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Re: Beginner's Marathon

Let $AD \cap \Omega = P$. Now, note that $AP = BB_1, AA_1 = CA_2$ and $BB_1 = CB_2$. Now, power of point implies that $AP.AD = AC.AA_1 \Rightarrow BB_1.BC = CA_2.AC \Rightarrow CA_2.AC=CB_2.BC$. Therefore, $A,B,A_2,B_2$ are cyclic.

Somebody post the next problem.
by dshasan
Wed Jun 28, 2017 4:23 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3371

Re: Beginner's Marathon

$\text{Problem 24}$
Let $n$ be a positive integer and let $a_1, a_2,.....a_k$(here $k$ > 1) be distinct integers in the set {${1,2.....n}$} such that $n$ divides $a_i(a_{i+1}-1)$ for $i = 1,2,.....k-1$. Prove that $n$ does not divide $a_k(a_1 - 1)$
by dshasan
Fri Jun 23, 2017 6:17 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3371

Re: Beginner's Marathon

$\text{Problem 23}$ Let \[a^2 + b + c = (a + x)^2\]\[b^2 + c + a = (b + y)^2\]\[c^2 + a + b = (c + z)^2\]where $x,y,z$ are positive integers.Now, from this 3 equations we get \[b + c = x(2a + x)\]\[c + a = y(2b + y)\]\[a + b = z(2c + z)\]Adding them yi...
by dshasan
Fri Jun 23, 2017 6:07 pm
 
Forum: Junior Level
Topic: Beginner's Marathon
Replies: 68
Views: 3371

$x^2 \equiv x (mod n)$

Let $n$ be a positive integer. Determine, in terms of n, the number of $x$ such that $x \in {1,2,...n}$ and \[x^2 \equiv x(mod n)\]
by dshasan
Sat Jun 10, 2017 11:52 pm
 
Forum: Number Theory
Topic: $x^2 \equiv x (mod n)$
Replies: 1
Views: 157

Re: MPMS Problem Solving Marathon

$\text{Problem 3}$ Let $4n^2 - 6n + 45 = (2k+1)^2$ $\Rightarrow (2n-3)^2 +36 + 6n = (2k + 1)^2 $ $\Rightarrow 36 + 6n = (2k + 1)^2 - (2n-3)^2$ $\Rightarrow 6(6 + n) = (2k + 2n -2)(2k -2n + 4)$ $\Rightarrow 3(6 + n) = (k + n ...
by dshasan
Mon Jun 05, 2017 10:32 am
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 515

Re: MPMS Problem Solving Marathon

$\text{Problem 1}$ Stronger claim: There exists an integer $a$ and prime $p$ such that $p|a^2 + 1$ if and only if $ p\equiv 1(mod 4)$ Proof: For the first part, we have $a^2 + 1 \equiv -1(mod p)$ $\Rightarrow (a^2)^{\dfrac{p-1}{2}} \equiv -1^{\dfrac{p-1}{2}}(mod p)$ $...
by dshasan
Sun Jun 04, 2017 10:45 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 515

APMO 2017 P5

Let $n$ be a positive integer. A pair of $n$-tuples $(a_1,....a_n)$ and $(b_1,...b_n)$ with integer entries is called an exquisite pair if $|a_1b_1 +...+ a_nb_n| \leq 1$. Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair.
by dshasan
Sat May 27, 2017 1:45 pm
 
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2017 P5
Replies: 0
Views: 116

APMO 2017 P4

Call a rational number $r$ powerful if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p,q$ and some integer $k > 1$. Let $a,b,c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x,y,z$ such that $a^x + b^y + c^z$...
by dshasan
Sat May 27, 2017 1:40 pm
 
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2017 P4
Replies: 0
Views: 91

APMO 2017 P3

Let $A(n)$ denote the number of sequences $a_1 \geq a_2 \geq ....\geq a_k$ of positive integers for which $a_1 + a_2 + ... + a_k = n$ and each $a_i + 1$ is a power of two . let $B(m)$ denote the number of sequences $b_1 \geq b_2 \geq ....\geq b_m$ of positive integers for which $b_1 + b_2 + .... b_m...
by dshasan
Sat May 27, 2017 1:36 pm
 
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2017 P3
Replies: 0
Views: 96

APMO 2017 P2

Let $\bigtriangleup ABC$ be a triangle with $AB < AC$. Let $D$ be the intersection point of the internal bisector of $\angle BAC$ and the circumcircle of $\bigtriangleup ABC$. Let $Z$ be the intersection point of the perpendicular bisector of $AC$ with the external bisector of $\angle BAC$. Prove th...
by dshasan
Sat May 27, 2017 1:11 pm
 
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2017 P2
Replies: 2
Views: 259
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