MathDB

1

a^2+b^2+c^2 +4abc<= \frac12, for sidelengths with a+b+c=1 (likely classic)

a,b,c

be side lengths of a triangle with

a+b+c = 1

. Prove that

a^2 +b^2 +c^2 +4abc \le \frac12

.

6

1

some marbles are distributed over 2n + 1 bags, each no of marbles the same

Some marbles are distributed over

2n + 1

bags. Suppose that, whichever bag is removed, it is possible to divide the remaining bags into two groups of

n

bags such that the number of marbles in each group is the same. Prove that all the bags contain the same number of marbles.

5

1

x^2 +5x+16 not divisible by 169

Prove that, for any integer

x

,

x^2 +5x+16

is not divisible by

169

.

3

1

P(x) upon division by (x-a)(x-b)(x-c) when P(a)=a, P(b)=b, P(c)=c

Let

a,b,c

be distinct real numbers and

P(x)

a polynomial with real coefficients. Suppose that the remainders of

P(x)

upon division by

(x-a), (x-b)

and

(x-c)

are

a,b

and

c

, respectively. Find the polynomial that is obtained as the remainder of

P(x)

upon division by

(x-a)(x-b)(x-c)

.

2

1

computational geometry, incenter lies on circumcircle

In a triangle

ABC

, the bisectors of the angles at

B

and

A

meet the opposite sides at

P

and

Q

, respectively. Suppose that the circumcircle of triangle

PQC

passes through the incenter

R

of

\vartriangle ABC

. Given that

PQ = l

, find all sides of triangle

PQR

.

1

no of lines passing through 2 or 3 centers of 27 unit cubes of a 3x3 cube

A cube of edge length

3

consists of

27

unit cubes. Find the number of lines passing through exactly three centers of these

27

cubes, as well as the number of those passing through exactly two such centers.

1990 ITAMO

Subcontests

a^2+b^2+c^2 +4abc<= \frac12, for sidelengths with a+b+c=1 (likely classic)

some marbles are distributed over 2n + 1 bags, each no of marbles the same

x^2 +5x+16 not divisible by 169

P(x) upon division by (x-a)(x-b)(x-c) when P(a)=a, P(b)=b, P(c)=c

computational geometry, incenter lies on circumcircle

no of lines passing through 2 or 3 centers of 27 unit cubes of a 3x3 cube

1990 ITAMO

Subcontests

a^2+b^2+c^2 +4abc&lt;= \frac12, for sidelengths with a+b+c=1 (likely classic)

some marbles are distributed over 2n + 1 bags, each no of marbles the same

x^2 +5x+16 not divisible by 169

P(x) upon division by (x-a)(x-b)(x-c) when P(a)=a, P(b)=b, P(c)=c

computational geometry, incenter lies on circumcircle

no of lines passing through 2 or 3 centers of 27 unit cubes of a 3x3 cube

a^2+b^2+c^2 +4abc<= \frac12, for sidelengths with a+b+c=1 (likely classic)