Ημέρα 1η
1. Solve the equation
in prime numbers.
2. Given two circles 
and 
with centers 
and 
that intersect at the points 
and 
.Passes through A two lines that intersect the circle at the points 
and 
, and the circle at the points 
and 
(points 
in colinear). Denote the midpoints of the segments 
and 
, through 
and 
.Prove that:
and with centers and that intersect at the points and .Passes through A two lines that intersect the circle at the points and , and the circle at the points and (points in colinear). Denote the midpoints of the segments and , through and .Prove that:
Points 
and lie on a circle
The center of the circle passing through and lies in the middle of the segment 
3. Let 
for which the following conditions::
for which the following conditions:: 
Find the minimum of expression 
.
. Ημέρα 2η
2. Given an inscribed quadrilateral 
, which marked the midpoints of the points 
in this order. Let diagonals 
and 
intersect at point 
. Prove that the triangle 
have the same radius of the circles.
, which marked the midpoints of the points in this order. Let diagonals and intersect at point . Prove that the triangle have the same radius of the circles.3. The cell of a 
board are painted in two colors - white and black. The unit cell of a row (column) is called dominant on the row (the column) if more than half of the cells that row (column) have the same color as this cell. Prove that at least 
cells on the board are dominant in both their row and column.
board are painted in two colors - white and black. The unit cell of a row (column) is called dominant on the row (the column) if more than half of the cells that row (column) have the same color as this cell. Prove that at least cells on the board are dominant in both their row and column. Grade 10
Ημέρα 1η
1. For a positive reals 
prove inequlity:
prove inequlity:
.2. Let be an inscribed quadrilateral, in which 
. On the rays 
and 
are selected points 
and 
, respectively, such that
. Let - the midpoint of .Prove that if 
,so 
.
be an inscribed quadrilateral, in which . On the rays and are selected points and , respectively, such that. Let - the midpoint of .Prove that if ,so .3. There are 
balls numbered from 
to , and 
boxes numbered from to . For each 
, ball number can only be put in the boxes with numbers from to 
. Let 
be an integer from to . In how many ways we can choose balls, boxes and put these balls in the selected boxes so that each box has exactly one ball?
balls numbered from to , and boxes numbered from to . For each , ball number can only be put in the boxes with numbers from to . Let be an integer from to . In how many ways we can choose balls, boxes and put these balls in the selected boxes so that each box has exactly one ball? Ημέρα 2η
1. Let 
-Excircles triangle 
with area 
. touch side 
at the point 
Direct 
intersect at the points and 
.Let 
- area of the quadrilateral 
Similarly, we define 
. Prove that
-Excircles triangle with area . touch side at the point Direct intersect at the points and .Let - area of the quadrilateral Similarly, we define . Prove that
.
Find the smallest 
such that 
divided 
such that divided Grade 11
Ημέρα 1η
2. We call a 
table consisting of zeros and ones right if the sum of the numbers in each row and each column is equal to 
. Two right tables are called similar if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.
table consisting of zeros and ones right if the sum of the numbers in each row and each column is equal to . Two right tables are called similar if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables. 3. Line 
is tangent to the incircle of triangle 
in such a way that the points 
and 
lie on the sides and , respectively. On the sides and are selected points and , respectively, so that 
and 
. Prove that all lines constructed in this manner 
pass through one point
is tangent to the incircle of triangle in such a way that the points and lie on the sides and , respectively. On the sides and are selected points and , respectively, so that and . Prove that all lines constructed in this manner pass through one point Ημέρα 2η
2. Given the rays 
and 
.Inside the smaller angle 
selected points and , such that 
and 
The circle, which concern the rays and 
, intersects the second circle, which concern the rays 
and at the points and 
. Prove that
.
and .Inside the smaller angle selected points and , such that and The circle, which concern the rays and , intersects the second circle, which concern the rays and at the points and . Prove that. 3. Consider the equation 
, where 
are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.
, where are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions. Κάντε κλικ εδώ, για να κατεβάσετε τα θέματα.
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