User Lê Thành Đạt

6 votes

Accepted

Find all possible integers $n$ such that $\sqrt{n + 2} + \sqrt{n + \sqrt{n + 2}}$ is an integer.

answered Jun 13, 2019 at 10:23

5 votes

Accepted

Solve $x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$ over the integers.

answered Jun 9, 2019 at 16:30

3 votes

Accepted

Given that $a$ and $b$ are integers satisfied $3 \mid ab(a + b) + 2$, prove that $9 \mid ab(a + b) + 2$.

answered Jun 6, 2019 at 10:39

3 votes

Accepted

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

answered Jun 8, 2019 at 2:15

2 votes

Accepted

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

answered Jun 6, 2019 at 10:07

2 votes

Accepted

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

answered Jun 14, 2019 at 16:11

2 votes

Accepted

If $x$ and $y$ are integers such that $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$.

answered Jun 15, 2019 at 16:34

2 votes

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Solve for $x, y \in \mathbb R$: $(x - y)^2 + 5y - 3x + 4 = 2\sqrt{(x + 1)(y - 1)}$ and $\dfrac{3xy - 6x - 5y + 11}{\sqrt{x^2 + 1}} = 5$.

answered Jun 9, 2019 at 1:41

2 votes

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$a, b, c$ are positives such that $a + b + c = 1$. Determine the maximal value of $\sum\limits_{cyc}\frac1{a(b + c)} - \frac{a^2 + b^2 + c^2}{2abc}$.

answered Sep 10, 2019 at 15:14

2 votes

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Prove that $\sum_{i = 0}^{n - 1}\lfloor\sqrt{a + \frac{i}{n}}\rfloor = n\lfloor a \rfloor + \lfloor n(a - \lfloor \sqrt a \rfloor) \rfloor$.

answered Dec 22, 2019 at 4:22

2 votes

For △ABC, prove $\frac a{h_a} + \frac b{h_b} + \frac c{h_c} \ge 2 (\tan\frac{\alpha}2+ \tan\frac{\beta}2 + \tan\frac{\gamma}2)$

answered Jan 27, 2020 at 15:35

1 vote

Accepted

$x_{n + 3} = \frac{(n + 1)(n^2 + n + 1)}{n}x_{n + 2} + (n^2 + n + 1)x_{n + 1} - \frac{n + 1}{n}x_n, n \ge 1$, $\sqrt{x_n} \in \mathbb N, n \ge 0$.

answered Feb 2, 2020 at 10:11

1 vote

Given positives $x$ and $y$ such that $x^2 + y^2 = xy + 1$, prove that $\frac{x}{x^2 + y} + \frac{y}{y^2 + x} \le 1$.

answered Jan 13, 2020 at 14:31

1 vote

Accepted

Given reals $x, y , z \ne 0$ satisfying $x^2 - xy + yz$ $= y^2 - yz + zx = z^2 - zx + xy$, calculate the value of $\dfrac{(x + y + z)^3}{xyz}$.

answered Jan 18, 2020 at 10:36

1 vote

Accepted

Given reals $a_1, a_2, \cdots, a_{n - 1}, a_n$ such that $\sum_{i = 1}^na_1^2 = 1$. Calculate the maximum value of $\sum_{cyc}|a_1 - a_2|$.

answered Jan 19, 2020 at 14:45

1 vote

Accepted

Given $F_m$ be the $m^\text{th}$ number in the Fibonacci sequence. Prove that for all natural $n$, $|F_n^2 + F_nF_{n + 1} - F_{n + 1}^2| = 1$.

answered Dec 30, 2019 at 6:02

1 vote

Accepted

Find all function $f \colon \mathbb R \to \mathbb R$ such that $f(xf(y)) + f(yf(x)) = 2xy, \forall x, y \in \mathbb R$.

answered Oct 13, 2019 at 6:08

1 vote

Accepted

Let $\gcd({p, q)} = 1$. Prove that $\sum_{k = 1}^{q - 1}\left\lfloor k \cdot \frac{p}{q} \right\rfloor = \frac{(p - 1)(q - 1)}{2}$.

answered Oct 21, 2019 at 15:14

1 vote

Accepted

Solve the system of equations: $ \sqrt{3(x - y)^2 - 4x + 8y + 5} - \sqrt x = \sqrt{y + 1}$ and $x^2y + y^2 - 3xy - 3x + 7y + 8 = 2x\sqrt{y + 3}$.

answered Aug 21, 2019 at 11:41

1 vote

Accepted

Solve the system of equations: $ \sqrt{y^2 - 8x + 9} - \sqrt[3]{xy - 6x + 12} = 1$ and $\sqrt{2(x - y)^2 + 10x - 6y + 12} - \sqrt{y} = \sqrt{x + 2}$.

answered Aug 26, 2019 at 5:53

1 vote

Accepted

$m$ and $n$ are odd numbers such that $|m^2 - n^2 + 1| \mid (n^2 - 1)$. Prove that $|m^2 - n^2 + 1|$ is a square number.

answered Sep 3, 2019 at 14:52

1 vote

Accepted

Prove that $PQ \perp EF$.

answered Jun 15, 2019 at 5:45

1 vote

Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$

answered Jul 18, 2019 at 14:41

1 vote

How to prove this inequality 4

answered Jun 27, 2019 at 9:03

1 vote

$|xy(x^2-y^2)+yz(y^2-z^2)+xz(z^2-x^2)|\le J (x^2 + y^2 + z^2)^2$, What is the smallest value of J that allows inequality

answered Jul 4, 2019 at 2:49

1 vote

Prove that $3(a^5b+b^5c+c^5a)\geq(a^2c+b^2a+c^2b)^2$

answered Jun 26, 2019 at 15:38

1 vote

Accepted

Solve $(y^2 + xy)(x^2 - x + 1) = 3x - 1$ over the integers.

answered Jun 10, 2019 at 6:01

1 vote

Accepted

Calculate the minimum value of $\frac{x^2 + 4}{y^2 + 1}$ where $1 \le y \le 2$ and $2y \le xy + 2$.

answered Jun 10, 2019 at 13:08

1 vote

Accepted

Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

answered Jun 13, 2019 at 17:14

1 vote

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

answered Jun 14, 2019 at 15:50

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61 Answers

Find all possible integers $n$ such that $\sqrt{n + 2} + \sqrt{n + \sqrt{n + 2}}$ is an integer.

Solve $x^2y^2 - 4x^2y + y^3 + 4x^2 - 3y^2 + 1 = 0$ over the integers.

Given that $a$ and $b$ are integers satisfied $3 \mid ab(a + b) + 2$, prove that $9 \mid ab(a + b) + 2$.

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

If $x$ and $y$ are integers such that $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$.

Solve for $x, y \in \mathbb R$: $(x - y)^2 + 5y - 3x + 4 = 2\sqrt{(x + 1)(y - 1)}$ and $\dfrac{3xy - 6x - 5y + 11}{\sqrt{x^2 + 1}} = 5$.

$a, b, c$ are positives such that $a + b + c = 1$. Determine the maximal value of $\sum\limits_{cyc}\frac1{a(b + c)} - \frac{a^2 + b^2 + c^2}{2abc}$.

Prove that $\sum_{i = 0}^{n - 1}\lfloor\sqrt{a + \frac{i}{n}}\rfloor = n\lfloor a \rfloor + \lfloor n(a - \lfloor \sqrt a \rfloor) \rfloor$.

For △ABC, prove $\frac a{h_a} + \frac b{h_b} + \frac c{h_c} \ge 2 (\tan\frac{\alpha}2+ \tan\frac{\beta}2 + \tan\frac{\gamma}2)$

$x_{n + 3} = \frac{(n + 1)(n^2 + n + 1)}{n}x_{n + 2} + (n^2 + n + 1)x_{n + 1} - \frac{n + 1}{n}x_n, n \ge 1$, $\sqrt{x_n} \in \mathbb N, n \ge 0$.

Given positives $x$ and $y$ such that $x^2 + y^2 = xy + 1$, prove that $\frac{x}{x^2 + y} + \frac{y}{y^2 + x} \le 1$.

Given reals $x, y , z \ne 0$ satisfying $x^2 - xy + yz$ $= y^2 - yz + zx = z^2 - zx + xy$, calculate the value of $\dfrac{(x + y + z)^3}{xyz}$.

Given reals $a_1, a_2, \cdots, a_{n - 1}, a_n$ such that $\sum_{i = 1}^na_1^2 = 1$. Calculate the maximum value of $\sum_{cyc}|a_1 - a_2|$.

Given $F_m$ be the $m^\text{th}$ number in the Fibonacci sequence. Prove that for all natural $n$, $|F_n^2 + F_nF_{n + 1} - F_{n + 1}^2| = 1$.

Find all function $f \colon \mathbb R \to \mathbb R$ such that $f(xf(y)) + f(yf(x)) = 2xy, \forall x, y \in \mathbb R$.

Let $\gcd({p, q)} = 1$. Prove that $\sum_{k = 1}^{q - 1}\left\lfloor k \cdot \frac{p}{q} \right\rfloor = \frac{(p - 1)(q - 1)}{2}$.

Solve the system of equations: $ \sqrt{3(x - y)^2 - 4x + 8y + 5} - \sqrt x = \sqrt{y + 1}$ and $x^2y + y^2 - 3xy - 3x + 7y + 8 = 2x\sqrt{y + 3}$.

Solve the system of equations: $ \sqrt{y^2 - 8x + 9} - \sqrt[3]{xy - 6x + 12} = 1$ and $\sqrt{2(x - y)^2 + 10x - 6y + 12} - \sqrt{y} = \sqrt{x + 2}$.

$m$ and $n$ are odd numbers such that $|m^2 - n^2 + 1| \mid (n^2 - 1)$. Prove that $|m^2 - n^2 + 1|$ is a square number.

Prove that $PQ \perp EF$.

Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$

How to prove this inequality 4

$|xy(x^2-y^2)+yz(y^2-z^2)+xz(z^2-x^2)|\le J (x^2 + y^2 + z^2)^2$, What is the smallest value of J that allows inequality

Prove that $3(a^5b+b^5c+c^5a)\geq(a^2c+b^2a+c^2b)^2$

Solve $(y^2 + xy)(x^2 - x + 1) = 3x - 1$ over the integers.

Calculate the minimum value of $\frac{x^2 + 4}{y^2 + 1}$ where $1 \le y \le 2$ and $2y \le xy + 2$.

Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.