MathDB

1

Part of MathLinks Contest 3rd

Problems(7)

0311 f: points -> lines 1-1 3rd edition Round 1 p1

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5/9/2021
Let PP be the set of points in the Euclidean plane, and let LL be the set of lines in the same plane. Does there exist an one-to-one mapping (injective function) f:L→Pf : L \to P such that for each ℓ∈L\ell \in L we have f(ℓ)∈ℓf(\ell) \in \ell?
geometryalgebra3rd edition
0321 functional system 3rd edition Round 2 p1

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5/9/2021
Find all functionsf,g:(0,∞)→(0,∞) f, g : (0,\infty) \to (0,\infty) such that for all x>0x > 0 we have the relations:
f(g(x))=xxf(x)−2f(g(x)) = \frac{x}{xf(x) - 2} and g(f(x))=xxg(x)−2g(f(x)) = \frac{x}{xg(x) - 2} .
algebra3rd edition
0331 combo geo 3rd edition Round 3 p1

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5/9/2021
Let SS be a nonempty set of points of the plane. We say that SS determines the distance d>0d > 0 if there are two points A,BA, B in SS such that AB=dAB = d. Assuming that SS does not contain 88 collinear points and that it determines not more than 9191 distances, prove that SS has less than 20042004 elements.
geometrycombinatorics3rd edition
0341 functional 3rd edition Round 4 p1

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5/9/2021
Find all functions f:(0,+∞)→(0,+∞)f : (0, +\infty) \to (0, +\infty) which are increasing on [1,+∞)[1, +\infty) and for all positive reals a,b,ca, b, c they fulfill the following relation f(ab)f(bc)f(ca)=f(a2b2c2)+f(a2)+f(b2)+f(c2)f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2).
functionalalgebra3rd edition
0361 geometry 3rd edition Round 6 p1

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5/9/2021
For a triangle ABCABC and a point MM inside the triangle we consider the lines AM,BM,CMAM, BM,CM which intersect the sides BC,CA,ABBC, CA, AB in A1,B1,C1A_1, B_1, C_1 respectively. Take A′,B′,C′A', B', C' to be the intersection points between the lines AA1,BB1,CC1AA_1, BB_1, CC_1 and B1C1,C1A1,A1B1B_1C_1, C_1A_1, A_1B_1 respectively. a) Prove that the lines BC′,CB′BC', CB' and AA′AA' intersect in a point A2A_2; b) Define similarly points B2,C2B_2, C_2. Find the loci of MM such that the triangle A1B1C1A_1B_1C_1 is similar with the triangle A2B2C2A_2B_2C_2.
geometry3rd edition
0371 combinatorics 3rd edition Round 7 p1

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5/9/2021
In a soccer championship 20042004 teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows: a) any two teams can play against one each other at most one game; b) from any 44 teams, 33 of them play against one each other. How many days are necessary to make such a championship, knowing that each team can play at most one game per day?
combinatorics3rd edition
0351 inequalities 3rd edition Round 5 p1

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5/9/2021
Let a,b,ca, b, c be positive reals. Prove that abc(a+b+c)+(a+b+c)2≥43abc(a+b+c).\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.
inequalities3rd edition