MathDB

2011 IMO Shortlist

Part of IMO Shortlist

Subcontests

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IMO Shortlist 2011, Combinatorics 6

Let nn be a positive integer, and let W=…x−1x0x1x2…W = \ldots x_{-1}x_0x_1x_2 \ldots be an infinite periodic word, consisting of just letters aa and/or bb. Suppose that the minimal period NN of WW is greater than 2n2^n.
A finite nonempty word UU is said to appear in WW if there exist indices k≤ℓk \leq \ell such that U=xkxk+1…xℓU=x_k x_{k+1} \ldots x_{\ell}. A finite word UU is called ubiquitous if the four words UaUa, UbUb, aUaU, and bUbU all appear in WW. Prove that there are at least nn ubiquitous finite nonempty words.
Proposed by Grigory Chelnokov, Russia

IMO Shortlist 2011, G6

Let ABCABC be a triangle with AB=ACAB=AC and let DD be the midpoint of ACAC. The angle bisector of ∠BAC\angle BAC intersects the circle through D,BD,B and CC at the point EE inside the triangle ABCABC. The line BDBD intersects the circle through A,EA,E and BB in two points BB and FF. The lines AFAF and BEBE meet at a point II, and the lines CICI and BDBD meet at a point KK. Show that II is the incentre of triangle KABKAB.
Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea

IMO Shortlist 2011, Number Theory 6

Let P(x)P(x) and Q(x)Q(x) be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both P(x)P(x) and Q(x).Q(x). Suppose that for every positive integer nn the integers P(n)P(n) and Q(n)Q(n) are positive, and 2Q(n)−12^{Q(n)}-1 divides 3P(n)−1.3^{P(n)}-1. Prove that Q(x)Q(x) is a constant polynomial.
Proposed by Oleksiy Klurman, Ukraine
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IMO Shortlist 2011, G1

Let ABCABC be an acute triangle. Let ω\omega be a circle whose centre LL lies on the side BCBC. Suppose that ω\omega is tangent to ABAB at B′B' and ACAC at C′C'. Suppose also that the circumcentre OO of triangle ABCABC lies on the shorter arc B′C′B'C' of ω\omega. Prove that the circumcircle of ABCABC and ω\omega meet at two points.
Proposed by Härmel Nestra, Estonia

IMO Shortlist 2011, Number Theory 1

For any integer d>0,d > 0, let f(d)f(d) be the smallest possible integer that has exactly dd positive divisors (so for example we have f(1)=1,f(5)=16,f(1)=1, f(5)=16, and f(6)=12f(6)=12). Prove that for every integer k≥0k \geq 0 the number f(2k)f\left(2^k\right) divides f(2k+1).f\left(2^{k+1}\right).
Proposed by Suhaimi Ramly, Malaysia
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IMO Shortlist 2011, Algebra 5

Prove that for every positive integer n,n, the set {2,3,4,…,3n+1}\{2,3,4,\ldots,3n+1\} can be partitioned into nn triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle.
Proposed by Canada

IMO Shortlist 2011, Combinatorics 5

Let mm be a positive integer, and consider a m×mm\times m checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time 00, each ant starts moving with speed 11 parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn 90∘90^{\circ} clockwise and continue moving with speed 11. When more than 22 ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.
Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist.
Proposed by Toomas Krips, Estonia

IMO Shortlist 2011, G5

Let ABCABC be a triangle with incentre II and circumcircle ω\omega. Let DD and EE be the second intersection points of ω\omega with AIAI and BIBI, respectively. The chord DEDE meets ACAC at a point FF, and BCBC at a point GG. Let PP be the intersection point of the line through FF parallel to ADAD and the line through GG parallel to BEBE. Suppose that the tangents to ω\omega at AA and BB meet at a point KK. Prove that the three lines AE,BDAE,BD and KPKP are either parallel or concurrent.
Proposed by Irena Majcen and Kris Stopar, Slovenia
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IMO Shortlist 2011, Algebra 3

Determine all pairs (f,g)(f,g) of functions from the set of real numbers to itself that satisfy g(f(x+y))=f(x)+(2x+y)g(y)g(f(x+y)) = f(x) + (2x + y)g(y) for all real numbers xx and yy.
Proposed by Japan

IMO Shortlist 2011, G3

Let ABCDABCD be a convex quadrilateral whose sides ADAD and BCBC are not parallel. Suppose that the circles with diameters ABAB and CDCD meet at points EE and FF inside the quadrilateral. Let ωE\omega_E be the circle through the feet of the perpendiculars from EE to the lines AB,BCAB,BC and CDCD. Let ωF\omega_F be the circle through the feet of the perpendiculars from FF to the lines CD,DACD,DA and ABAB. Prove that the midpoint of the segment EFEF lies on the line through the two intersections of ωE\omega_E and ωF\omega_F.
Proposed by Carlos Yuzo Shine, Brazil

IMO Shortlist 2011, Number Theory 3

Let n≥1n \geq 1 be an odd integer. Determine all functions ff from the set of integers to itself, such that for all integers xx and yy the difference f(x)−f(y)f(x)-f(y) divides xn−yn.x^n-y^n.
Proposed by Mihai Baluna, Romania
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