MathDB

Triangle Properties

Source:

10/21/2010

Let

a,b,c

denote the sides of a triangle and

[ABC]

the area of the triangle as usual.

(a)

If

6[ABC] = 2a^2+bc

, determine

A,B,C

.

(b)

For all triangles, prove that

3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]

.

geometrytrigonometrygeometry unsolved

Students getting the top mark in n subjects

Source:

11/7/2010

Students have taken a test paper in each of

n \ge 3

subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest

n

for which the above conditions imply that exactly one student got the best score in each of the

n

subjects.

combinatorics unsolvedcombinatorics

Sparse Subsets

Source:

12/9/2010

Let

n > 1

be an integer.
A set

S \subseteq \{ 0, 1, 2, \cdots , 4n - 1 \}

is called ’sparse’ if for any

k \in \{ 0, 1, 2, \cdots , n - 1 \}

the following two conditions are satisﬁed:

(a)

The set

S \cap \{4k - 2, 4k - 1, 4k, 4k + 1, 4k + 2 \}

has at most two elements;

(b)

The set

S \cap \{ 4k +1, 4k +2, 4k +3 \}

has at most one element. Prove that there are exactly

8 \cdot 7^{n-1}

sparse subsets.

combinatorics unsolvedcombinatorics

Nice Floors

Source:

12/9/2010

Solve the equation for positive integers

m, n

:

\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn

floor functionnumber theory unsolvednumber theory

Find all Polynomials

Source:

12/9/2010

Find all polynomials

P

with integer coeﬃcients which satisfy the property that, for any relatively prime integers

a

and

b

, the sequence

\{P (an + b) \}_{n \ge 1}

contains an inﬁnite number of terms, any two of which are relatively prime.

algebrapolynomialmodular arithmeticinductionnumber theoryrelatively primealgebra unsolved

6

Problems(5)