MathDB

2014 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

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a^2 + b^2 + c^2 + a^2b^2 + b^2c^2 + c^2a^2 \geq 15/16 if 8abc=1

Let a,b,ca, b, c be positive real numbers such that abc=18abc = \dfrac {1} {8}. Prove the inequality:a2+b2+c2+a2b2+b2c2+c2a2≥1516a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16} When the equality holds?

flights, rootes and two mins

In a country with nn towns, all the direct flights are of double destinations (back and forth). There are r>2014r>2014 rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of nn and the minimum possible rr for that value of nn.

2014 JBMO Shortlist G2

Acute-angled triangle ABC{ABC} with AB<AC<BC{AB<AC<BC} and let be c(O,R){c(O,R)} it’s circumcircle. Diameters BD{BD} and CE{CE} are drawn. Circle c1(A,AE){c_1(A,AE)} interescts AC{AC} at K{K}. Circle c2(A,AD){{c}_{2}(A,AD)} intersects BA{BA} at L{L} .(A{A} lies between B{B} and L{L}). Prove that lines EK{EK} and DL{DL} intersect at circle cc .
by Evangelos Psychas (Greece)
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AZE JBMO TST

Let a,b,ca,b,c be positive real numbers. Prove that ((3a2+1)2+2(1+3b)2)((3b2+1)2+2(1+3c)2)((3c2+1)2+2(1+3a)2)≥483\left((3a^2+1)^2+2\left(1+\frac{3}{b}\right)^2\right)\left((3b^2+1)^2+2\left(1+\frac{3}{c}\right)^2\right)\left((3c^2+1)^2+2\left(1+\frac{3}{a}\right)^2\right)\geq 48^3

Nice and interesting JBMO problem

Let ABCDABCD be a quadrilateral whose diagonals are not perpendicular and whose sides ABAB and CDCD are not parallel.Let OO be the intersection of its diagonals.Denote with H1H_1 and H2H_2 the orthocenters of triangles AOBAOB and COD,COD, respectively.If MM and NN are the midpoints of the segment lines ABAB and CD,CD, respectively.Prove that the lines H1H2H_1H_2 and MNMN are parallel if and only if AC=BD.AC=BD.

find 3 consecutive positive integers given 5 conditions

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.(nn is perfect if σ(n)=2n\sigma\left(n\right)=2n). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to 11 in their decimal representation". Dusan:"If we add 1111 to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than 1010". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?
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AZE JBMO TST

Let x,yx,y and zz be non-negative real numbers satisfying the equation x+y+z=xyzx+y+z=xyz. Prove that 2(x2+y2+z2)≥3(x+y+z)2(x^2+y^2+z^2)\geq3(x+y+z).

2014 JBMO Shortlist G5

Let ABCABC be a triangle with AB≠BC{AB\ne BC}; and let BD{BD} be the internal bisector of ∠ABC, \angle ABC,\ , (D∈AC)\left( D\in AC \right). Denote by M{M} the midpoint of the arc AC{AC} which contains point B{B}. The circumscribed circle of the triangle △BDM{\vartriangle BDM} intersects the segment AB{AB} at point K≠B{K\neq B}. Let J{J} be the reflection of A{A} with respect to K{K}. If DJ∩AM={O}{DJ\cap AM=\left\{O\right\}}, prove that the points J,B,M,O{J, B, M, O} belong to the same circle.

AZE JBMO TST

Find all non-negative solutions to the equation 2013x+2014y=2015z2013^x+2014^y=2015^z
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