MathDB

Real variable inequality, cyc sum a^5+a^3c^2

Source: 2022 Israel TST 3 P2

5/22/2022

The numbers

a

,

b

, and

c

are real. Prove that

(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2\geq 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5a^3)

inequalities

Rings containing integer points

Source: 2022 Israel TST 8 P2

5/21/2022

Define a ring in the plane to be the set of points at a distance of at least

r

and at most

R

from a specific point

O

, where

r<R

are positive real numbers. Rings are determined by the three parameters

(O, R, r)

. The area of a ring is labeled

S

. A point in the plane for which both its coordinates are integers is called an integer point.
a) For each positive integer

n

, show that there exists a ring not containing any integer point, for which

S>3n

and

R<2^{2^n}

.
b) Show that each ring satisfying

100\cdot R<S^2

contains an integer point.

number theorygeometry

Polynomial function on Z^2

Source: 2022 Israel TST test 10 P2

7/18/2022

Let

f: \mathbb{Z}^2\to \mathbb{R}

be a function. It is known that for any integer

C

the four functions of

x

f(x,C), f(C,x), f(x,x+C), f(x, C-x)

are polynomials of degree at most

100

. Prove that

f

is equal to a polynomial in two variables and find its maximal possible degree.
Remark: The degree of a bivariate polynomial

P(x,y)

is defined as the maximal value of

i+j

over all monomials

x^iy^j

appearing in

P

with a non-zero coefficient.

functionalgebrapolynomial

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