MathDB

tangent line wanted, 2 lines symmetric wrt common chord 2 circles

Source: 2016 Saudi Arabia IMO TST , level 4, I p3

7/27/2020

Given two circles

(O_1)

and

(O_2)

intersect at

A

and

B

. Let

d_1

and

d_2

be two lines through

A

and be symmetric with respect to

AB

. The line

d_1

cuts

(O_1)

and

(O_2)

at

G, E

(

\ne A

), respectively, the line

d_2

cuts

(O_1)

and

(O_2)

at

F, H

(

\ne A

), respectively, such that

E

is between

A, G

and

F

is between

A, H

. Let

J

be the intersection of

EH

and

FG

. The line

BJ

cuts

(O_1), (O_2)

at

K, L

(

\ne B

), respectively. Let

N

be the intersection of

O_1K

and

O_2L

. Prove that the circle

(NLK)

is tangent to

AB

.

geometrycirclestangent

x [f(x + y) - f (x - y)] = 4y f (x)

Source: 2016 Saudi Arabia IMO TST , level 4, II p3

7/29/2020

Find all functions

f : R \to R

such that

x[f(x + y) - f (x - y)] = 4y f (x)

for any real numbers

x, y

.

algebrafunctionalfunctional equation

product of each pair of 2 non-adjacent numbers is divisible by 2015x2016

Source: 2016 Saudi Arabia IMO TST , level 4, III p3

7/29/2020

Let

n \ge 4

be a positive integer and there exist

n

positive integers that are arranged on a circle such that:

\bullet

The product of each pair of two non-adjacent numbers is divisible by

2015 \cdot 2016

.

\bullet

The product of each pair of two adjacent numbers is not divisible by

2015 \cdot 2016

. Find the maximum value of

n

combinatoricsnumber theorydivisible

Line passes through orthocenter 2

Source: Own- Arab Saudi TST 2016

7/29/2016

Let

ABC

be a triangle inscribed in

(O)

. Two tangents of

(O)

at

B,C

meets at

P

. The bisector of angle

BAC

intersects

(P,PB)

at point

E

lying inside triangle

ABC

. Let

M,N

be the midpoints of arcs

BC

and

BAC

. Circle with diameter

BC

intersects line segment

EN

at

F

. Prove that the orthocenter of triangle

EFM

lies on

BC

.

geometry

P300. Saudi Arabia IMO TST

Source:

6/30/2018

Find the number of permutations

( a_1, a_2, . \ . \ , a_{2016})

of the first

2016

positive integers satisfying the following two conditions:
1.

a_{i+1} - a_i \leq 1

for all

i = 1, 2, . \ . \ . , 2015

, and 2. There are exactly two indices

i < j

with

1 \leq i < j \leq 2016

such that

a_i = i

and

a_j = j

.

SAUmiscellaneous

infinite family of sets, each of size r, no 2 of which share < s elements

Source: 2016 Saudi Arabia IMO TST , level 4+, II p3

7/29/2020

Given two positive integers

r > s

, and let

F

be an infinite family of sets, each of size

r

, no two of which share fewer than

s

elements. Prove that there exists a set of size

r -1

that shares at least

s

elements with each set in

F

.

combinatoricssetSubsets

P in Q[x] a polynomial of degree 2016, m = n^3 + 3n + 1$

Source: 2016 Saudi Arabia IMO TST , level 4+, III p3

7/29/2020

Let

P \in Q[x]

be a polynomial of degree

2016

whose leading coefficient is

1

. A positive integer

m

is nice if there exists some positive integer

n

such that

m = n^3 + 3n + 1

. Suppose that there exist infinitely many positive integers

n

such that

P(n)

are nice. Prove that there exists an arithmetic sequence

(n_k)

of arbitrary length such that

P(n_k)

are all nice for

k = 1,2, 3

,

algebrapolynomialarithmetic sequence

3

Problems(7)