 #1
fresh_42
Mentor
 12,647
 9,166
Summary:

We have a variety of areas this month: calculus, trigonometry, topology, algebra (geometry, linear, abstract), so I hope you all will find an interesting one.
Authors: QQ (QuantumQuest), IR (Infrared), FR (fresh_42)
See also https://www.physicsforums.com/threads/mathchallengemarch2020partii.985311/
Main Question or Discussion Point
Questions
1. (solved by @Antarres, @Not anonymous ) Prove the inequality ##\cos(\theta)^p\leq\cos(p\theta)## for ##0\leq\theta\leq\pi/2## and ##0<p<1##. (IR)
2. (solved by @suremarc ) Let ##F:\mathbb{R}^n\to\mathbb{R}^n## be a continuous function such that ##F(x)F(y)\geq xy## for all ##x,y\in\mathbb{R}^n##. Show that ##F## is a homeomorphism. (IR)
3. (solved by @Fred Wright ) Evaluate the integral ##\int_0^{\infty}\frac{e^{t}\sin(t)}{t}dt\,.## (IR)
4. (solved by @suremarc ) Let ##k## be a field that is not algebraically closed. Let ##n\geq 1##. Show that there exists a polynomial ##p\in k[x_1,\ldots,x_n]## that vanishes only at the origin ##(0,\ldots,0)\in k^n##. (IR)
5. (solved by @Not anonymous ) Find the area of the shape ##T## which is surrounded by the line ##a_1 x + b_1 y + c_1 + a_2 x + b_2 y + c_2 = m , m \gt 0 ## (given that ##a_1 b_2  a_2 b_1 \neq 0##). (QQ)
6. (solved by @Antarres ) Calculate ##I = \iint\limits_{R^2} e^{y x^2} dx dy## (QQ)
7. (solved by @Antarres ) Let's take the vector space of continuous real functions. Let also ##g: V \ni f \rightarrow g(f) \in V## be a linear mapping with ##(g(f))(x) = \int_{0}^{x} f(t) dt\,.## Show that ##g## has no eigenvalues. (QQ)
8. Let ##\mathfrak{g}=\operatorname{lin}_\mathbb{R}\{\,e_1,e_2,e_3,e_4\,\}## on which we define the following multiplication:
$$
[e_1,e_4]=2e_1\; , \;[e_2,e_4]=3e_2e_3\; , \;[e_3,e_4]=e_2+3e_3
$$
and ##[e_i,e_j]=0## otherwise, as well as ##[e_i,e_i]=0##.
Show that
a.) ##\mathfrak{g}## is a Lie algebra.
b.) There exists an ##\alpha_0 \in A(\mathfrak{g})## where $$
A(\mathfrak{g}):=\{\,\alpha : \mathfrak{g}\stackrel{\text{linear}}{\longrightarrow} \mathfrak{g}\,\,\forall\,X,Y \in \mathfrak{g}: [\alpha(X),Y]+[X,\alpha(Y)]=0\,\}
$$
such that ##[\operatorname{ad}X,\alpha_0] \in \mathbb{R}\cdot \alpha_0## for all ##X\in \mathfrak{g}\,.##
c.) The center ##Z(\mathfrak{g})=\{\,0\,\}\,.##
d.) ##\mathfrak{g}## has a one dimensional ideal.
(FR)
9. (solved by @suremarc ) Let ##A,B\in \mathbb{M}(m,\mathbb{R})## and ##\A\,\B\\leq 1## with a submultiplicative matrix norm, then $$\left\\,e^{A+B}e^A\cdot e^B\,\right\\leq 6e^2\cdot \left\\,[A,B]\,\right\$$
(FR)
10. (solved by @julian ) Show that for ##m \times m## matrices ##A,B##
$$
e^{t\,(A+B)} = \lim_{n \to \infty}\left(e^{t\,\frac{A}{n}} \cdot e^{t\,\frac{B}{n}}\right)^n
$$
in a submultiplicative matrix norm.
Hint: You may use the estimation in problem #9. (FR)
High Schoolers only
11. (solved by @etotheipi, @Not anonymous, @lekh2003 ) If ##\tan^2 a = 1 + 2 \tan^2 b## show that it also holds ##\cos 2b  2 \cos 2a = 1## (QQ)
12. (solved by @etotheipi ) Inside a steel sphere of radius ##R## we construct a spherical cavity which is tangent to the steel sphere and passes through its center (##Fig. 1##). Before this construction, the mass of the steel sphere was ##M##. Find the force ##F## (which is due to the Newton's law of gravitation) with which the steel sphere pulls a small sphere of mass ##m## which is at a distance ##d## from its center over the straight line of centers and on the side of the cavity.
##Fig. 1##
Now, I think this way: I find the center of gravity of the sphere with the cavity: its distance from the center of the sphere I would have without constructing the cavity (i.e. concrete sphere), can be found from the equation ##Mgx = \frac{Mg}{8} (\frac{R}{2} + x)## from which I find ##x = \frac{R}{14}##. Then, I find the force ##F## which is exerted on the sphere of mass ##m## by the sphere with the cavity (its mass is the ##\frac{7}{8}## of the mass of the concrete steel sphere), like we had two spheres with a distance ##d + \frac{R}{14}## between them, so ##F = G\frac{\frac{8}{7}Mm}{(d + \frac{R}{14})^2}##.
Is this correct? Try to recreate the solution in detail and justify your answer. If the above solution is not correct, give your solution. (QQ)
13. (solved by @Not anonymous ) Let ##f:\mathbb{Q}\to\mathbb{Q}## be the function ##f(x)=x^32x##. Show that ##f## is injective. (IR)
14. (solved by @Not anonymous, @krns21 ) Given two integers ##n,m## with ##nm\neq 0##. Show that there is a integer expression ##1=sn+tm## if and only if ##n## and ##m## are coprime, i.e. have no proper common divisor. (FR)
15. (solved by @Not anonymous ) Division of an integer by a prime number ##p## leaves us with the possible remainders ##C:=\{\,0,1,2,\ldots ,p1\,\}\,.## We can define an addition and a multiplication on ##C## if we wrap it around ##p##, i.e. we identify ##0=p=2p=3p= \ldots \, , \,1=1+p=1+2p=1+3p=\ldots\, , \,\ldots ## This is called modular arithmetic (modulo ##p##).
Show that for any given numbers ##a,b\in C## the equations ##a+x=b## and ##a\cdot x =b## (##a\neq 0##) have a unique solution.
Is this still true if we drop the requirement that ##p## is prime? (FR)
Remark: This problem is about proof techniques, so be as accurate (not long) as possible, i.e. note which property or condition you use at each step.
1. (solved by @Antarres, @Not anonymous ) Prove the inequality ##\cos(\theta)^p\leq\cos(p\theta)## for ##0\leq\theta\leq\pi/2## and ##0<p<1##. (IR)
2. (solved by @suremarc ) Let ##F:\mathbb{R}^n\to\mathbb{R}^n## be a continuous function such that ##F(x)F(y)\geq xy## for all ##x,y\in\mathbb{R}^n##. Show that ##F## is a homeomorphism. (IR)
3. (solved by @Fred Wright ) Evaluate the integral ##\int_0^{\infty}\frac{e^{t}\sin(t)}{t}dt\,.## (IR)
4. (solved by @suremarc ) Let ##k## be a field that is not algebraically closed. Let ##n\geq 1##. Show that there exists a polynomial ##p\in k[x_1,\ldots,x_n]## that vanishes only at the origin ##(0,\ldots,0)\in k^n##. (IR)
5. (solved by @Not anonymous ) Find the area of the shape ##T## which is surrounded by the line ##a_1 x + b_1 y + c_1 + a_2 x + b_2 y + c_2 = m , m \gt 0 ## (given that ##a_1 b_2  a_2 b_1 \neq 0##). (QQ)
6. (solved by @Antarres ) Calculate ##I = \iint\limits_{R^2} e^{y x^2} dx dy## (QQ)
7. (solved by @Antarres ) Let's take the vector space of continuous real functions. Let also ##g: V \ni f \rightarrow g(f) \in V## be a linear mapping with ##(g(f))(x) = \int_{0}^{x} f(t) dt\,.## Show that ##g## has no eigenvalues. (QQ)
8. Let ##\mathfrak{g}=\operatorname{lin}_\mathbb{R}\{\,e_1,e_2,e_3,e_4\,\}## on which we define the following multiplication:
$$
[e_1,e_4]=2e_1\; , \;[e_2,e_4]=3e_2e_3\; , \;[e_3,e_4]=e_2+3e_3
$$
and ##[e_i,e_j]=0## otherwise, as well as ##[e_i,e_i]=0##.
Show that
a.) ##\mathfrak{g}## is a Lie algebra.
b.) There exists an ##\alpha_0 \in A(\mathfrak{g})## where $$
A(\mathfrak{g}):=\{\,\alpha : \mathfrak{g}\stackrel{\text{linear}}{\longrightarrow} \mathfrak{g}\,\,\forall\,X,Y \in \mathfrak{g}: [\alpha(X),Y]+[X,\alpha(Y)]=0\,\}
$$
such that ##[\operatorname{ad}X,\alpha_0] \in \mathbb{R}\cdot \alpha_0## for all ##X\in \mathfrak{g}\,.##
c.) The center ##Z(\mathfrak{g})=\{\,0\,\}\,.##
d.) ##\mathfrak{g}## has a one dimensional ideal.
(FR)
9. (solved by @suremarc ) Let ##A,B\in \mathbb{M}(m,\mathbb{R})## and ##\A\,\B\\leq 1## with a submultiplicative matrix norm, then $$\left\\,e^{A+B}e^A\cdot e^B\,\right\\leq 6e^2\cdot \left\\,[A,B]\,\right\$$
(FR)
10. (solved by @julian ) Show that for ##m \times m## matrices ##A,B##
$$
e^{t\,(A+B)} = \lim_{n \to \infty}\left(e^{t\,\frac{A}{n}} \cdot e^{t\,\frac{B}{n}}\right)^n
$$
in a submultiplicative matrix norm.
Hint: You may use the estimation in problem #9. (FR)
High Schoolers only
11. (solved by @etotheipi, @Not anonymous, @lekh2003 ) If ##\tan^2 a = 1 + 2 \tan^2 b## show that it also holds ##\cos 2b  2 \cos 2a = 1## (QQ)
12. (solved by @etotheipi ) Inside a steel sphere of radius ##R## we construct a spherical cavity which is tangent to the steel sphere and passes through its center (##Fig. 1##). Before this construction, the mass of the steel sphere was ##M##. Find the force ##F## (which is due to the Newton's law of gravitation) with which the steel sphere pulls a small sphere of mass ##m## which is at a distance ##d## from its center over the straight line of centers and on the side of the cavity.
##Fig. 1##
Now, I think this way: I find the center of gravity of the sphere with the cavity: its distance from the center of the sphere I would have without constructing the cavity (i.e. concrete sphere), can be found from the equation ##Mgx = \frac{Mg}{8} (\frac{R}{2} + x)## from which I find ##x = \frac{R}{14}##. Then, I find the force ##F## which is exerted on the sphere of mass ##m## by the sphere with the cavity (its mass is the ##\frac{7}{8}## of the mass of the concrete steel sphere), like we had two spheres with a distance ##d + \frac{R}{14}## between them, so ##F = G\frac{\frac{8}{7}Mm}{(d + \frac{R}{14})^2}##.
Is this correct? Try to recreate the solution in detail and justify your answer. If the above solution is not correct, give your solution. (QQ)
13. (solved by @Not anonymous ) Let ##f:\mathbb{Q}\to\mathbb{Q}## be the function ##f(x)=x^32x##. Show that ##f## is injective. (IR)
14. (solved by @Not anonymous, @krns21 ) Given two integers ##n,m## with ##nm\neq 0##. Show that there is a integer expression ##1=sn+tm## if and only if ##n## and ##m## are coprime, i.e. have no proper common divisor. (FR)
15. (solved by @Not anonymous ) Division of an integer by a prime number ##p## leaves us with the possible remainders ##C:=\{\,0,1,2,\ldots ,p1\,\}\,.## We can define an addition and a multiplication on ##C## if we wrap it around ##p##, i.e. we identify ##0=p=2p=3p= \ldots \, , \,1=1+p=1+2p=1+3p=\ldots\, , \,\ldots ## This is called modular arithmetic (modulo ##p##).
Show that for any given numbers ##a,b\in C## the equations ##a+x=b## and ##a\cdot x =b## (##a\neq 0##) have a unique solution.
Is this still true if we drop the requirement that ##p## is prime? (FR)
Remark: This problem is about proof techniques, so be as accurate (not long) as possible, i.e. note which property or condition you use at each step.
Last edited: