MathDB

Easy inequality.

Source: Kyrgyz TST 2005.(also Romanian olympiad)

4/20/2005

If reals

a,b,c

satisfy

a^2+b^2+c^2=3

then prove that

|a|+|b|+|c|-abc\leq4

.

6-digit integers with sum of digits 9 2004 Romania District VII p1

Source:

8/15/2024

Find the number of positive

6

digit integers, such that the sum of their digits is

9

, and four of its digits are

2,0,0,4.

[hide= original wording] before finding a typo ..
Find the number of positive

6

digit integers, such that the sum of their digits is

9

, and four of its digits are

1,0,0,4.

Posts 2 and 3 reply to this wording

number theorysum of digits

a^2+b, b^2+a are rationals

Source: 2004 Romania District VIII p1

8/15/2024

We say that the real numbers

a

and

b

have property

P

if:

a^2+b \in Q

and

b^2 + a \in Q

.Prove that:
a) The numbers

a= \frac{1+\sqrt2}{2}

and

b= \frac{1-\sqrt2}{2}

are irrational and have property

P

b) If

a, b

have property

P

and

a+b \in Q -\{1\}

, then

a

and

b

are rational numbers c) If

a, b

have property

P

and

\frac{a}{b} \in Q

, then

a

and

b

are rational numbers.

algebrarational

Romania District Olympiad 2004 - Grade XI

Source:

4/10/2011

Let

(x_n)_{n\ge 0}

a sequence of real numbers defined by

x_0>0

and

x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}

. Compute

\lim_{n\to \infty}x_n

and

\lim_{n\to \infty} \frac{x_n^3}{n^2}

.

limitPutnamreal analysisreal analysis unsolved

How many tripartite arithmetic sequences do we find in {1,...,n} ?

Source: Romanian District Olympiad 2004, Grade X, Problem 1

10/7/2018

From a fixed set formed by the first consecutive natural numbers, find the number of subsets having exactly three elements, and these in arithmetic progression.

arithmetic sequencealgebra

sum of the matrices with fixed rank

Source: RMO district 2004,grade 12, 1

7/12/2005

Let

n\geq 2

and

1 \leq r \leq n

. Consider the set

S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)

. Compute the sum

\sum_{X \in S_r}X

linear algebramatrixlinear algebra unsolved

1

Problems(6)