MathDB

2

Part of 2013 Iran MO (3rd Round)

Problems(5)

Finding Max!

Source: Iran 3rd round 2013 - Algebra Exam - Problem 2

9/11/2013
Real numbers a1,a2,…,ana_1 , a_2 , \dots, a_n add up to zero. Find the maximum of a1x1+a2x2+⋯+anxna_1 x_1 + a_2 x_2 + \dots + a_n x_n in term of aia_i's, when xix_i's vary in real numbers such that (x1−x2)2+(x2−x3)2+⋯+(xn−1−xn)2≤1(x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_{n-1} - x_n)^2 \leq 1. (15 points)
inequalities proposedinequalities
2ab+1 | a^2 + b^2 + 1

Source: Iran 3rd round 2013 - Number Theory Exam - Problem 2

9/11/2013
Suppose that a,ba,b are two odd positive integers such that 2ab+1∣a2+b2+12ab+1 \mid a^2 + b^2 + 1. Prove that a=ba=b. (15 points)
algebrapolynomialVietanumber theory proposednumber theoryVieta Jumping
b+c=2a Make Cyclic Quadrilateral

Source: Iran Third Round 2013 - Geometry Exam - Problem 2

9/7/2013
Let ABCABC be a triangle with circumcircle (O)(O). Let M,NM,N be the midpoint of arc AB,ACAB,AC which does not contain C,BC,B and let M′,N′M',N' be the point of tangency of incircle of △ABC\triangle ABC with AB,ACAB,AC. Suppose that X,YX,Y are foot of perpendicular of AA to MM′,NN′MM',NN'. If II is the incenter of △ABC\triangle ABC then prove that quadrilateral AXIYAXIY is cyclic if and only if b+c=2ab+c=2a.
geometrycircumcircleincentergeometric transformationgeometry unsolved
Rooks Threatening Each Other

Source: Iran 3rd round 2013- Combinatorics exam problem 2

9/18/2014
How many rooks can be placed in an n×nn\times n chessboard such that each rook is threatened by at most 2k2k rooks? (15 points) Proposed by Mostafa Einollah zadeh
modular arithmeticcombinatorics unsolvedcombinatorics
Distance of Circles

Source: Iran 3rd round 2013 - final exam problem 2

9/25/2014
We define the distance between two circles ω,ω′\omega ,\omega 'by the length of the common external tangent of the circles and show it by d(ω,ω′)d(\omega , \omega '). If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius 00 and the distance between two cirlces can be zero. (a) Centroid. nn circles ω1,…,ωn\omega_1,\dots, \omega_n are fixed on the plane. Prove that there exists a unique circle ω‾\overline \omega such that for each circle ω\omega on the plane the square of distance between ω\omega and ω‾\overline \omega minus the sum of squares of distances of ω\omega from each of the ωi\omega_is 1≤i≤n1\leq i \leq n is constant, in other words:d(ω,ω‾)2−1n∑i=1nd(ωi,ω)2=constantd(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant (b) Perpendicular Bisector. Suppose that the circle ω\omega has the same distance from ω1,ω2\omega_1,\omega_2. Consider ω3\omega_3 a circle tangent to both of the common external tangents of ω1,ω2\omega_1,\omega_2. Prove that the distance of ω\omega from centroid of ω1,ω2\omega_1 , \omega_2 is not more than the distance of ω\omega and ω3\omega_3. (If the distances are all defined) (c) Circumcentre. Let CC be the set of all circles that each of them has the same distance from fixed circles ω1,ω2,ω3\omega_1,\omega_2,\omega_3. Prove that there exists a point on the plane which is the external homothety center of each two elements of CC. (d) Regular Tetrahedron. Does there exist 4 circles on the plane which the distance between each two of them equals to 11?
Time allowed for this problem was 150 minutes.
geometrygeometric transformationhomothetygeometry unsolved