 11,581
 8,045
 Summary

number theory
geometry
stochastics
calculus
topology
functional analysis
Questions
1. (solved by @MathematicalPhysicist ) Show that the difference of the square roots of two consecutive natural numbers which are greater than ##k^2##, is less than ##\dfrac{1}{2k}##, ##k \in \mathbb{N}  \{0\}##. (@QuantumQuest )
2. (solved by @tnich ) Let A, B, C and D be four points in space such that at most one of the distances AB, AC, AD, BC, BD and CD is greater than ##1##. Find the maximum value of the sum of the six distances. (@QuantumQuest )
3. (solved by @PeroK ) A random number chooser can choose only one of the nine integers ##1, 2, \dots, 9## each time, with equal probability for each choice. What is the probability, after n times of choosing (##n \gt 1##), the product of the ##n## chosen numbers be evenly divisible by ##10##. (@QuantumQuest )
4. (solved by @tnich ) Let ##A## be an ##n\times n## matrix with real entries such that ##Ax\cdot x=0## for all ##x\in\mathbb{R}^n##. Show that ##A^2x\cdot x\leq 0## for all ##x\in\mathbb{R}^n##. (@Infrared )
5. Let ##A=\sum_{k=0}^\infty a_k\, , \,B=\sum_{k=0}^\infty b_k## be two convergent series one of which absolutely.
Prove: The Cauchyproduct ##C=\sum_{k=0}^\infty c_k## with ##c_k=\sum_{j=0}^ka_jb_{kj}## converges to ##AB##.
Give an example that absolute convergence of at least one factor is necessary.
6. Prove that a ##T_0## topological group (Kolmogorov space) is already ##T_2## (Hausdorff space).
Show that an infinite linear algebraic group with the Zariski topology is always ##T_0## but never ##T_2##.
Why this discrepancy?
7. Let ##\mathcal{O}(n)## be the group of orthogonal real ##n \times n## matrices. For ##f \in L^p = L^p(\mathbb{R}^n)## we set
$$
A.f(x) = f(A^{1}x)
$$
Show that ##\mathcal{O}_f=\{\,A.f\,\,A\in \mathcal{O}(n)\,\}\subseteq L^p(\mathbb{R}^n)## is compact.
8.
a.) (solved by @Delta2 ) Let ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## be a convex function and ##\lambda_1,\ldots ,\lambda_n## positive weights, i.e. ##\sum_{i=1}^n \lambda_i = 1##. Show that $$f\left(\sum_{i=1}^n \lambda_ix_i \right) \leq \sum_{i=1}^n \lambda_i f(x_i)$$
b.) (solved by @Delta2 ) Let ##g\, : \,[0,1]\longrightarrow \mathbb{R}## be an integrable function such that the continuous function ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## is convex on the image of ##g\,.## Prove
$$
f\left(\dfrac{1}{ba}\int_a^b g(x)\,dx \right) \leq \dfrac{1}{ba}\int_a^b f(g(x))\,dx
$$
c.) (solved by @StoneTemplePython ) Prove without differentiation that the cylinder with the least surface area among the ones with given volume ##V## is the cylinder whose height equals the diameter of its base.
d.) (solved by @Delta2 ) Prove that for any sequence ##a_n \geq \ldots \geq a_1 > 0## of positive real numbers
$$
\dfrac{1}{\dfrac{1}{a_1}}+\dfrac{2}{\dfrac{1}{a_1}+\dfrac{1}{a_2}}+\ldots +\dfrac{n}{\dfrac{1}{a_1}+\ldots+\dfrac{1}{a_n}} < 2 (a_1+\ldots +a_n)
$$
9. Let ##p(x)=x^n+a_{n1}x^{n1}+\ldots +a_1x+a_0## be a nonlinear polynomial with ##a_n=1## and suppose ##(x1)^{k+1}\,\,p(x)## for some positive integer ##k\,.## Prove that
$$
\sum_{j=0}^{n1} a_j> 1+ \dfrac{2k^2}{n}
$$
Hint: At some stage of the proof you will need Chebyshev polynomials.
10. (solved by @aheight ) Consider the triangle ##A=(0,0)\, , \,B=(2\sqrt{3},0)\, , \,C=(3\sqrt{3}\, , \,3+3\sqrt{3})##. Now choose on each side a point, ##M_a,M_b,M_c##, such that the new triangle built by those points is of minimal perimeter.
What is the area of the ##\triangle (M_a,M_b,M_c)\,?##
High Schoolers only.
11. (solved by @etotheipi ) Show that ##\lim_{x\to 0}\dfrac{\tan(\sin x)}{\sin(\tan x)} = 1##. (@QuantumQuest )
12. (solved by @etotheipi ) Calculate the masses of Sun, Earth and Jupiter. You may assume circular orbits. We further calculate with the following data:
13. (solved by @etotheipi ) A car drives at ##72## km/h directly past a resting observer when the driver presses its horn. By what interval does the pitch of the horn change as the car passes the observer? (Speed of sound ##s= 340## m/s.)
14. (solved by @archaic and @lekh2003 ) Consider the sphere ##\mathbb{S}^2=\{\,(x,y,z)\,\,x^2+y^2+z^2=r^2\,\}## and a point ##P\in \mathbb{S}^2##.
Determine the set of all center points of all chords starting in ##P##.
15. (solved by @YoungPhysicist ) At the monthly meeting of former mathematics students, six members choose a real number ##a##, which has to be guessed by a seventh mathematician who had left the room before. He gets the following information after he returned:
(1) ##a## is rational.
(2) ##a## is an integer divisible by ##14##.
(3) ##a## is real and its square equals ##13##.
(4) ##a## is an integer divisible by ##7##.
(5) ##a## is real and the inequality ##0<a^3+a<8,000## holds.
(6) ##a## is even.
He is told, that all pairs ##(1,2),(3,4),(5,6)## always consist of a true and a false statement. What is ##a##?
1. (solved by @MathematicalPhysicist ) Show that the difference of the square roots of two consecutive natural numbers which are greater than ##k^2##, is less than ##\dfrac{1}{2k}##, ##k \in \mathbb{N}  \{0\}##. (@QuantumQuest )
2. (solved by @tnich ) Let A, B, C and D be four points in space such that at most one of the distances AB, AC, AD, BC, BD and CD is greater than ##1##. Find the maximum value of the sum of the six distances. (@QuantumQuest )
3. (solved by @PeroK ) A random number chooser can choose only one of the nine integers ##1, 2, \dots, 9## each time, with equal probability for each choice. What is the probability, after n times of choosing (##n \gt 1##), the product of the ##n## chosen numbers be evenly divisible by ##10##. (@QuantumQuest )
4. (solved by @tnich ) Let ##A## be an ##n\times n## matrix with real entries such that ##Ax\cdot x=0## for all ##x\in\mathbb{R}^n##. Show that ##A^2x\cdot x\leq 0## for all ##x\in\mathbb{R}^n##. (@Infrared )
5. Let ##A=\sum_{k=0}^\infty a_k\, , \,B=\sum_{k=0}^\infty b_k## be two convergent series one of which absolutely.
Prove: The Cauchyproduct ##C=\sum_{k=0}^\infty c_k## with ##c_k=\sum_{j=0}^ka_jb_{kj}## converges to ##AB##.
Give an example that absolute convergence of at least one factor is necessary.
6. Prove that a ##T_0## topological group (Kolmogorov space) is already ##T_2## (Hausdorff space).
Show that an infinite linear algebraic group with the Zariski topology is always ##T_0## but never ##T_2##.
Why this discrepancy?
7. Let ##\mathcal{O}(n)## be the group of orthogonal real ##n \times n## matrices. For ##f \in L^p = L^p(\mathbb{R}^n)## we set
$$
A.f(x) = f(A^{1}x)
$$
Show that ##\mathcal{O}_f=\{\,A.f\,\,A\in \mathcal{O}(n)\,\}\subseteq L^p(\mathbb{R}^n)## is compact.
8.
a.) (solved by @Delta2 ) Let ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## be a convex function and ##\lambda_1,\ldots ,\lambda_n## positive weights, i.e. ##\sum_{i=1}^n \lambda_i = 1##. Show that $$f\left(\sum_{i=1}^n \lambda_ix_i \right) \leq \sum_{i=1}^n \lambda_i f(x_i)$$
b.) (solved by @Delta2 ) Let ##g\, : \,[0,1]\longrightarrow \mathbb{R}## be an integrable function such that the continuous function ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## is convex on the image of ##g\,.## Prove
$$
f\left(\dfrac{1}{ba}\int_a^b g(x)\,dx \right) \leq \dfrac{1}{ba}\int_a^b f(g(x))\,dx
$$
c.) (solved by @StoneTemplePython ) Prove without differentiation that the cylinder with the least surface area among the ones with given volume ##V## is the cylinder whose height equals the diameter of its base.
d.) (solved by @Delta2 ) Prove that for any sequence ##a_n \geq \ldots \geq a_1 > 0## of positive real numbers
$$
\dfrac{1}{\dfrac{1}{a_1}}+\dfrac{2}{\dfrac{1}{a_1}+\dfrac{1}{a_2}}+\ldots +\dfrac{n}{\dfrac{1}{a_1}+\ldots+\dfrac{1}{a_n}} < 2 (a_1+\ldots +a_n)
$$
9. Let ##p(x)=x^n+a_{n1}x^{n1}+\ldots +a_1x+a_0## be a nonlinear polynomial with ##a_n=1## and suppose ##(x1)^{k+1}\,\,p(x)## for some positive integer ##k\,.## Prove that
$$
\sum_{j=0}^{n1} a_j> 1+ \dfrac{2k^2}{n}
$$
Hint: At some stage of the proof you will need Chebyshev polynomials.
10. (solved by @aheight ) Consider the triangle ##A=(0,0)\, , \,B=(2\sqrt{3},0)\, , \,C=(3\sqrt{3}\, , \,3+3\sqrt{3})##. Now choose on each side a point, ##M_a,M_b,M_c##, such that the new triangle built by those points is of minimal perimeter.
What is the area of the ##\triangle (M_a,M_b,M_c)\,?##
High Schoolers only.
11. (solved by @etotheipi ) Show that ##\lim_{x\to 0}\dfrac{\tan(\sin x)}{\sin(\tan x)} = 1##. (@QuantumQuest )
12. (solved by @etotheipi ) Calculate the masses of Sun, Earth and Jupiter. You may assume circular orbits. We further calculate with the following data:
 the gravitational constant ##G=6.67 \cdot 10^{11}\,\dfrac{m^3}{kg\cdot s^2}##
 the Kepler constant for our solar system ##C= \dfrac{T^2}{R^3}=0.29 \cdot 10^{18}\,\dfrac{s^2}{m^3}##
 the acceleration by gravity on earth ##\gamma = 9.81\,\dfrac{m}{s^2}##
 the earth's radius ##R=6,370\,km##
 Io's orbital radius ##R_I=4.22 \cdot 10^{8}\,m##
 Io's orbital period ##T_I=1.77 \,d##
13. (solved by @etotheipi ) A car drives at ##72## km/h directly past a resting observer when the driver presses its horn. By what interval does the pitch of the horn change as the car passes the observer? (Speed of sound ##s= 340## m/s.)
14. (solved by @archaic and @lekh2003 ) Consider the sphere ##\mathbb{S}^2=\{\,(x,y,z)\,\,x^2+y^2+z^2=r^2\,\}## and a point ##P\in \mathbb{S}^2##.
Determine the set of all center points of all chords starting in ##P##.
15. (solved by @YoungPhysicist ) At the monthly meeting of former mathematics students, six members choose a real number ##a##, which has to be guessed by a seventh mathematician who had left the room before. He gets the following information after he returned:
(1) ##a## is rational.
(2) ##a## is an integer divisible by ##14##.
(3) ##a## is real and its square equals ##13##.
(4) ##a## is an integer divisible by ##7##.
(5) ##a## is real and the inequality ##0<a^3+a<8,000## holds.
(6) ##a## is even.
He is told, that all pairs ##(1,2),(3,4),(5,6)## always consist of a true and a false statement. What is ##a##?
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