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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: 25

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: 18

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: 18

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: 1
Views: 48

APMO 2017 P1

We call a $5$-tuple of integers arrangeable if its elements can be labeled $a, b, c, d, e$ in some order so that $a-b+c-d+e = 29$. Determine all $2017$-tuples of integers $n_1,n_2,...,n_{2017}$ such that if we place them in a circle in clockwise order, then any $5$-tuple of numbers in consecutive po...
by dshasan
Sat May 27, 2017 1:07 pm
 
Forum: Asian Pacific Math Olympiad (APMO)
Topic: APMO 2017 P1
Replies: 0
Views: 25

Re: ISL 2006 G3

My solution Let $BD$ and $CD$ meet $AC$ and $AD$ at $X$ and $Y$ resp. lets define $\angle ACE = \beta$, $\angle DCE = \alpha$, $\angle BDA = \theta$, $\angle CDB = \gamma$ Now, we can easily get that $\bigtriangleup ABD \cong \bigtriangleup ACE$ and $\bigtriangleup BCD \cong \bigtriangleup CDE$ this...
by dshasan
Mon May 15, 2017 6:02 pm
 
Forum: Geometry
Topic: ISL 2006 G3
Replies: 2
Views: 91

ISL 2006 G3

let $ABCDE$ be a convex pentagon such that

$\angle BAC =\angle CAD =\angle DAE$ and $\angle ABC =\angle ACD =\angle ADE$.

Diagonals $BD$ and $CE$ meet at $P$. Prove that ray $AP$ bisects $CD$.
by dshasan
Mon May 15, 2017 5:30 pm
 
Forum: Geometry
Topic: ISL 2006 G3
Replies: 2
Views: 91

Re: BDMO 2017/09

double post. the topic is discussed here viewtopic.php?f=13&t=3908
by dshasan
Thu May 11, 2017 5:34 pm
 
Forum: Secondary Level
Topic: BDMO 2017/09
Replies: 1
Views: 83

USA(J)MO 2017 #2

Consider the equation $(3x^3 + xy^2)(x^2y + 3y^3) = (x - y)^7$

a. Prove that there are infinitely many pairs$(x,y)$ of positive integers satisfying the equation.
b. Describe all pairs $(x,y)$ of positive integers satisfying the equation.
by dshasan
Sat Apr 22, 2017 6:23 pm
 
Forum: Algebra
Topic: USA(J)MO 2017 #2
Replies: 1
Views: 86

USA(J)MO 2017 #3

Let $ABC$ be an equilateral triangle, and point $P$ on it's circumcircle. Let $PA$ and $BC$ intersect at $D$, $PB$ and $AC$ intersect at $E$, and $PC$ and $AB$ intersect at $F$. Prove that the area of $\bigtriangleup DEF$ is twice the area of $\bigtriangleup ABC$
by dshasan
Sat Apr 22, 2017 6:06 pm
 
Forum: Geometry
Topic: USA(J)MO 2017 #3
Replies: 6
Views: 204
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