MathDB

3

Part of 2013 Tournament of Towns

Problems(7)

2013 ToT Spring Junior O p3, 11 integer weights

Source:

3/4/2020
Each of 1111 weights is weighing an integer number of grams. No two weights are equal. It is known that if all these weights or any group of them are placed on a balance then the side with a larger number of weights is always heavier. Prove that at least one weight is heavier than 3535 grams.
weighingsnumber theorycombinatorics
2013 ToT Spring Junior A p3 mark 1x1 squares in 19x19 with 10x10 condition

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3/4/2020
There is a 19×1919\times19 board. Is it possible to mark some 1×11\times 1 squares so that each of 10×1010\times 10 squares contain different number of marked squares?
combinatorial geometrycombinatoricssquare table
2013 ToT Spring Senior A p3 lattice triangle with 2 lattice points interior v2

Source:

3/5/2020
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing at least two nodes inside. Prove that there exists a pair of internal nodes such that a straight line connecting them either passes through a vertex or is parallel to side of the triangle.
geometrylattice pointsparallelcombinatorial geometry
2013 ToT Fall Junior A p3 AN divides the bisector of angle C in half.

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3/22/2020
Assume that CC is a right angle of triangle ABCABC and NN is a midpoint of the semicircle, constructed on CBCB as on diameter externally. Prove that ANAN divides the bisector of angle CC in half.
angle bisectorright trianglesemicirclegeometry
2013 ToT Fall Junior O p3 (n, n + 1) < (n, n + 2) <... < (n,n + 35) gcd

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3/22/2020
Denote by (a,b)(a, b) the greatest common divisor of aa and bb. Let nn be a positive integer such that (n,n+1)<(n,n+2)<...<(n,n+35)(n, n + 1) < (n, n + 2) <... < (n,n + 35). Prove that (n,n+35)<(n,n+36)(n, n + 35) < (n,n + 36).
GCDinequalitiesnumber theorygreatest common divisor
2013 ToT Fall Senior O p3 [n, n + 1] &gt; [n, n + 2] &gt;...&gt; [n, n + 35], lcm

Source:

3/22/2020
Denote by [a,b][a, b] the least common multiple of aa and bb. Let nn be a positive integer such that [n,n+1]>[n,n+2]>...>[n,n+35][n, n + 1] > [n, n + 2] >...> [n, n + 35]. Prove that [n,n+35]>[n,n+36][n, n + 35] > [n,n + 36].
LCMnumber theoryinequalitiesleast common multiple
2013 ToT Fall Senior O p3, 4 points concyclic, equilateral, circumcircle

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3/22/2020
Let ABCABC be an equilateral triangle with centre OO. A line through CC meets the circumcircle of triangle AOBAOB at points DD and EE. Prove that points A,OA, O and the midpoints of segments BD,BEBD, BE are concyclic.
geometryEquilateralcircumcircleConcyclic