MathDB

2

Part of 2001 Tournament Of Towns

Problems(8)

Bad Grades

Source: ToT - 2001 Spring Senior A-Level #2

8/17/2011
At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.
combinatorics unsolvedcombinatorics
Midlines of a Triangle

Source: ToT - 2001 Spring Junior O-Level #2

8/17/2011
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.
geometry unsolvedgeometry
Piles of Stones

Source: ToT - 2001 Spring Junior A-Level #2

8/17/2011
In three piles there are 51,4951, 49, and 55 stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get 105105 piles with one stone in each?
number theorygreatest common divisornumber theory unsolved
Can you make 2001?

Source: ToT - 2001 Spring Senior O-Level #2

8/17/2011
The decimal expression of the natural number aa consists of nn digits, while that of a3a^3 consists of mm digits. Can n+mn + m be equal to 2001?
geometry3D geometrynumber theory unsolvednumber theory
Prove a Mistake

Source: ToT - 2001 Fall Junior O-Level #2

8/17/2011
Clara computed the product of the first nn positive integers, and Valerie computed the product of the first mm even positive integers, where m≥2m\ge2. They got the same answer. Prove that one of them had made a mistake.
number theory unsolvednumber theory
n-gons

Source: ToT - 2001 Fall Junior A-Level #2

8/17/2011
Let n≥3n\ge3 be an integer. A circle is divided into 2n2n arcs by 2n2n points. Each arc has one of three possible lengths, and no two adjacent arcs have the same lengths. The 2n2n points are colored alternately red and blue. Prove that the nn-gon with red vertices and the nn-gon with blue vertices have the same perimeter and the same area.
geometryperimetergeometry unsolved
Consecutive Primes

Source: ToT - 2001 Fall Senior O-Level #2

8/17/2011
There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, 1001!+2,1001!+3,...,1001!+10011001!+2,1001!+3,...,1001!+1001. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?
number theoryprime numbersnumber theory unsolved
Greater than the LCM

Source: ToT - 2001 Fall Senior A-Level #2

8/17/2011
Do there exist positive integers a1<a2<…<a100a_1<a_2<\ldots<a_{100} such that for 2≤k≤1002\le k\le100, the least common multiple of ak−1a_{k-1} and aka_k is greater than the least common multiple of aka_k and ak+1a_{k+1}?
number theoryleast common multipleinductionnumber theory unsolved