MathDB

3

Part of 2005 IMO Shortlist

Problems(4)

p + q + r + s = 9 and p^2 + q^2 + r^2 + s^2 = 21

Source: IMO Shortlist 2005 problem A3

7/8/2006
Four real numbers p p, q q, r r, s s satisfy p+q+r+s=9 p+q+r+s = 9 and p2+q2+r2+s2=21 p^{2}+q^{2}+r^{2}+s^{2}= 21. Prove that there exists a permutation (a,b,c,d) \left(a,b,c,d\right) of (p,q,r,s) \left(p,q,r,s\right) such that ab−cd≥2 ab-cd \geq 2.
inequalitiesalgebraIMO Shortlistcalculus
Colorings of a rectangle - prove N^2 >= M * 2^(mn)

Source: IMO Shortlist 2005 Combinatorics problem C3; 2nd German pre-TST 2006, problem 3

12/29/2006
Consider a m×nm\times n rectangular board consisting of mnmn unit squares. Two of its unit squares are called adjacent if they have a common edge, and a path is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called non-intersecting if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a coloring of the board if all its mnmn unit squares are colored. Let NN be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let MM be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that N2≥M⋅2mnN^{2}\geq M\cdot 2^{mn}.
combinatoricsrectanglematrixcountinggraph theoryIMO ShortlistHi
Geo quest. [excenters in a parallelogram; < KCL fixed]

Source: IMOTC ST1 Q1; IMO shortlist 2005 problem G3

5/16/2006
Let ABCDABCD be a parallelogram. A variable line gg through the vertex AA intersects the rays BCBC and DCDC at the points XX and YY, respectively. Let KK and LL be the AA-excenters of the triangles ABXABX and ADYADY. Show that the angle ∡KCL\measuredangle KCL is independent of the line gg.
Proposed by Vyacheslev Yasinskiy, Ukraine
geometryparallelogramhomothetyincenterexterior angleIMO ShortlistSpiral Similarity
Composite sum

Source: INDIA IMOTC-2006 TST1 PROBLEM-2; IMO Shortlist 2005 problem N3

6/3/2006
Let a a, b b, c c, d d, e e, f f be positive integers and let S=a+b+c+d+e+f S = a+b+c+d+e+f. Suppose that the number S S divides abc+def abc+def and ab+bc+ca−de−ef−df ab+bc+ca-de-ef-df. Prove that S S is composite.
algebrapolynomialnumber theorycomposite numberprime numbersIMO ShortlistHi