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Most Evil and Brutal Integral Ever Officially Proposed for an Integration Bee

Silver08 3

N 2 hours ago by Silver08

Source: UK University Integration Bee 2024-2025 Round 2 Relay (Singapore)

Compute: $\int_{1}^{2}e^{x( x+\sqrt{x^2-1} )}dx$

3 replies

Silver08

Mar 6, 2025

Silver08

2 hours ago

IMC 2018 P4

ThE-dArK-lOrD 17

N 4 hours ago by sangsidhya

Source: IMC 2018 P4

Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that
$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$
Proposed by Orif Ibrogimov, National University of Uzbekistan

17 replies

ThE-dArK-lOrD

Jul 24, 2018

sangsidhya

4 hours ago

f(x)=x-xe^(-1/x)

Sayan 6

N 4 hours ago by kamatadu

Source: ISI (BS) 2006 #6

(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$ . Show that $f(x)$ is an increasing function on $(0,\infty)$ , and $\lim_{x\to\infty} f(x)=1$ .

(b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$ , and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.

6 replies

Sayan

Jun 2, 2012

kamatadu

4 hours ago

Square of a rational matrix of dimension 2

loup blanc 7

N 6 hours ago by ysharifi

The following exercise was posted -two months ago- on the Website StackExchange; cf.
https://math.stackexchange.com/questions/5006488/image-of-the-squaring-function-on-mathcalm-2-mathbbq
There was no solution on Stack.

-Statement of the exercise-
We consider the matrix function $f:X\in M_2(\mathbb{Q})\mapsto X^2\in M_2(\mathbb{Q})$ .
Find the image of $f$ .
In other words, give a method to decide whether a given matrix has or does not have at least a square root
in $M_2(\mathbb{Q})$ ; if the answer is yes, then give a method to calculate at least one of its roots.

7 replies

loup blanc

Feb 17, 2025

ysharifi

6 hours ago

find the isomorphism

nguyenalex 10

N 6 hours ago by Royrik123456

I have the following exercise:

Let $E$ be an algebraic extension of $K$ , and let $F$ be an algebraic closure of $K$ containing $E$ . Prove that if $\sigma : E \to F$ is an embedding such that $\sigma(c) = c$ for all $c \in K$ , then $\sigma$ extends to an automorphism of $F$ .

My attempt:

Theorem (*): Suppose that $E$ is an algebraic extension of the field $K$ , $F$ is an algebraically closed field, and $\sigma: K \to F$ is an embedding. Then, there exists an embedding $\tau: E \to F$ that extends $\sigma$ . Moreover, if $E$ is an algebraic closure of $K$ and $F$ is an algebraic extension of $\sigma(K)$ , then $\tau$ is an isomorphism.

Back to our main problem:

Since $K \subset E$ and $F$ is an algebraic extension of $K$ , it follows that $F$ is an algebraic extension of $E$ . Assume that there exists an embedding $\sigma : E \to F$ such that $\sigma(c) = c$ for all $c \in K$ . By Theorem (*), there exists an embedding $\tau : F \to F$ that extends $\sigma$ . Since $F$ is algebraically closed, $\tau(F)$ is also an algebraically closed field.

Furthermore, because $\sigma(c) = c$ for all $c \in K$ and $\tau$ is an extension of $\sigma$ , we have
$K = \sigma(K) \subset K \subset \sigma(E) \subset \tau(F) \subset F.$
This implies that $F$ is an algebraic extension of $\tau(F)$ . We conclude that $F = \tau(F)$ , meaning that $\tau$ is an automorphism. (Finished!!)

Let choose $F = A$ be the field of algebraic numbers, $K=\mathbb{Q}$ . Consider the embedding $\sigma: \mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{2}) \subset A$ defined by
$a + b\sqrt{2} \mapsto a - b\sqrt{2}.$ Then, according to the exercise above, $\sigma$ extends to an isomorphism
$\bar{\sigma}: A \to A.$ How should we interpret $\bar{\sigma}$ ?

10 replies

nguyenalex

Yesterday at 3:58 PM

Royrik123456

6 hours ago

Generating functions and recursions smelling from 1000 km

Assassino9931 12

N Today at 11:22 AM by sangsidhya

Source: IMC 2022 Day 1 Problem 3

Let $p$ be a prime number. A flea is staying at point $0$ of the real line. At each minute,
the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right.
After $p-1$ minutes, it wants to be at $0$ again. Denote by $f(p)$ the number of its strategies to do this
(for example, $f(3) = 3$ : it may either stay at $0$ for the entire time, or go to the left and then to the
right, or go to the right and then to the left). Find $f(p)$ modulo $p$ .

12 replies

Assassino9931

Aug 5, 2022

sangsidhya

Today at 11:22 AM

ISI 2018 #3

integrated_JRC 34

N Today at 7:08 AM by anudeep

Source: ISI 2018 B.Stat / B.Math Entrance Exam

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and for all $t\geqslant 0$ , $f(x)=f(e^tx)$ Show that $f$ is a constant function.

34 replies

integrated_JRC

May 13, 2018

anudeep

Today at 7:08 AM

Integration Bee Kaizo

Calcul8er 40

N Today at 7:07 AM by Figaro

Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!

40 replies

Calcul8er

Mar 2, 2025

Figaro

Today at 7:07 AM

A polynomial problem which originates from a combinatorical problem

BlueCloud12 0

Today at 12:58 AM

- Suppose $t > 1$ is irrational, and $\{ \alpha_n \}$ and $\{ \beta_n \}$ are increasing positive integer sequences such that $\operatorname{gcd}(\alpha_n , \beta_n ) = 1$ and $\beta_n = [t \alpha_n]$ . Prove that:
- (1) There are exactly $\beta_n - 1$ roots of $z^{\alpha_n + \beta_n} - 2z^{\beta_n} + 1 = 0$ in $\{ z : |z| < 1\}$ ;
- (2)Denote these roots as $\gamma_i(\alpha_n , \beta_n) (i = 1,2,\ldots , \beta_n - 1)$ . Then $\tfrac{1}{\alpha_n}\prod_{i=1}^{\beta_n - 1} (1 - \gamma_i (\alpha_n , \beta_n ))$ converges.

0 replies

BlueCloud12

Today at 12:58 AM

0 replies

Galois group

ILOVEMYFAMILY 4

N Yesterday at 8:25 PM by ishan.panpaliya

Let $K$ be a field. Find the Galois groups

$a) \text{Gal}(K(x), K)$

$b) \text{Gal}(K(x,y), K)$

4 replies

ILOVEMYFAMILY

Mar 11, 2025

ishan.panpaliya

Yesterday at 8:25 PM

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