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LHF
  • Member for 2 years, 5 months
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42 votes
Accepted

Problem from the 2020 Latvian "Sophomore's Dream" competition

18 votes
Accepted

Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives

12 votes

Prove that $p(x)=x^4-x+\frac{1}{2}$ has no real roots.

12 votes

$f(x)=\frac{\sin x}{x}$, prove that $|f^{(n)}(x)|\le \frac{1}{n+1}$

10 votes
Accepted

Prove that $\int_0^1\sqrt{f^4(x)+(\int_0^1f(t)\, dt)^4}\, dx\le \sqrt{2}\int_0^1f^2(x)\,dx$

10 votes

Evaluate $\lim_{n\to\infty}\sum_{k=1}^n\frac{k^3}{\sqrt{n^8+k}}$ & $\lim_{n\to\infty}n\bigg(\sum_{k=1}^n \frac{k^3}{\sqrt{n^8+k}}-\frac{1}{4}\bigg)$.

8 votes

Show that $\int_0^{\pi}\sqrt{9\sin^2t+16\cos^2t}\leq4\pi$

8 votes

Find $\lim_{n\to \infty}\int _0^{\frac{\pi}{2}} \sqrt{1+\sin^nx}$

7 votes

If $f\big(f(x)\big)=2^x-1$ for every $x\in\mathbb{R}$, then what is $f(0)+f(1)$?

7 votes
Accepted

The minimum value of $ x_{1}+\frac{x_{2}^{2}}{2}+\frac{x_{3}^{3}}{3}+\ldots+\frac{x_{n}^{n}}{n}$ if $n$ is a positive integer

7 votes
Accepted

Evaluate the value of $\lim_{n \to \infty}\frac{x_n}{\prod_{i=1}^{n-1} x_i}$ where $x_n=x_{n-1}^2-2, x_1=5$

7 votes
Accepted

Find $\lim_{x\to0} \frac{\log(1+3x)}{f(x)}$ given $\lim_{x\to0} \frac{f(x)}{\sin(x)} = 2$

7 votes
Accepted

How to prove $\lim\limits_{n \to \infty} \frac{n}{\log_2 n!} = 0$

7 votes
Accepted

Find $\lim_{n \to \infty}\sum_{k=0}^n \binom{n}{k}\frac{1}{n^k (k+3)}$

6 votes
Accepted

GRE math question: $ \lim_{x \to 0} \left[ \frac{1}{x^2} \int_0^x \frac{t + t^2}{1 + \sin t}\, \mathrm{d} t \right] $

6 votes
Accepted

Finding xy+yz+zx such that the given determinant = 0

6 votes

Integers that fit for $x$ and $y$: $x^3 + x^2 + x + 1 = y^3$

6 votes
Accepted

High School Olympiad - System of Equations

6 votes
Accepted

Find $\lim\limits_{n \to \infty} n \int_2^e (\ln x)^n \, dx$.

6 votes
Accepted

Prove there exists a $c \in (a,b)$ such that $\frac{f'(c)}{f(c)} = \frac{1}{a-c}+\frac{1}{b-c}.$

6 votes

Showing that $\frac{1}{18} + \frac{1}{19} + \cdots + \frac{1}{47} < 1$ without brute force calculation

6 votes

Inequality with x,y,z fractions $\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$

6 votes
Accepted

Roots of the equation $(x^2+3x+4)^2+3(x^2+3x+4)+4=x$

6 votes
Accepted

Let a,b and c be the roots of the equation $x^3-9x^2+11x-1=0$ and $s=\sqrt a +\sqrt b +\sqrt c$

6 votes
Accepted

Value of $ \lim_{n \to \infty} \int \limits_{0}^{1}nx^n e^{ x^2} ?$

6 votes
Accepted

Weird limit of a product $ \lim_{n\to \infty} \frac{3n+1}{3n}\frac{3n+2019+1}{3n+2019}...\frac{3n+2019n+1}{3n+2029n} $

5 votes

If $x^4+12x-5$ has roots $x_1,x_2,x_3,x_4$ find polynomial with roots $x_1+x_2,x_1+x_3,x_1+x_4,x_2+x_3,x_2+x_4,x_3+x_4$

5 votes

Find all the functions $ f\left( x+y\right) +xy=f\left( x\right) f\left( y\right) $

5 votes

Prove there is $c\in (0,1)$ such that $c^3f(c)+cf(c)-1=0$

5 votes

Find limit of $\frac{n(nx_n-\frac{1}{3})}{\ln n}$ knowing that $x_{n+1}=x_n-3x_n^2$

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