MathDB

2012 JBMO ShortLists

Part of JBMO ShortLists

Subcontests

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Find area!

Let O1O_1 be a point in the exterior of the circle ω\omega of center OO and radius RR , and let O1NO_1N , O1DO_1D be the tangent segments from O1O_1 to the circle. On the segment O1NO_1N consider the point BB such that BN=RBN=R .Let the line from BB parallel to ONON intersect the segment O1DO_1D at CC . If AA is a point on the segment O1DO_1D other than CC so that BC=BA=aBC=BA=a , and if the incircle of the triangle ABCABC has radius rr , then find the area of △ABC\triangle ABC in terms of a,R,ra ,R ,r.

Isn't a perfect square!

If aa , bb , cc , dd are integers and A=2(a−2b+c)4+2(b−2c+a)4+2(c−2a+b)4A=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4 , B=d(d+1)(d+2)(d+3)+1B=d(d+1)(d+2)(d+3)+1 , then prove that (A+1)2+B\left (\sqrt{A}+1 \right )^2 +B cannot be a perfect square.
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Inequality transformed in an equation!

Solve the following equation for x,y,z∈Nx , y , z \in \mathbb{N} : (1+xy+z)2+(1+yz+x)2+(1+zx+y)2=274\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}

Isogonal conjugates!

Let ABCABC be an acute-angled triangle with circumcircle ω\omega , and let OO , HH be the triangle's circumcenter and orthocenter respectively . Let also A′A^{'} be the point where the angle bisector of the angle BACBAC meets ω\omega . If A′H=AHA^{'}H=AH , then find the measure of the angle BACBAC.

Diophantine equation !

Determine all triples (m,n,p)(m , n , p) satisfying : n2p=m2+n2+p+1n^{2p}=m^2+n^2+p+1 where mm and nn are integers and pp is a prime number.
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Similar with Simson's line !

Let ABCABC be an equilateral triangle , and PP be a point on the circumcircle of the triangle but distinct from AA ,BB and CC. The lines through PP and parallel to BCBC , CACA , ABAB intersect the lines CACA , ABAB , BCBC at MM , NN and QQ respectively .Prove that MM , NN and QQ are collinear .

Name tags !

Along a round table are arranged 1111 cards with the names ( all distinct ) of the 1111 members of the 16th16^{th} JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its 1111 members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?

JBMO Shortlist 2012 N1

If aa , bb are integers and s=a3+b3−60ab(a+b)≥2012s=a^3+b^3-60ab(a+b)\geq 2012 , find the least possible value of ss.