MathDB

ratio of sides, square on hypothenuse of a right triangle

Source: Tournament Of Towns Spring 1999 Junior 0 Level p2

5/7/2020

ABC

is a right-angled triangle. A square

ABDE

is constructed on the opposite side of the hypothenuse

AB

from

C

. The bisector of

\angle C

cuts

DE

at

F

. If

AC = 1

and

BC = 3

, compute

\frac{DF}{EF}

.
(A Blinkov)

ratiosquaregeometryright triangle

TOT 1999 Spring AJ2 tangent to 2 circumcircles, parallelogram

Source:

5/11/2020

Let

O

be the intersection point of the diagonals of a parallelogram

ABCD

. Prove that if the line

BC

is tangent to the circle passing through the points

A, B

, and

O

, then the line

CD

is tangent to the circle passing through the points

B, C

and

O

.
(A Zaslavskiy)

geometryparallelogramcircumcircletangent

TOT 1999 Spring AS2 concyclic wanted, cyclic ABCD given

Source:

5/11/2020

Let all vertices of a convex quadrilateral

ABCD

lie on the circumference of a circle with center

O

. Let

F

be the second intersection point of the circumcircles of the triangles

ABO

and

CDO

. Prove that the circle passing through the points

A, F

and

D

also passes through the intersection point of the segments

AC

and

BD

.
(A Zaslavskiy)

geometrycircumcircleConcyclicCyclicCircumcenter

TOT 1999 Autumn OJ2 d = a^{1999} + b^{1999} + c^{1999} when a + b + c = 0

Source:

5/11/2020

Let

d = a^{1999} + b^{1999} + c^{1999}

, where

a, b

and

c

are integers such that

a + b + c = 0

. (a) May it happen that

d = 2

? (b) May it happen that

d

is prime?
(V Senderov)

number theoryprimeSum of powersSum

junior areas inequalites

Source: Tournament Of Towns Spring 1999 Junior A Level p2

7/19/2024

Let

ABC

be an acute-angled triangle,

C'

and

A'

be arbitrary points on the sides

AB

and

BC

respectively, and

B'

be the midpoint of the side

AC

.
(a) Prove that the area of triangle

A'B'C'

is at most half the area of triangle

ABC

.
(b) Prove that the area of triangle

A'B'C'

is equal to one fourth of the area of triangle

ABC

if and only if at least one of the points

A'

,

C'

is the midpoint of the corresponding side.
(E Cherepanov)

geometryareasgeometric inequality

TOT 1999 Autumn OS2 2^n + n is composite of odd infinite n

Source:

5/11/2020

Prove that there exist infinitely many odd positive integers

n

for which the number

2^n + n

is composite.
(V Senderov)

Compositenumber theoryodd

TOT 1999 Autumn AS2 marked points on a rectangle paper, folding

Source:

5/11/2020

On a rectangular piece of paper there are (a) several marked points all on one straight line, (b) three marked points (not necessarily on a straight line). We are allowed to fold the paper several times along a straight line not containing marked points and then puncture the folded paper with a needle. Show that this can be done so that after the paper has been unfolded all the marked points are punctured and there are no extra holes.
(A Shapovalov)

geometryrectanglefoldingcombinatoricscombinatorial geometry

2

Problems(7)