MathDB

PQ is parallel to BC

Source: 2013 China TST Quiz 1 Day 1 P1

3/30/2013

The quadrilateral

ABCD

is inscribed in circle

\omega

.

F

is the intersection point of

AC

and

BD

.

BA

and

CD

meet at

E

. Let the projection of

F

on

AB

and

CD

be

G

and

H

, respectively. Let

M

and

N

be the midpoints of

BC

and

EF

, respectively. If the circumcircle of

\triangle MNG

only meets segment

BF

at

P

, and the circumcircle of

\triangle MNH

only meets segment

CF

at

Q

, prove that

PQ

is parallel to

BC

.

geometrycircumcirclegeometric transformationreflectiongeometry proposed

Maximum of sum given two conditions

Source: Chinese TST 1 2012 Day 2 Q1

3/15/2013

Let

n

and

k

be two integers which are greater than

1

. Let

a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m

be non-negative real numbers such that i)

a_1\ge a_2\ge\ldots\ge a_n

and

a_1+a_2+\ldots+a_n=1

; ii) For any integer

m\in\{1,2,\ldots,n\}

, we have that

c_1+c_2+\ldots+c_m\le m^k

. Find the maximum of

c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k

.

inequalities proposedinequalitiesChina TST

A collection of lattice points and their distances

Source: Chinese TST 2 2013 Day 1 Q1

4/1/2013

For a positive integer

k\ge 2

define

\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}

to be a collection of

k^2

lattice points on the cartesian coordinate plane. Let

d_1(k)>d_2(k)>\cdots

be the decreasing sequence of the distinct distances between any two points in

T_k

. Suppose

S_i(k)

be the number of distances equal to

d_i(k)

. Prove that for any three positive integers

m>n>i

we have

S_i(m)=S_i(n)

.

analytic geometryinductionfloor functioncombinatorics proposedcombinatorics

Sum of exponents of primes in factorisation

Source: Chinese TST 2 2013 Day 2 Q1

3/29/2013

For a positive integer

N>1

with unique factorization

N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}

, we define

\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.

Let

a_1,a_2,\dotsc, a_n

be positive integers and

p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)

such that for all positive integers

k

,

\Omega(P(k))

is even. Show that

n

is an even number.

symmetrynumber theory proposednumber theory

2013 China IMO Team Selection Test 3 Day 2 Q1

Source: 25 Mar 2013

3/29/2013

Let

p

be a prime number and

a, k

be positive integers such that

p^a<k<2p^a

. Prove that there exists a positive integer

n

such that

n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.

modular arithmeticgeometrygeometric transformationreal analysisnumber theory proposednumber theory

2013 China IMO Team Selection Test 3 Day 1 Q1

Source:

4/1/2013

Let

n\ge 2

be an integer.

a_1,a_2,\dotsc,a_n

are arbitrarily chosen positive integers with

(a_1,a_2,\dotsc,a_n)=1

. Let

A=a_1+a_2+\dotsb+a_n

and

(A,a_i)=d_i

. Let

(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n

. Find the minimum of

\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}

number theory proposednumber theory

1

Problems(6)