MathDB

2000 IMO Shortlist

Part of IMO Shortlist

Subcontests

(7)
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At least how many gangsters will be killed?

Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?

Find all ints n for which n-independent polynomials exist

For a polynomial P P of degree 2000 with distinct real coefficients let M(P) M(P) be the set of all polynomials that can be produced from P P by permutation of its coefficients. A polynomial P P will be called n n-independent if P(n) \equal{} 0 and we can get from any Q∈M(P) Q \in M(P) a polynomial Q1 Q_1 such that Q_1(n) \equal{} 0 by interchanging at most one pair of coefficients of Q. Q. Find all integers n n for which n n-independent polynomials exist.
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Double functional equation

Find all pairs of functions f:R→R f : \mathbb R \to \mathbb R, g:R→Rg : \mathbb R \to \mathbb R such that f \left( x + g(y) \right) = xf(y) - y f(x) + g(x)  \text{for all } x, y\in\mathbb{R}.

Lines AD, BE, and CF are concurrent

Let OO be the circumcenter and HH the orthocenter of an acute triangle ABCABC. Show that there exist points DD, EE, and FF on sides BCBC, CACA, and ABAB respectively such that OD+DH=OE+EH=OF+FH OD + DH = OE + EH = OF + FH and the lines ADAD, BEBE, and CFCF are concurrent.

Vertices of a convex polygon if and only if m(S) = f(n)

Let n≥4 n \geq 4 be a fixed positive integer. Given a set S \equal{} \{P_1, P_2, \ldots, P_n\} of n n points in the plane such that no three are collinear and no four concyclic, let at, a_t, 1≤t≤n, 1 \leq t \leq n, be the number of circles PiPjPk P_iP_jP_k that contain Pt P_t in their interior, and let m(S)=a1+a2+⋯+an.m(S)=a_1+a_2+\cdots + a_n. Prove that there exists a positive integer f(n), f(n), depending only on n, n, such that the points of S S are the vertices of a convex polygon if and only if m(S)=f(n). m(S) = f(n).
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nice [symmedians in a triangle, < ABM = < BAN]

The tangents at BB and AA to the circumcircle of an acute angled triangle ABCABC meet the tangent at CC at TT and UU respectively. ATAT meets BCBC at PP, and QQ is the midpoint of APAP; BUBU meets CACA at RR, and SS is the midpoint of BRBR. Prove that ∠ABQ=∠BAS\angle ABQ=\angle BAS. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.

Semiperimeter, inradius of triangle with int side lengths

Prove that there exist infinitely many positive integers n n such that p \equal{} nr, where p p and r r are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

Bound for sum of vertices of regions cut by rectangles

In the plane we have nn rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is nice if it has at least one of the vertices of the nn rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than 40n40n. (There can be nonconvex regions as well as regions with more than one boundary curve.)
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3 numbers have their fractional parts lying in the interval

Let a,b,c a, b, c be positive integers satisfying the conditions b>2a b > 2a and c>2b. c > 2b. Show that there exists a real number λ \lambda with the property that all the three numbers λa,λb,λc \lambda a, \lambda b, \lambda c have their fractional parts lying in the interval (13,23]. \left(\frac {1}{3}, \frac {2}{3} \right].

Staircase-brick with 3 steps of width 2 using 12 unit cubes

A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers n n for which it is possible to build a cube of side n n using such bricks.

number of divisors

For a positive integer nn, let d(n)d(n) be the number of all positive divisors of nn. Find all positive integers nn such that d(n)3=4nd(n)^3=4n.
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