MathDB

3

Part of 2006 Tournament of Towns

Problems(8)

How Many Integer Solutions?

Source: Spring 2006 Tournament of Towns Junior O-Level #3

4/15/2015
Let aa be some positive number. Find the number of integer solutions xx of inequality 2<xa<32 < xa < 3 given that inequality 1<xa<21 < xa < 2 has exactly 33 integer solutions. Consider all possible cases.
(4 points)
inequalities
Intersection of Three Lines

Source: Spring 2006 Tournament of Towns Junior A-Level #3

4/15/2015
On sides ABAB and BCBC of an acute triangle ABCABC two congruent rectangles ABMNABMN and LBCKLBCK are constructed (outside of the triangle), so that AB=LBAB = LB. Prove that straight lines AL,CMAL, CM and NKNK intersect at the same point.
(5 points)
geometry
How Many Integer Solutions?

Source: Spring 2006 Tournament of Towns Senior O-Level #3

9/9/2015
Let aa be some positive number. Find the number of integer solutions xx of inequality 100<xa<1000100 < xa < 1000 given that inequality 10<xa<10010 < xa < 100 has exactly 55 integer solutions. Consider all possible cases.
(4 points)
inequalities
TOT 2006 Spring - Senior A-Level p3 P(x)=x^4+x^3-3x^2+x+2

Source:

2/25/2020
Consider a polynomial P(x)=x4+x3−3x2+x+2P(x) = x^4+x^3-3x^2+x+2. Prove that at least one of the coefficients of (P(x))k(P(x))^k, (kk is any positive integer) is negative. (5)
polynomialalgebra
TOT 2006 Fall - Junior O-Level p3 product of numbers is a^2 - b^2

Source:

2/25/2020
(a) Prove that from 20072007 given positive integers, one of them can be chosen so the product of the remaining numbers is expressible in the form a2−b2a^2 - b^2 for some positive integers aa and bb. (2) (b) One of 20072007 given positive integers is 20062006. Prove that if there is a unique number among them such that the product of the remaining numbers is expressible in the form a2−b2a^2 - b^2 for some positive integers aa and bb, then this unique number is 20062006. (2)
number theoryDifference
TOT 2006 Fall - Senior O-Level p3 2006 x 2006 board

Source:

2/25/2020
Each of the numbers 1,2,3,...,200621, 2, 3,... , 2006^2 is placed at random into a cell of a 2006\times 2006 board. Prove that there exist two cells which share a common side or a common vertex such that the sum of the numbers in them is divisible by 44. (4)
combinatorics
TOT 2006 Fall - Junior A-Level p3 magic sguare

Source:

2/25/2020
A 3×33 \times 3 square is filled with numbers: a,b,c,d,e,f,g,h,ia, b, c, d, e, f, g, h, i in the following way: https://cdn.artofproblemsolving.com/attachments/8/9/737c41e9d0dbfdc81be1b986b8e680290db55e.png Given that the square is magic (sums of the numbers in each row, column and each of two diagonals are the same), show that a) 2(a+c+g+i)=b+d+f+h+4e2(a + c + g + i) = b + d + f + h + 4e. (3) b) 2(a3+c3+g3+i3)=b3+d3+f3+h3+4e32(a^3 + c^3 + g^3 + i^3) = b^3 + d^3 + f^3 + h^3 + 4e^3. (3)
combinatorics
TOT 2006 Fall - Senior A-Level p3 integer part of n\sqrt2, irrational

Source:

2/25/2020
The nn-th digit of number a=0.12457...a = 0.12457... equals the first digit of the integer part of the number n2n\sqrt2. Prove that aa is irrational number. (6)
Integer PartalgebraDigitirrational