MathDB

3

Part of 2012 Tournament of Towns

Problems(7)

2012 ToT Spring Junior O p3 rectangle wanted, 4 incircles, parallelogram

Source:

3/4/2020
In the parallelogram ABCDABCD, the diagonal ACAC touches the incircles of triangles ABCABC and ADCADC at WW and YY respectively, and the diagonal BDBD touches the incircles of triangles BADBAD and BCDBCD at XX and ZZ respectively. Prove that either W,X,YW,X, Y and ZZ coincide, or WXYZWXYZ is a rectangle.
geometryrectangleparallelogramincircle
2012 ToT Spring Junior A p3, guard on duty for n days in a row

Source:

3/4/2020
In a team of guards, each is assigned a different positive integer. For any two guards, the ratio of the two numbers assigned to them is at least 3:13:1. A guard assigned the number nn is on duty for nn days in a row, off duty for nn days in a row, back on duty for nn days in a row, and so on. The guards need not start their duties on the same day. Is it possible that on any day, at least one in such a team of guards is on duty?
combinatorics
2012 ToT Spring Senior O p3 intersections of y = cos x and x = 100cos (100y)

Source:

3/5/2020
Consider the points of intersection of the graphs y=cos⁡xy = \cos x and x=100cos⁡(100y)x = 100 \cos (100y) for which both coordinates are positive. Let aa be the sum of their xx-coordinates and bb be the sum of their yy-coordinates. Determine the value of ab\frac{a}{b}.
trigonometrySumcoordinatesalgebra
2012 ToT Spring Senior A p3 2n-1| (...(((x^2+a_1)^2 +a_2)^2+...)^2+a_{n-1})^2

Source:

3/5/2020
Let nn be a positive integer. Prove that there exist integers a1,a2,...,ana_1, a_2,..., a_n such that for any integer xx, the number (...(((x2+a1)2+a2)2+...)2+an−1)2+an(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n is divisible by 2n−12n - 1.
number theorydividespolynomialdivisible
2012 ToT Fall Junior A p3 even 2 x 2 sub-tables in 11x 11 table

Source:

3/22/2020
Some cells of a 11×1111 \times 11 table are filled with pluses. It is known that the total number of pluses in the given table and in any of its 2×22 \times 2 sub-tables is even. Prove that the total number of pluses on the main diagonal of the given table is also even. (2×22 \times 2 sub-table consists of four adjacent cells, four cells around a common vertex).
combinatorics
2012 ToT Fall Junior O p3 game “Bomb Squad'' in a 10x10 table

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3/22/2020
A table 10×1010 \times 10 was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex.
combinatoricsgame
2012 ToT Fall Senior O p3 several field trips for a class of 20 students

Source:

3/22/2020
For a class of 2020 students several field trips were arranged. In each trip at least four students participated. Prove that there was a field trip such that each student who participated in it took part in at least 1/171/17-th of all field trips.
combinatorics