4 votes
Accepted

Regarding the solution of finding the remainder of $g(x^{12})$ divided by $g(x)$

I confess i was, am, and will be also hating "artificial" solutions for problems that hide the structure. As a college student, many competitions i took part at had problems that were easily ...
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  • 24.3k
4 votes
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A alone takes $a$ more days than A and B, and B alone takes $b$ more days than A and B. Find how long for A and B.

Let A and B together = $x$ days A alone = $a + x$ days B alone = $b + x$ days Per day work of A + Per day work of B = per day work of A + B $\frac{1}{a+x} + \frac{1}{b+x} = \frac{1}{x}$ We get $(a + b ...
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  • 1,978
4 votes
Accepted

Solve for reals $[2022+x(2+\sqrt{x})][111-x(2+\sqrt{x})]=-52^3$

I would use a substitution to keep degrees low, like so: with $a^3=2022+x(2+\sqrt x), b^3 = 111-x(2+\sqrt x)$, we get the simpler system $a^3+b^3=2133, a+b=9$ to solve first. $\implies a^2-ab+b^2 = ...
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  • 42.3k
4 votes
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How many factors of $2400$ are not factors of $3600$?

If you do a prime number decomposition you find $$2400 = 2^5\cdot 3 \cdot 5^2$$ $$3600 = 2^4\cdot 3^2 \cdot 5^2$$ So the only time you can have a factor of $2400$ and not of $3600$ is when $2^5=32$ ...
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  • 1,198
3 votes
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Prove that there is a circle containing exactly $2018$ points

Your idea is good, and you just need to find the simplest rigorous way forward. One way is as follows: Take any ray $L$ with $2018$ points on its left. (I leave you to figure out how to rigorously ...
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  • 53.2k
3 votes

Prove that $\frac{2022}{n} + 4n$ is a perfect square iff $\frac{2022}{n} - 8n$ is a perfect square

You can greatly reduce the number of cases to check by working modulo $4$. We have $$2022 = 2\cdot 3 \cdot 337 \equiv 2\cdot 3\cdot 1 \pmod 4.$$ Since a square must be $0$ or $1$ mod $4$, we can ...
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  • 5,493
3 votes
Accepted

Help in proving $\sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{a(bc+1)}$

Hello NiceMathophile welcome to M.S.E. Here's a hint to solve the problem; Can you show that $\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{bc},$ and then show that $\sqrt{a-1}+\sqrt{bc}\leq \sqrt{a(bc+1)}$? Use ...
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3 votes

APMO 2020 Geometry Problem | Proving lines to be concurrent

You have a beautiful, neat proof for this problem, and it is absolutely correct, in fact it looks much more neat than the official solutions. There is nothing wrong with your solution. But when exams ...
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2 votes
Accepted

Evaluating $\arctan(u)+\arctan(v)+\arctan(w)$, where $u$, $v$, $w$ are the zeros of $P(x) = x^3 - 10x+11$

\begin{multline} \operatorname{Im}[\ln(1+iu)+\ln(1+iv)+\ln(1+iw)] = \operatorname{Im}[\ln((1+iu)(1+iv)(1+iw))] \\= \operatorname{Im}[\ln((-i)^3(i-u)(i-v)(i - w))]= \operatorname{Im}[\ln(iP(i))] \end{...
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  • 16.7k
2 votes

APMO 2020 Geometry Problem | Proving lines to be concurrent

Your proof is fine. By the way, it use with great ingenuity the hypothesis that e 𝐵, 𝐷, 𝐹, 𝐸 are concyclic via the property of radical center.
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2 votes

Prove that there exist primes $p_{i}$ such that $\prod_{i=1}^{k}p_{i} \mid \sum_{i=1}^{k}(p_{i})^{a_{i}}$

Choose $q_{1}$ to be a prime that divides $k-1$ By lemma 3 (proved below) exists $j_{1}$ so that $q_{1}^{j_{1}}+k-2$ is a multiple of some prime $q_{2}>q_{1}$. $\text{ }$ By lemma 3 there exists $...
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2 votes
Accepted

Prove the zeroes of a polynomial are all real and distinct

Assuming $n>1.$ $P$ and $P'$ have no common zeroes so $0=P'(x)-kP(x)\iff k=\frac {P'(x)}{P(x)}=\sum_{j=1}^n\frac {1}{x-x_j}.$ Let $1\le j<n.$ For each $i$ such that $j\ne i\ne j+1,$ the function ...
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2 votes
Accepted

Find the smallest value of the product $ab$

Starting from $2a^2+3ab+b^2+b=0$, use the quadratic formula to obtain an expression for $a$: $$a=\frac{-3b\pm \sqrt{9b^2-8(b^2+b)}}{4}$$ This only has integer solutions when $(b^2-8b)$ is a perfect ...
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1 vote

Prove that ABCD is a rectangle

Since you have shown that $\angle{ABC} = 90°$, what remains to prove is that $\angle{ADC} = 90°$. Let $H_A$ and $H_C$ be the feet of the perpendiculars dropped from $A$ and $C$ to $EF$, respectively. ...
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1 vote
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Prove that ABCD is a rectangle

Let $A':=ED\cap C_2\neq E, C':=FD\cap C_3\neq F$. In virtue of Thales's Theorem, we infer $$\angle EA'B=90^\circ\implies \angle BA'D=90^\circ, \quad \angle EDF=\angle A'DC'=90^\circ, \quad \angle BC'F=...
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  • 8,418
1 vote

Problems in understanding 2008 AMC 12B Problem 19

What I'd do: Consider the real and the imaginary parts of $\alpha$ and $\gamma$ separately $\alpha=\alpha_r+\alpha_i i$, $\gamma=\gamma_r+\gamma_i i $. We have then \begin{align} f(1)&=4 +i+\...
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  • 1,365
1 vote

What is the value of $a_1a_2\cdots a_{2019}$?

Observe $$a_n - a_{n-1} = \frac{2n+1}{n^2(n+1)^2} = \frac{(n+1)^2 - n^2}{n^2 (n+1)^2} = \frac{1}{n^2} - \frac{1}{(n+1)^2}. \tag{1}$$ So $$a_m - a_1 = \sum_{n=2}^m a_n - a_{n-1} = \frac{1}{2^2} - \...
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  • 112k
1 vote

Help in proving $\sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{a(bc+1)}$

Try to apply the Cauchy-Schwarz inequality in the following way: $$ \sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}=\sqrt{a-1}\cdot 1+1\cdot (\sqrt{b-1}+\sqrt{c-1})\le \\ \le\sqrt{(\sqrt{a-1})^2+1^2}\cdot\sqrt{1^2+(\...
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  • 3,797
1 vote

Prove that points E, H, and F are collinear

Here's a trig-based (but not exactly -bashed) solution, which OP is free not to accept. Perhaps someone will find clues to a fully-synthetic approach within. First, a relation based on orthocenter $H$...
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  • 69.2k
1 vote

Prove that points E, H, and F are collinear

Let $T_1$ be the foot from $H$ to $AI$, and $T_2$ be the foot from $E$ to $AI$. Since $D$ is on line $AI$, $AE=AF$, and so $T_2$ is also the foot from $F$ to $AI$. This means that line $EF$ is the ...
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1 vote

Maximum number of students who have passed exactly in one subject : $4$ set Venn diagram problem

This is best understood by a "line diagram" rather than a Venn diagram. From one end, draw a line of length $45$ for Hindi, and one from the other end, avoiding the blank space, one of ...
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1 vote

find the probability both planes can park at a gate

The intended interpretation seems to be that if one airplane is still parked at the gate when the other airplane arrives, the second airplane cannot park at the gate until the first airplane leaves. I ...
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  • 85.3k
1 vote

Solve for reals $[2022+x(2+\sqrt{x})][111-x(2+\sqrt{x})]=-52^3$

This is more of a comment than an answer, but I wanted to show a method. If $a,b,c$ are the roots of the cubic $z^3+pz-q=0$ (so that $a+b+c=0$ and $q=abc$) then $$0=(a^3+pa-q)+(b^3+pb-q)+(c^3+pc-q)=a^...
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  • 96.3k

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