MathDB

1

Part of 2000 Tournament Of Towns

Problems(7)

Easy-Medium warm-up :)

Source:

3/17/2014
Can the product of 22 consecutive natural numbers equal the product of 22 consecutive even natural numbers? (natural means positive integers)
inequalities
TOT 2000 Spring AJ1 (x+1)^{21}+(x+1)^{20}(x-1)+...+(x-1)^{21}=0

Source:

5/10/2020
Determine all real numbers that satisfy the equation (x+1)21+(x+1)20(x−1)+(x+1)19(x−1)2+...+(x−1)21=0(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0
(RM Kuznec)
algebraequation
TOT 2000 Spring OS1 sum of areas PAB, PCD = sum of areas PAD,PCB

Source:

5/11/2020
The diagonals of a convex quadrilateral ABCDABCD meet at PP. The sum of the areas of triangles PABPAB and PCDPCD is equal to the sum of areas of triangles PADPAD and PCBPCB. Prove that PP is the midpoint of either ACAC or BDBD.
(Folklore)
areasareageometrymidpoint
TOT 2000 Spring AS1 max of gcd of m + 2000n and n + 2000m

Source:

5/11/2020
Positive integers mm and nn have no common divisor greater than one. What is the largest possible value of the greatest common divisor of m+2000nm + 2000n and n+2000mn + 2000m ?
(S Zlobin)
number theorygreatest common divisor
TOT 2000 Autumn OJ1 numbers in a 4x4 table

Source:

5/10/2020
Each of the 1616 squares in a 4×44 \times 4 table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to 11. Determine the sum of all 1616 numbers in the table.
(R Zhenodarov)
Sumcombinatoricstablesquare table
TOT 2000 Autumn AJ1 numbers on a nxn table

Source:

5/10/2020
Each 1×11 \times 1 square of an n×nn \times n table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical.
(V Klepcyn)
combinatoricssquare tabletable
TOT 2000 Autumn OS1 isosceles wanted, circles related

Source:

5/11/2020
Triangle ABCABC is inscribed in a circle. Chords AMAM and ANAN intersect side BCBC at points KK and LL respectively. Prove that if a circle passes through all of the points K,L,MK, L, M and NN, then ABCABC is an isosceles triangle.
(V Zhgun)
geometryisoscelescircles