MathDB

2013 ToT Spring Junior O p2, x9, delete digit 1, obtain A+1 from A?

Source:

3/4/2020

There is a positive integer

A

. Two operations are allowed: increasing this number by

9

and deleting a digit equal to

1

from any position. Is it always possible to obtain

A+1

by applying these operations several times?

combinatoricsDigitnumber theory

2013 ToT Spring Junior A p2, 10 boys and 10 girls in a line

Source:

3/4/2020

Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.

Sumalgebra

2013 ToT Spring Senior O p2 squares on sides of right triangle

Source:

3/5/2020

Let

C

be a right angle in triangle

ABC

. On legs

AC

and

BC

the squares

ACKL, BCMN

are constructed outside of triangle. If

CE

is an altitude of the triangle prove that

LEM

is a right angle.

geometrySquaresright angleright triangle

2013 ToT Spring Senior A p2 brave boys and girls

Source:

3/5/2020

A boy and a girl were sitting on a long bench. Then twenty more children one after another came to sit on the bench, each taking a place between already sitting children. Let us call a girl brave if she sat down between two boys, and let us call a boy brave if he sat down between two girls. It happened, that in the end all girls and boys were sitting in the alternating order. Is it possible to uniquely determine the number of brave children?

combinatorics

2013 ToT Fall Junior O p2 10-digit number with different digits

Source:

3/22/2020

Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?

Digitsnumber theoryComposite

2013 ToT Fall Junior A p2 10 consequtive numbers , sum of products

Source:

3/22/2020

A math teacher chose

10

consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.

consecutivenumber theorySumProduct

2013 ToT Fall Senior O p2 3 similar triangles on sides of a triangle

Source:

3/22/2020

On the sides of triangle

ABC

, three similar triangles are constructed with triangle

YBA

and triangle

ZAC

in the exterior and triangle

XBC

in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle

YBA

and triangle

ZAC

takes

Y

to

Z, B

to

A

and

A

to

C

). Prove that

AYXZ

is a parallelogram

geometrysimilar trianglesparallelogram

2013 ToT Fall Senior O p2 P(x) + ax^k, Q(x) + bx^{ell} no common

Source:

3/22/2020

Find all positive integers

n

for which the following statement holds: For any two polynomials

P(x)

and

Q(x)

of degree

n

there exist monomials

ax^k

and

bx^{ell}, 0 \le k,\ ell \le n

, such that the graphs of

P(x) + ax^k

and

Q(x) + bx^{ell}

have no common points.

polynomialalgebra

2

Problems(8)