MathDB

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Part of 2002 Moldova National Olympiad

Problems(10)

Sending postcards

Source: Moldova NMO 2002 grade 7 problem nr.1

10/31/2008
Before going to vacation, each of the 7 7 pupils decided to send to each of the 3 3 classmates one postcard. Is it possible that each student receives postcards only from the classmates he has sent postcards?
Ratio between volumes

Source: Moldova NMO 2002 grade 7 problem nr.5

10/31/2008
Volume A A equals one fourth of the sum of the volumes B B and C C, while volume B B equals one sixth of the sum of the volumes C C and A A. Find the ratio of the volume C C to the sum of the volumes A A and B B.
ratio
Solution for the equation

Source: Moldova NMO 2002 grade 8 problem nr.1

10/31/2008
Find all real solutions of the equation: [x]\plus{}[x\plus{}\dfrac{1}{2}]\plus{}[x\plus{}\dfrac{2}{3}]\equal{}2002
Rectangle with integer side lenghts and diagonal of 2002....

Source: Moldova NMO 2002 grade 8 problem nr.5

11/2/2008
Several pupils wrote a solution of a math problem on the blackboard on the break. When the teacher came in, a pupil was just clearing the blackboard, so the teacher could only observe that there was a rectangle with the sides of integer lenghts and a diagonal of lenght 2002 2002. Then the teacher pointed out that there was a computation error in pupils' solution. Why did he conclude that?
geometryrectangle
Greatest value of |a|+|b|+|c|

Source: Moldova NMO 2002 grade 9 problem nr.1

11/2/2008
Consider the real numbers a≠0,b,c a\ne 0,b,c such that the function f(x) \equal{} ax^2 \plus{} bx \plus{} c satisfies ∣f(x)∣≤1 |f(x)|\le 1 for all x∈[0,1] x\in [0,1]. Find the greatest possible value of |a| \plus{} |b| \plus{} |c|.
functioninequalitiestriangle inequality
Existance of a commom divisor greater than 1

Source: Moldova NMO 2002 grade 9 problem nr.5

11/2/2008
Integers a1,a2,…a9 a_1,a_2,\ldots a_9 satisfy the relations a_{k\plus{}1}\equal{}a_k^3\plus{}a_k^2\plus{}a_k\plus{}2 for k\equal{}1,2,...,8. Prove that among these numbers there exist three with a common divisor greater than 1 1.
modular arithmetic
Sequence

Source: Moldova NMO 2002 grade 11 problem nr.1

11/3/2008
The sequence (an) (a_n) is defined by a1∈(0,1) a_1\in (0,1) and a_{n\plus{}1}\equal{}a_n(1\minus{}a_n) for n≥1 n\ge 1. Prove that \lim_{n\rightarrow \infty} na_n\equal{}1
limit
nuggets containing gold

Source: Moldova NMO 2002 grade 10 problem nr.1

11/3/2008
We are given three nuggets of weights 1 1 kg, 2 2 kg and 3 3 kg, containing different percentages of gold, and need to cut each nugget into two parts so that the obtained parts can be alloyed into two ingots of weights 1 1 kg ande 5 5 kg containing the same proportion of gold. How we can do that?
Triplets of primes (very easy)

Source: Moldova NMO 2002 grade 10 problem nr.5

11/3/2008
Find all triplets of primes in the form (p, 2p\plus{}1, 4p\plus{}1).
modular arithmetic
Solve a equation

Source: Moldova NMO 2002 grade 11 problem nr.5

11/3/2008
Solve in R \mathbb R the equation \sqrt{1\minus{}x}\equal{}2x^2\minus{}1\plus{}2x\sqrt{1\minus{}x^2}.
trigonometry