As this is the Beginner's Marathon, I request everyone to not give shortlist problems. Problem $25$ Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $ \Omega $ is a circle passing through $A,B,C,D$. Let $ \omega $ be the circle passing through $C,D$ and intersecting with $CA,CB$ at $A_1$, $B_1$ ...

- Mon Jun 26, 2017 4:04 am
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**66** - Views:
**2178**

Solution to Problem $24$ ( NOT OVERKILL ) As the previous solution we solve by contradiction. So, assume that $a_1a_k \equiv a_k (mod n)$. $ a_1\equiv a_1\cdot a_2\equiv a_1\cdot a_2\cdot a_3\equiv \dots \equiv a_1\cdot\dots\cdot a_k\equiv a_1\cdot\dots\cdot a_{k-2}\cdot a_k\equiv\dots\equiv a_1\cdo...

- Mon Jun 26, 2017 3:58 am
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**66** - Views:
**2178**

Solution to problem $24$ ( OVERKILL ) We prove by contradiction. Assume that $n$ divides $a_k(a_1-1)$. Assume that $(a_i, n)=1$. Then $n| a_i(a_{i+1}-1)$ implies $n|a_{i+1}-1$ which implies $a_{i+1}=1$. Then $n|1(a_{i+2}-1)$ which is only possible if $a_{i+2}=1$ but this contradicts the distinction....

- Mon Jun 26, 2017 3:44 am
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**66** - Views:
**2178**

By AM-GM inequality, $ac+bc+ab \geq 3(abc)^{2/3}$. Also by AM-GM inequality, $a+b+c \geq 3(abc)^{1/3}$. So, $3 \geq 2+ \frac{3 (abc)^{1/3}}{(a+b+c)}$. Or, $ac+bc+ab \geq 3(abc)^{2/3} \geq (abc)^{2/3} (2+ \frac{3 (abc)^{1/3}}{(a+b+c)})$. Or, $ac+bc+ab \geq 2(abc)^{2/3} + \frac {3abc}{a+b+c}$. Or, $ac...

- Mon Jun 26, 2017 1:51 am
- Forum: Algebra
- Topic: Good inequality..
- Replies:
**1** - Views:
**59**

If we denote $f(n)$ the desired number, then CRT implies $f(mn)=f(m)f(n)$ whenever $(m;n)=1$. So it suffices to find $f(n)$ when $n$ is a prime power. It's quite trivial that there are $2$ solutions for $x $ if $n $ is a prime power. So, the answer is $2^z $ where $z $ is the number of distinct prim...

- Thu Jun 22, 2017 6:51 am
- Forum: Number Theory
- Topic: $x^2 \equiv x (mod n)$
- Replies:
**1** - Views:
**92**

We rearrange the equation as, $n^5+n^4+1= 7^m $. Observe that $n^5+n^4+1=(n^2+n+1)(n^3-n+1)$. So, each of $n^2+n+1$ and $n^3-n+1$ is $7^x $ for some integer $x \geq 0$. But using the euclidean algorithm, we find that $gcd(n^2+n+1, n^3-n+1)$ divides $7$. So we consider the $4$ equations $n^2+n+1=1$ $...

- Thu Jun 22, 2017 6:20 am
- Forum: Number Theory
- Topic: n^5+n^4
- Replies:
**1** - Views:
**55**

The left hand side is even, so $ x^4+y^2$ is even, and $ x, y$ has the same parity. However, if $ x, y$ are odd, we have that $ 4$ divides $ 7^x-3^y$, but $ 4$ doesn't divide $ x^4+y^2$. So $ x, y$ are even, and let's say $ x=2x_1,y=2y_1$. You'll get $ (7^{x_1}-3^{y_1})(7^{x_1}+3^{y_1})$ divides $ 4...

- Thu Jun 22, 2017 5:42 am
- Forum: Number Theory
- Topic: Divisibility... with x,y
- Replies:
**1** - Views:
**60**

Ugly problem tbh. Let us assign $\angle BAC = 4a$. Then quick angle chases give us, $\angle CAK = a $, $\angle DCI = 45-a $, $\angle CEK = 3a $. We can apply sine law on $\triangle IEK $ and $\triangle CEK $ to get $ \frac{IK}{KC} = \frac{sin(45)*sin (45-a)}{sin(3a)*sin(90-2a)}$ ........(relation 1)...

- Thu Jun 22, 2017 4:53 am
- Forum: Geometry
- Topic: IMO 2009 SL(G-1)
- Replies:
**1** - Views:
**66**

Problem $23$

Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

- Wed Jun 21, 2017 5:26 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**66** - Views:
**2178**

They start, from Mirpur, one with the bike($ A$) and the other($ B$) on foot, and from Lalmatia($ C$) on foot of course. After one hour $ A$ will give the bike to $ C$, and $ B$ will stop, and wait for the bike. $ A$ will reach Lamatia after 2 hours. $ C$ rides the bike until he meets $ B$ , and giv...

- Tue Jun 20, 2017 10:39 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies:
**66** - Views:
**2178**