User MathMagician

4 votes

4 answers

108 views

$f(x)$ and $g(x)$ are monic cubic polynomials, with $f(x)-g(x)=r$. If $f$ has roots $r+1$ and $r+7$, and $g$ has roots $r+3$ and $r+9$, then find $r$.

asked Mar 4 at 16:11

3 votes

3 answers

121 views

A cube is dropped on the floor, and the triangular hole created has sides of lengths $68,$ $75,$ and $77.$ Find the depth of the hole.

asked Mar 11 at 16:51

3 votes

2 answers

72 views

Range of $\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}$ on $0 \le a,$ $b,$ $c,$ $d \le 1.$

asked Apr 5 at 1:01

3 votes

1 answer

74 views

Find $b_{32}$ if $\prod_{n=1}^{32}(1-z^n)^{b_n}\equiv 1-2z \pmod{z^{33}}$ and $b_n\in\mathbb{Z}^{+}$

asked May 4 at 16:35

3 votes

3 answers

167 views

Weird Problem on Polynomial Roots

asked Feb 27 at 19:12

2 votes

2 answers

72 views

Weird Maximization Problem

asked Feb 26 at 23:03

2 votes

1 answer

126 views

$f(f(x - y)) = f(x) f(y) - f(x) + f(y) - xy$

asked Feb 17 at 16:38

2 votes

0 answers

49 views

Minimize $\displaystyle\sum_{1 \le i < j \le 95} a_i a_j$.

asked Feb 24 at 16:39

2 votes

1 answer

92 views

$f(x)=\sum_{k=0}^{n}\binom{n}{k}\sin^{k}x\sec^{k-n}{x}.$

asked Jun 10 at 21:18

1 vote

3 answers

61 views

How many ways are there to select a three digit number $\underline{A}\ \underline{B}\ \underline{C}$ so that $A \neq B$, $B \leq C$, and $A < C$?

asked Mar 8 at 18:21

1 vote

3 answers

159 views

Prove that the solutions to $(z+1)^7 = z^7$ have same real parts

asked Apr 8 at 23:21

1 vote

3 answers

89 views

Maximize $z$ over $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$

asked Apr 9 at 1:56

1 vote

0 answers

70 views

Roots of $x^4+2x^3+2=0$ [duplicate]

asked Apr 9 at 17:38

1 vote

2 answers

134 views

Find $n\ge 3$ such that $\displaystyle\sum_{k=1}^n x_k = 0$ implies $\displaystyle\sum_{\mathrm{cyc}} x_1x_2\le 0$

asked Mar 5 at 15:49

1 vote

1 answer

83 views

$\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} > 6!.$

asked Feb 25 at 16:47

1 vote

1 answer

94 views

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ Find the number of possible values of $f(x)$ for $0 \le x \le 10.$

asked Jan 23 at 21:52

1 vote

0 answers

45 views

Minimize $\max\{xy, 1-x-y+xy, x+y-2xy\}$ over $0\leq x \leq y \leq 1$

asked Feb 9 at 23:34

1 vote

2 answers

123 views

Bounding of $\frac{x-y}{x+y}$ over $x \in [-5,-3]$ and $y \in [2,4]$ [closed]

asked Feb 27 at 16:29

1 vote

1 answer

41 views

Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

asked Feb 27 at 18:45

1 vote

1 answer

53 views

Minimizing problem.

asked Mar 1 at 18:46

1 vote

2 answers

233 views

Minimize $xy$ over $x^2+y^2+z^2=7$ and $xy+xz+yz=4$.

asked Mar 2 at 2:13

1 vote

0 answers

71 views

Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$

asked Feb 28 at 17:22

1 vote

1 answer

29 views

Number of $l$-tuples $(A_1, A_2,\cdots, A_l)$ such that $∅ ⊆ A_1 ⊆ A_2 ⊆ · · · ⊆ A_l ⊆ S$.

asked Aug 19 at 21:43

0 votes

1 answer

44 views

Ways to assign each of the integers $1$ to $n$ to $a_1, a_2,\cdots, a_n$ such that $|a_k − k| ≤ 1$ for $k = 1, 2, . . . , n$

asked Aug 19 at 22:11

0 votes

1 answer

69 views

Find $\displaystyle\min_{y \in \mathbb{R}} \displaystyle\max_{0 \le x \le 1} |x^2 - xy|.$ [closed]

asked Feb 28 at 23:09

0 votes

1 answer

248 views

Parametric Equations on $x= 1 + \sin (\pi t),$ $y= 3 \sin (\pi t)$

asked Mar 1 at 4:11

0 votes

0 answers

43 views

$\max_{x,y>0}\left(\min\left\{x,\dfrac{1}{y},y+\dfrac{1}{x}\right\}\right)$. [duplicate]

asked Mar 1 at 4:25

0 votes

1 answer

65 views

Find $a$ if $4x^4 - ax^3 + bx^2 - cx + 5=0,\frac{r_1}{2} + \frac{r_2}{4} + \frac{r_3}{5} + \frac{r_4}{8} = 1.$

asked Mar 4 at 2:20

0 votes

0 answers

86 views

Prove $(a^2 + b^2)(c^2 + d^2)(e^2 + f^2) \ge (ace + bdf)^2.$

asked Feb 16 at 19:06

0 votes

1 answer

52 views

Rational Function Problem [closed]

asked Feb 16 at 20:40

Stack Exchange Network

44 Questions

$f(x)$ and $g(x)$ are monic cubic polynomials, with $f(x)-g(x)=r$. If $f$ has roots $r+1$ and $r+7$, and $g$ has roots $r+3$ and $r+9$, then find $r$.

A cube is dropped on the floor, and the triangular hole created has sides of lengths $68,$ $75,$ and $77.$ Find the depth of the hole.

Range of $\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}$ on $0 \le a,$ $b,$ $c,$ $d \le 1.$

Find $b_{32}$ if $\prod_{n=1}^{32}(1-z^n)^{b_n}\equiv 1-2z \pmod{z^{33}}$ and $b_n\in\mathbb{Z}^{+}$

Weird Problem on Polynomial Roots

Weird Maximization Problem

$f(f(x - y)) = f(x) f(y) - f(x) + f(y) - xy$

Minimize $\displaystyle\sum_{1 \le i < j \le 95} a_i a_j$.

$f(x)=\sum_{k=0}^{n}\binom{n}{k}\sin^{k}x\sec^{k-n}{x}.$

How many ways are there to select a three digit number $\underline{A}\ \underline{B}\ \underline{C}$ so that $A \neq B$, $B \leq C$, and $A < C$?

Prove that the solutions to $(z+1)^7 = z^7$ have same real parts

Maximize $z$ over $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$

Roots of $x^4+2x^3+2=0$ [duplicate]

Find $n\ge 3$ such that $\displaystyle\sum_{k=1}^n x_k = 0$ implies $\displaystyle\sum_{\mathrm{cyc}} x_1x_2\le 0$

$\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} > 6!.$

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$ Find the number of possible values of $f(x)$ for $0 \le x \le 10.$

Minimize $\max\{xy, 1-x-y+xy, x+y-2xy\}$ over $0\leq x \leq y \leq 1$

Bounding of $\frac{x-y}{x+y}$ over $x \in [-5,-3]$ and $y \in [2,4]$ [closed]

Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

Minimizing problem.

Minimize $xy$ over $x^2+y^2+z^2=7$ and $xy+xz+yz=4$.

Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$

Number of $l$-tuples $(A_1, A_2,\cdots, A_l)$ such that $∅ ⊆ A_1 ⊆ A_2 ⊆ · · · ⊆ A_l ⊆ S$.

Ways to assign each of the integers $1$ to $n$ to $a_1, a_2,\cdots, a_n$ such that $|a_k − k| ≤ 1$ for $k = 1, 2, . . . , n$

Find $\displaystyle\min_{y \in \mathbb{R}} \displaystyle\max_{0 \le x \le 1} |x^2 - xy|.$ [closed]

Parametric Equations on $x= 1 + \sin (\pi t),$ $y= 3 \sin (\pi t)$

$\max_{x,y>0}\left(\min\left\{x,\dfrac{1}{y},y+\dfrac{1}{x}\right\}\right)$. [duplicate]

Find $a$ if $4x^4 - ax^3 + bx^2 - cx + 5=0,\frac{r_1}{2} + \frac{r_2}{4} + \frac{r_3}{5} + \frac{r_4}{8} = 1.$

Prove $(a^2 + b^2)(c^2 + d^2)(e^2 + f^2) \ge (ace + bdf)^2.$

Rational Function Problem [closed]