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Fill in a "magic" 6x6 square
v_Enhance 12
N
by zuat.e
Source: Taiwan 2014 TST3 Quiz 1, P1
Consider a 
grid. Define a diagonal to be the six squares whose coordinates 
(
satisfy 
for some 
. Hence there are six diagonals.
Determine if it is possible to fill it with the numbers
(each exactly once) such that each row, each column, and each of the six diagonals has the same sum.
grid. Define a diagonal to be the six squares whose coordinates 
(
satisfy 
for some 
. Hence there are six diagonals.Determine if it is possible to fill it with the numbers
(each exactly once) such that each row, each column, and each of the six diagonals has the same sum.
12 replies
Trapezium inscribed in a circle
shivangjindal 27
N
by andrewthenerd
Source: Balkan Mathematics Olympiad 2014 - Problem-3
Let 
be a trapezium inscribed in a circle 
with diameter 
. Let 
be the intersection point of the diagonals 
and 
. The circle with center 
and radius 
meets at the points 
and 
(where is on the same side of as 
). The line perpendicular to at intersects 
at 
. Prove that 
is perpendicular to 
.
Greece - Silouanos Brazitikos
be a trapezium inscribed in a circle 
with diameter 
. Let 
be the intersection point of the diagonals 
and 
. The circle with center 
and radius 
meets at the points 
and 
(where is on the same side of as 
). The line perpendicular to at intersects 
at 
. Prove that 
is perpendicular to 
.Greece - Silouanos Brazitikos
27 replies
Intersection of a cevian with the incircle
djb86 24
N
by Ilikeminecraft
Source: South African MO 2005 Q4
The inscribed circle of triangle 
touches the sides 
, 
and at 
, and 
respectively. Let 
denote the other point of intersection of 
and the inscribed circle. Prove that 
extended passes through the midpoint of 
if and only if 
.
touches the sides 
, 
and at 
, and 
respectively. Let 
denote the other point of intersection of 
and the inscribed circle. Prove that 
extended passes through the midpoint of 
if and only if 
.
24 replies
Convex quadrilateral and midpoints [RMO2-2011, India]
Potla 14
N
by mqoi_KOLA
Let be a convex quadrilateral. Let 
be the midpoints of 
respectively. If 
concur at a point 
prove that is a parallelogram.
be a convex quadrilateral. Let 
be the midpoints of 
respectively. If 
concur at a point 
prove that is a parallelogram.
14 replies
IMO 2014 Problem 4
ipaper 167
N
by bjump
Let 
and be on segment of an acute triangle such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .
Proposed by Giorgi Arabidze, Georgia.
and be on segment of an acute triangle such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .Proposed by Giorgi Arabidze, Georgia.
167 replies
geometry coordinates
CHESSR1DER 0
Source: simplified version of Belarus TST
Points 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
, 
, 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
, 
, 
0 replies
Folding...
Rushil 14
N
by Jupiterballs
Source: RMO 1990 Problem 3
A square sheet of paper is so folded that falls on the mid point of of . Prove that the crease will divide in the ration 
.
is so folded that falls on the mid point of of . Prove that the crease will divide in the ration 
.
14 replies
Nearby lattice point with relatively prime coordinates
MellowMelon 3
N
by quantam13
Source: USA TSTST 2011/2012 P3
Prove that there exists a real constant 
such that for any pair 
of real numbers, there exist relatively prime integers 
and 
satisfying the relation
![\[
\sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2).
\]](//latex.artofproblemsolving.com/7/9/4/7944ded70c80b8788ea6a2ece4dce7c16693f87b.png)
such that for any pair 
of real numbers, there exist relatively prime integers 
and 
satisfying the relation![\[
\sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2).
\]](http://latex.artofproblemsolving.com/7/9/4/7944ded70c80b8788ea6a2ece4dce7c16693f87b.png)
3 replies
Rotating segment by 45 degrees and interchanging endpoints.
Goutham 9
N
by Mathandski
A needle (a segment) lies on a plane. One can rotate it 
round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?
round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?
9 replies
A hard coordinate problem
Dftioscjg 4
N
by jasperE3
A chip moves in a plane provided with an orthonormal coordinate system.
Initially the chip is located at the point of coordinate (0,0). The chip can only perform jumps : of length 5 units, in a straight line and in any direction. Moreover, the chip can only reach points whose two coordinates are integers. The chip wants to reach the point of coordinates(2023,0).
What is the minimum number of jumps she will do?
Initially the chip is located at the point of coordinate (0,0). The chip can only perform jumps : of length 5 units, in a straight line and in any direction. Moreover, the chip can only reach points whose two coordinates are integers. The chip wants to reach the point of coordinates(2023,0).
What is the minimum number of jumps she will do?
4 replies
