MathDB

1

Part of 1998 Tournament Of Towns

Problems(7)

TOT 1998 Spring OJ1 points for forming words from given letter set

Source:

5/11/2020
Anya, Borya, and Vasya listed words that could be formed from a given set of letters. They each listed a different number of words : Anya listed the most, Vasya the least . They were awarded points as follows. Each word listed by only one of them scored 22 points for this child. Each word listed by two of them scored 11 point for each of these two children. Words listed by all three of them scored 00 points. Is it possible that Vasya got the highest score, and Anya the lowest?
(A Shapovalov)
Wordscombinatorics
TOT 1998 Spring AJ1 10 pos. integers, square of each divided by others

Source:

5/11/2020
Do there exist 1010 positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers?
(Folklore)
number theorydivisibledivides
TOT 1998 Spring OS1 similar triangle to create a rectangle

Source:

5/11/2020
Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying?
(A Fedotov)
geometrysimilar trianglesTilingcombinatoricscombinatorial geometryrectangle
TOT 1998 Spring AS1 sum a^3/a^2+ab+b^2 >=(a+b+c)/3

Source:

5/11/2020
Prove that a3a2+ab+b2+b3b2+bc+c2+c3c2+ca+a2≥a+b+c3\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\geq \frac{a+b+c}{3} for positive reals a,b,ca,b,c
(S Tokarev)
inequalitiesalgebra
Numbers in a cube

Source: ToT Junior O Level Autumn 1998

10/5/2008
A 20×20×20 20\times20\times20 block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three 1×20×20 1\times20\times20 slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.
geometry3D geometrycombinatorics proposedcombinatorics
TOT 1998 Autumn OS1 19 weights of mass 1,2,...,19, 9 iron,9 bronze, 1 gold

Source:

5/11/2020
Nineteen weights of mass 11 gm, 22 gm, 33 gm, . . . , 1919 gm are given. Nine are made of iron, nine are of bronze and one is pure gold. It is known that the total mass of all the iron weights is 9090 gm more than the total mass of all the bronze ones. Find the mass of the gold weight .
(V Proizvolov)
weighingscombinatorics
TOT 1998 Autumn AJ1 AS1 lcm (a, b) = lcm (a + c, b + c)

Source:

5/11/2020
(a) Prove that for any two positive integers a and b the equation lcm(a,a+5)=lcm(b,b+5)lcm (a, a + 5) = lcm (b, b + 5) implies a=ba = b. (b) Is it possible that lcm(a,b)=lcm(a+c,b+c)lcm (a, b) = lcm (a + c, b + c) for positive integers a,ba, b and cc?
(A Shapovalov)
PS. part (a) for Juniors, both part for Seniors
number theoryleast common multipleLCM