MathDB

3

Part of MathLinks Contest 3rd

Problems(7)

0313 number theory 3rd edition Round 1 p3

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5/9/2021
Let aa and bb be different positive rational numbers such that the there exist an infinity of positive integers nn for which an−bna^n - b^n is an integer. Prove that aa and bb are also integers.
number theory3rd edition
0323 coloring points 3rd edition Round 2 p3

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5/9/2021
Each point in the Euclidean space is colored with one of n≥2n \ge 2 colors, and each of the nn colors is used. Prove that one can find a triangle such that the color assigned to the orthocenter is different from all the colors assigned to the vertices of the triangle.
combinatoricsColoring3rd edition
0333 number theory 3rd edition Round 3 p3

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5/9/2021
An integer zz is said to be a friendly integer if ∣z∣|z| is not the square of an integer. Determine all integers nn such that there exists an infinite number of triplets of distinct friendly integers (a,b,c)(a, b, c) such that n=a+b+cn = a+b+c and abcabc is the square of an odd integer.
number theory3rd edition
0343 combo geo 3rd edition Round 4 p3

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5/9/2021
An integer point of the usual Euclidean 33-dimensional space is a point whose three coordinates are all integers. A set SS of integer points is called a covered set if for all points A,BA, B in SS each integer point in the segment [AB][AB] is also in SS. Determine the maximum number of elements that a covered set can have if it does not contain 20042004 collinear points.
combinatoricsgeometry3rd edition
0353 tetrahedron 3rd edition Round 5 p3

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5/9/2021
We say that a tetrahedron is median if and only if for each vertex the plane that passes through the midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is called regular if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.
3D geometrygeometry3rd editiontetrahedron
0363 inequalities 3rd edition Round 6 p3

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5/9/2021
Let n≥3n \ge 3 be an integer. Find the minimal value of the real number knk_n such that for all positive numbers x1,x2,...,xnx_1, x_2, ..., x_n with product 11, we have 11+knx1+11+knx2+...+11+knxn≤n−1.\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.
inequalities3rd edition
0373 2004x2004 chessboard 3rd edition Round 7 p3

Source:

5/9/2021
On a 2004×20042004\times 2004 chessboard we place 20042004 white knights1^1 in the upper row, and 20042004 black ones in the lowest row. After a finite number of regular chess moves2^2 , we get the opposite situation where the black ones are on the top and the white ones on the bottom lines. In a turn we make a move with each of the pieces of a color. If you know that each square except those on which the knights originally lie, must not be used more than once in this process, and that after each turn no 22 knights of the same color can be attacking each other3^3 , determine the number of ways in which this can be accomplished.
1^1 also known as horses 2^2 the knight can be moved either one square horizontally and two vertically or two squares horizontally and one vertically, in any direction on both horizontal and vertical lines 3^3 a knight is attacking another knight, if in one chess move, the first one can be placed on the second one’s place
combinatorics3rd edition