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MathMagician
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8 votes
2 answers
124 views

Maximize $(1-a)(1-c)+(1-b)(1-d)$ over $a^2+b^2=c^2+d^2=1$. [closed]

4 votes
4 answers
159 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 $r$.

3 votes
3 answers
158 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.

3 votes
2 answers
93 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 $0 \le a,$ $b,$ $c,$ $d \le 1.$

3 votes
1 answer
145 views

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

3 votes
3 answers
192 views

Weird Problem on Polynomial Roots

3 votes
1 answer
146 views

Closed form of $a_1=x$, $a_n=\left\lfloor\dfrac{(n+2)a_{n-1}-2}{n}\right\rfloor,n\ge 2$

2 votes
2 answers
77 views

Weird Maximization Problem

2 votes
1 answer
167 views

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

2 votes
0 answers
57 views

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

1 vote
1 answer
105 views

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

1 vote
1 answer
127 views

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

1 vote
0 answers
48 views

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

1 vote
2 answers
162 views

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

1 vote
1 answer
67 views

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

1 vote
1 answer
57 views

Minimizing problem.

1 vote
2 answers
400 views

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

1 vote
0 answers
78 views

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

1 vote
1 answer
110 views

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

1 vote
3 answers
67 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 $A < C$?

1 vote
3 answers
424 views

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

1 vote
3 answers
96 views

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

1 vote
0 answers
74 views

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

1 vote
2 answers
150 views

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

1 vote
1 answer
42 views

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

1 vote
3 answers
219 views

Distributing 7 red balls and 2 blue balls into 3 containers such that in each container, there are 3 balls and at least 2 of them are red balls.

0 votes
1 answer
49 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 $k = 1, 2, . . . , n$

0 votes
0 answers
39 views

Counting problem involving diagonals (Not duplicate) [duplicate]

0 votes
2 answers
371 views

Proving $(1+a)(1+b)(1+c)\ge 8(1-a)(1-b)(1-c)$ if $a+b+c=1$ [duplicate]

0 votes
0 answers
42 views

Let $a_1=1$, for each $n\geq 1$ let $a_{n+1}=\sqrt{3+2a_n}$. Prove with induction that for all $n\in\mathbb{N}$, we have $a_{n}\leq a_{n+1}\leq 3.$ [duplicate]