geometry
- Replies: 2
- Views: 5606
Re: APMO 2021 P3 - Geometry
My Solution Firstly note that $ L$ is the incentre of $\triangle TAC ; T= AB \cap CD $ Let $ I_a ,I_d$ be the incentres of $ \triangle BAC ,\triangle DBC$ and let $ E_a ,E_d$ be the excentres of $ \triangle ABC ,\triangle DBC$ and $ P$ be the excentre of $\triangle EBC$. Notice $ \angle BI_aC= 90+\d...
- Wed Jun 16, 2021 10:38 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
- Replies: 5
- Views: 10593
Re: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Let $ (a,N)=x , (b,N)=y$ then $ (x,y)=1$ By Dirichlet's theorem there exists infinitely many primes $p,q$ such that $ p \equiv \dfrac{a}{x}$ (mod $\dfrac{N}{x}$) and $q \equiv \dfrac{b}{y}$ (mod $\dfrac {N}{y}$) or, $px \equiv a $ (mod N) and $ qy \equiv b $ (mod N) Setting $ m=px ,n = qy$ we are d...
- Wed Jun 16, 2021 10:35 pm
- Forum: National Math Camp
- Topic: National Math Camp Geometry Exam Problem-3
- Replies: 2
- Views: 8155
Re: National Math Camp Geometry Exam Problem-3
Restate the problem as : Let $P$ be a moving point on side $BC$ of $\triangle ABC$. Let the rotation of $PB$ with angle $B$ and centre $P$ meet $AB$ at $Q$. Define $R$ similarly.It suffices to show $R,Q,A,O$ is cyclic. Degree of $P =1$. So degree of line $PQ = d(PB)+d(p)=0+1=1$ since $PB(BC)$ is a f...
- Wed Jun 16, 2021 8:09 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
- Replies: 5
- Views: 10593
Re: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Let $ (a,N)=x , (b,N)=y$ then $ (x,y)=1$ By Dirichlet's theorem there exists infinitely many primes $p,q$ such that $ p \equiv \dfrac{a}{x}$ (mod $\dfrac{N}{x}$) and $q \equiv \dfrac{b}{y}$ (mod $\dfrac {N}{y}$) or, $px \equiv a $ (mod N) and $ qy \equiv b $ (mod N) Setting $ m=px ,n = qy$ we are done