1. Let 
and 
. Prove the inequality
2. Acute angled triangle
is given. A line 
parallel to side 
passing through vertex 
is drawn. Let the angle bisectors of 
and 
intersect the sides 
and 
at points 
and 
, and line at points 
and 
respectively. Prove that if 
then 
.
3. Find all natural numbers
for which each natural number written with 
'ones' and one 'seven' is prime.
4. Find all functions
which satisfy the equation:
5. A table of the type
is defined in the following way: 
squares are ordered horizontally one next to another, then 
squares are ordered horizontally beneath the already ordered squares. The procedure continues until a net composed of squares in the first row, in the second, 
in the 
-th row is obtained, such that there are totally 
squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from 
till in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type 
and one possible way of filling it is attached to the post. Find the number of ways the table of type 
can be filled.
and . Prove the inequality 2. Acute angled triangle
is given. A line parallel to side passing through vertex is drawn. Let the angle bisectors of and intersect the sides and at points and , and line at points and respectively. Prove that if then . 3. Find all natural numbers
for which each natural number written with 'ones' and one 'seven' is prime. 4. Find all functions
which satisfy the equation: 5. A table of the type
is defined in the following way: squares are ordered horizontally one next to another, then squares are ordered horizontally beneath the already ordered squares. The procedure continues until a net composed of squares in the first row, in the second, in the -th row is obtained, such that there are totally squares in the net. The ordered rows form a straight line on the left, as shown in the example. The obtained table is filled with the numbers from till in a way that the numbers in each row and column become greater from left to right and from top to bottom, respectively. An example of a table of the type and one possible way of filling it is attached to the post. Find the number of ways the table of type can be filled. 
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