MathDB

Representing positive integers in a certain form

Source: Russian TST 2021, Day 4 P1

3/21/2023

Do there exist infinitely many positive integers not expressible in the form

(a+b)+\log_2(b+c)-2^{c+a},

where

a,b,c

are positive integers?

number theoryrepresentation

Vending machine

Source: Russian TST 2021, Day 7 P1

3/21/2023

A machine accepts coins of

k{}

values

1 = a_1 <\cdots < a_k

and sells

k{}

different drinks with prices

0<b_1 < \cdots < b_k

. It is known that if we start inserting coins into the machine in an arbitrary way, sooner or later the total value of the coins will be equal to the price of a drink. For which sets of numbers

(a_1,\ldots,a_k;b_1,\ldots,b_k)

does this property hold?

combinatoricsnumber theory

Excircle geometry 2

Source: Russian TST 2021, Day 8 P1

3/21/2023

A point

P{}

is considered on the incircle of the triangle

ABC

. We draw the tangent segments from

P{}

to the three excircles of

ABC

. Prove that from the obtained three tangent segments it is possible to make a right triangle if and only if the point

P{}

lies on one of the lines connecting two of the midpoints of the sides of

ABC

.

excirclegeometry

P1

Problems(3)