MathDB

writing number in 7x7 board

Source: Ukraine 1999 Grade 8 P2

5/9/2021

Is it possible to write numbers in the cells of a

7\times7

board in such a way that the sum of numbers in every

2\times2

or

3\times3

square is divisible by

1999

, but the sum of all numbers in the board is not divisible by

1999

?

number theorycombinatorics

geometric inequality with sides of triangle and arbitrary reals

Source: Ukraine 1999 Grade 9 P2

5/10/2021

Let

x

and

y

be positive real numbers with

(x-1)(y-1)\ge1

. Prove that for sides

a,b,c

of an arbitrary triangle we have

a^2x+b^2y>c^2

.

geometric inequalitygeometryinequalities

area of triangle, parallel line intersection points given

Source: Ukraine 1999 Grade 10 P2

5/11/2021

Let

M

be a point inside a triangle

ABC

. The line through

M

parallel to

AC

meets

AB

at

N

and

BC

at

K

. The lines through

M

parallel to

AB

and

BC

meet

AC

at

D

and

L

, respectively. Another line through

M

intersects the sides

AB

and

BC

at

P

and

R

respectively such that

PM=MR

. Given that the area of

\triangle ABC

is

S

and that

\frac{CK}{CB}=a

, compute the area of

\triangle PQR

.

geometryTriangleTriangles

unique solution for system of inequalities with parameter

Source: Ukraine 1999 Grade 11 P2

5/11/2021

Find all values of the parameter

k

for which the system of inequalities \begin{align*} ky^2+4ky-2x+6k+3&\le0\\ kx^2-2y-2kx+3k-3&\le0 \end{align*}has a unique solution.

inequalities

Problem 2

Problems(4)