MathDB

Problem 6

Part of 2000 Moldova National Olympiad

Problems(6)

prove that n is not p+k^2 if n is 2 mod 3

Source: Moldova 2000 Grade 7 P6

4/23/2021
A natural number n≥5n\ge5 leaves the remainder 22 when divided by 33. Prove that the square of nn is not a sum of a prime number and a perfect square.
number theory
max of xy, weird eq. condition

Source: Moldova 2000 Grade 8 P6

4/25/2021
Assuming that real numbers xx and yy satisfy y(1+x2)=x(1−4y2−1)y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right), find the maximum value of xyxy.
Inequalityinequalities
72!|n^8-n^2 (Moldova 2000 Grade 9 P6)

Source:

4/26/2021
Find all nonnegative integers nn for which n8−n2n^8-n^2 is not divisible by 7272.
number theory
unique solution for logarithmic syseq with parameter

Source: Moldova 2000 Grade 10 P6

4/26/2021
Find all real values of the parameter aa for which the system \begin{align*} &1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\ &\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3) \end{align*}has a unique real solution. Solve the system for these values of aa.
algebra
sequence inequality

Source: Moldova 2000 Grade 11 P6

4/27/2021
Let (an)n≥0(a_n)_{n\ge0} be a sequence of positive numbers that satisfy the relations ai−1ai+1≤ai2a_{i-1}a_{i+1}\le a_i^2 for all i∈Ni\in\mathbb N. For any integer n>1n>1, prove the inequality a0+…+ann+1⋅a1+…+an−1n−1≥a0+…+an−1n⋅a1+…+ann.\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.
Inequalityinequalities
existence of solution of integral equality

Source: Moldova 2000 Grade 12 P6

4/28/2021
Show that there is a positive number pp such that ∫0πxpsin⁡xdx=200010\int^\pi_0x^p\sin xdx=\sqrt[10]{2000}.
calculusintegration