MathDB

2012 ToT Spring Senior A p4 sum of 6 lengths in cube >=6\sqrt2

Source:

3/5/2020

Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least

6\sqrt2

.

geometry3D geometrycubeSum

2012 ToT Spring Junior O p4 10 \div ... \div 2

Source:

3/4/2020

Brackets are to be inserted into the expression

10 \div 9 \div 8 \div 7 \div 6 \div 5 \div 4 \div 3 \div 2

so that the resulting number is an integer. (a) Determine the maximum value of this integer. (b) Determine the minimum value of this integer.

number theoryalgebraInteger

2012 ToT Spring Junior A p4 +, - into a nxn table, n steps

Source:

3/4/2020

Each entry in an n\times n table is either

+

or

-

. At each step, one can choose a row or a column and reverse all signs in it. From the initial position, it is possible to obtain the table in which all signs are

+

. Prove that this can be accomplished in at most

n

steps.

combinatoricssquare tabletableSigns

2012 ToT Spring Senior O p4 locus is a circle, cyclic ABCD given

Source:

3/5/2020

A quadrilateral

ABCD

with no parallel sides is inscribed in a circle. Two circles, one passing through

A

and

B

, and the other through

C

and

D

, are tangent to each other at

X

. Prove that the locus of

X

is a circle.

geometryCyclicLocuscirclestangent circles

2012 ToT Fall Junior A p4 incenters coincide, equilateral criterion?

Source:

3/22/2020

Given a triangle

ABC

. Suppose I is its incentre, and

X, Y, Z

are the incentres of triangles

AIB, BIC

and

AIC

respectively. The incentre of triangle

XYZ

coincides with

I

. Is it necessarily true that triangle

ABC

is regular?

geometryincenterEquilateral

2012 ToT Fall Junior O p4 KL bisects height of parallelogram

Source:

3/22/2020

A circle touches sides

AB, BC, CD

of a parallelogram

ABCD

at points

K, L, M

respectively. Prove that the line

KL

bisects the height of the parallelogram drawn from the vertex

C

to

AB

.

parallelogrambisects segmentgeometrycircle

2012 ToT Fall Senior O p4 no fo prime divisors, finite, C(a+b)=C(a)+C(b)

Source:

3/22/2020

Let

C(n)

be the number of prime divisors of a positive integer

n

. (a) Consider set

S

of all pairs of positive integers

(a, b)

such that

a \ne b

and

C(a + b) = C(a) + C(b)

. Is

S

finite or infinite? (b) Define

S'

as a subset of S consisting of the pairs

(a, b)

such that

C(a+b) > 1000

. Is

S'

finite or infinite?

number theoryprimeDivisors

2012 ToT Fall Senior A p4 A_1BC_1I is cyclic then <AKC is obtuse

Source:

3/22/2020

In a triangle

ABC

two points,

C_1

and

A_1

are marked on the sides

AB

and

BC

respectively (the points do not coincide with the vertices). Let

K

be the midpoint of

A_1C_1

and

I

be the incentre of the triangle

ABC

. Given that the quadrilateral

A_1BC_1I

is cyclic, prove that the angle

AKC

is obtuse.

Cyclicobtusemidpointincentergeometry

4

Problems(8)