MathDB

1

$a_1=n^2-10n+23, a_1=n^2-9n+31, a_1=n^2-12n+46.$

n{}

, consider the numbers

a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.

a) Prove that

a_1+a_2+a_3

is even. b) Find all positive integers

n

for which

a_1, a_2

and

a_3

are primes.

3

1

Find the remainder of division of $(m+3)^{1444}$ to $n$

\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n},

where

\frac{m}{n}

is irreducible. a) Find

m+n.

b) Find the remainder of division of

(m+3)^{1444}

to

n{}

.

2

1

find the measure of the angle $\angle MCN.$

The triangle

ABC

has

\angle BCA=90^{\circ}.

Bisector of angle

\angle CAB

intersects the side

BC

in point

P

and bisector of angle

\angle ABC

intersects the side

AC

in point

Q.

If

M

and

N

are projections of

P

and

Q

on side

AB

, find the measure of the angle

\angle MCN.

1

Compute the sum $a_1+a_2+\ldots+a_{2015}.$

For every positive integer

n{}

denote

a_n

as the last digit of the sum of the number from

1

to

n{}

. For example

a_5=5, a_6=1.

a) Find

a_{21}.

b) Compute the sum

a_1+a_2+\ldots+a_{2015}.

5

1

(3x-1)^2/x+(3y-1)^2/y >=1, for x,y>0, x+y=1 Austria Beginners' 2010 p3

Let

x

and

y

be positive real numbers with

x + y =1

. Prove that

\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.

For which

x

and

y

equality holds?
(K. Czakler, GRG 21, Vienna)

2015 Albania JBMO TST

Subcontests

$a_1=n^2-10n+23, a_1=n^2-9n+31, a_1=n^2-12n+46.$

Find the remainder of division of $(m+3)^{1444}$ to $n$

find the measure of the angle $\angle MCN.$

Compute the sum $a_1+a_2+\ldots+a_{2015}.$

(3x-1)^2/x+(3y-1)^2/y >=1, for x,y>0, x+y=1 Austria Beginners' 2010 p3

2015 Albania JBMO TST

Subcontests

$a_1=n^2-10n+23, a_1=n^2-9n+31, a_1=n^2-12n+46.$

Find the remainder of division of $(m+3)^{1444}$ to $n$

find the measure of the angle $\angle MCN.$

Compute the sum $a_1+a_2+\ldots+a_{2015}.$

(3x-1)^2/x+(3y-1)^2/y &gt;=1, for x,y&gt;0, x+y=1 Austria Beginners' 2010 p3

(3x-1)^2/x+(3y-1)^2/y >=1, for x,y>0, x+y=1 Austria Beginners' 2010 p3