MathDB

1

2 player game with a coin and 9 cells

Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?

6

1

ax^2+by^2+cz^2>=0 if x+y+z=0 , a+b+c>= 0, ab+bc+ca >=0

Let

a,b,c,x,y,z

be real numbers such that

x+y+z=0

,

a+b+c\geq 0

,

ab+bc+ca \ge 0

. Prove that

ax^2+by^2+cz^2\ge 0

5

1

best constant C such that S/(ab+bc+ca)<= C

Find a costant

C

, such that

\frac{S}{ab+bc+ca}\le C

where

a,b,c

are the side lengths of an arbitrary triangle, and

S

is the area of the triangle. (The maximal number of points is given for the best possible constant, with proof.)

2

1

(p+m)/(p+n)= m/n + 1/p^2

Let

p

be a prime number. Find all pairs of coprime positive integers

(m,n)

such that

\frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.

4

1

angle chasing, <AEC=45^o, altitude, angle bisector related

Let

D

be the foot of the altitude towards

BC

in the triangle

ABC

. Let

E

be the intersection of

AB

with the bisector of angle

\angle C

. Assume that the angle

\angle AEC = 45^o

. Determine the angle

\angle EDB

.

3

1

2 XY <= AD + BC, 3D geo inequality , X, Y midpoints of AB,CD

Given the segments

AB

and

CD

not necessarily on the same plane. Point

X

is the midpoint of the segment

AB

, and the point

Y

is the midpoint of

CD

. Given that point

X

is not on line

CD

, and that point

Y

is not on line

AB

, prove that

2 | XY | \le | AD | + | BC |

. When is equality achieved?

2017 Swedish Mathematical Competition

Subcontests

2 player game with a coin and 9 cells

ax^2+by^2+cz^2>=0 if x+y+z=0 , a+b+c>= 0, ab+bc+ca >=0

best constant C such that S/(ab+bc+ca)<= C

(p+m)/(p+n)= m/n + 1/p^2

angle chasing, <AEC=45^o, altitude, angle bisector related

2 XY <= AD + BC, 3D geo inequality , X, Y midpoints of AB,CD

2017 Swedish Mathematical Competition

Subcontests

2 player game with a coin and 9 cells

ax^2+by^2+cz^2&gt;=0 if x+y+z=0 , a+b+c&gt;= 0, ab+bc+ca &gt;=0

best constant C such that S/(ab+bc+ca)&lt;= C

(p+m)/(p+n)= m/n + 1/p^2

angle chasing, &lt;AEC=45^o, altitude, angle bisector related

2 XY &lt;= AD + BC, 3D geo inequality , X, Y midpoints of AB,CD

ax^2+by^2+cz^2>=0 if x+y+z=0 , a+b+c>= 0, ab+bc+ca >=0

best constant C such that S/(ab+bc+ca)<= C

angle chasing, <AEC=45^o, altitude, angle bisector related

2 XY <= AD + BC, 3D geo inequality , X, Y midpoints of AB,CD