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**Summary:**Differential Equations, Linear Algebra, Topology, Algebraic Geometry, Number Theory, Functional Analysis, Integrals, Hilbert Spaces, Algebraic Topology, Calculus.

**1.**(solved by @etotheipi ) Let ##T## be a planet's orbital period, ##a## the length of the semi-major axis of its orbit. Then

$$

T'(a)=\gamma \sqrt[3]{T(a)}\, , \,T(0)=0

$$

with a constant proportional factor ##\gamma >0.## Solve this equation for all ##a\in \mathbb{R}## and determine whether the solution is unique and why.

**2.**(solved by @Office_Shredder ) Show that the Hadamard (elementwise) product of two positive definite complex matrices is again positive definite.

**3.**A function ##f\, : \,S^k\longrightarrow X## is called antipodal if it is continuous and ##f(-x)=-f(x)## for all ##x\in S^k## and any topological space ##X\subseteq \mathbb{R}^m.## Show that the following statements are equivalent:

(a) For every antipodal map ##f: S^n\longrightarrow \mathbb{R}^n## there is a point ##x\in S^n## satisfying ##f(x)=0.##

(b) There is no antipodal map ##f: S^n\longrightarrow S^{n-1}.##

(c) There is no continuous mapping ##f : B^n\longrightarrow S^{n-1}## that is antipodal on the boundary.

Assume the conditions hold. Prove Brouwer's fixed point theorem:

Any continuous map ##f : B^n\longrightarrow B^n## has a fixed point.

**4.**(solved by @Office_Shredder ) Let ##Y## be an affine, complex variety. Prove that ##Y## is irreducible if and only if ##I(Y)## is a prime ideal.

**5.**Let ##p>5## be a prime number. Show that

$$

\left(\dfrac{6}{p}\right) = 1 \Longleftrightarrow p\equiv k \,(24) \text{ with } k\in \{1,5,19,23\}.

$$

The parentheses are the Legendre symbol.

**6.**(solved by @benorin ) Let ## f\in L^2 ( \mathbb{R} ) ## and ## g : \mathbb{R} \longrightarrow \overline{\mathbb{R}} ## be given as

$$

g(t):=t\int_\mathbb{R} \chi_{[t,\infty )}(|x|)\exp(-t^2(|x|+1))f(x)\,dx

$$

Show that ##g\in L^1(\mathbb{R}).##

**7.(a)**(solved by @etotheipi and @Office_Shredder ) Let ##V## be the pyramid with vertices ##(0,0,1),(0,1,0),(1,0,0)## and ##(0,0,0).## Calculate

$$

\int_V \exp(x+y+z) \,dV

$$

**7.(b)**(solved by @graphking and @julian ) ##A\in \operatorname{GL}(d,\mathbb{R}).## Calculate

$$

\int_{\mathbb{R}^d}\exp(-\|Ax\|_2^2)\,dx

$$

**8.**(solved by @etotheipi ) Consider the Hilbert space ##L^2([0,1])## and its subspace ##K:=\operatorname{span}_\mathbb{C}\{x,1\}##. Let ##\pi^\perp\, : \,H\longrightarrow K## be the orthogonal projection. Give an explicit formula for ##\pi^\perp## and calculate ##\pi^\perp(e^x).##

**9.**(solved by @Office_Shredder ) Prove ##\pi_1(S^n;x)=\{e\}## for ##n\geq 2.##

**10.**(solved by @Office_Shredder ) Let ##U\subseteq \mathbb{R}^{2n}## be an open set and ##f\in C^2(U,\mathbb{R})## a twice continuously differentiable function at a point ##\vec{a}\in U.## Prove that if ##f## has a critical point in ##\vec{a}## and the Hessian matrix ##Hf(\vec{a})## has a negative determinant, then ##f## has neither a local maximum nor a local minimum in ##\vec{a}.##

**High Schoolers only (until 26th)**

11.Show that every non-negative real polynomial ##p(x)## can be written as ##p(x)=a(x)^2+b(x)^2## with ##a(x),b(x)\in \mathbb{R}[x].##

11.

**12.**(solved by @Not anonymous ) Show that all Pythagorean triples ##x^2+y^2=z^2## can be found by

$$

(x,y,z)=d\cdot (u^2-v^2,2uv,u^2+v^2) \text{ with }u,v\in \mathbb{N}\, , \,u>v \quad (*)

$$

and which are primitive (no common divisor of ##x,y,z##) if and only if ##u,v## are coprime and one is odd and the other one even.

(*) corrected statement

**13.**(solved by @Not anonymous ) Write

$$

\sqrt[8]{2207-\dfrac{1}{2207-\dfrac{1}{2207-\dfrac{1}{2207-\ldots}}}}

$$

as ##\dfrac{a+b\sqrt{c}}{d}.##

**14.**(solved by @Not anonymous ) To each positive integer with ##n^2## decimal digits, we associate the determinant of the matrix obtained by writing

the digits in order across the rows. For example, for ##n =2##, to the integer ##8617## we associate ##\det\left(\begin{bmatrix}

8&6\\1&7 \end{bmatrix}\right)=50.## Find, as a function of ##n##, the sum of all the determinants associated with ##n^2-##digit integers. Leading digits are assumed to be nonzero; for example, for ##n = 2##, there are ##9000## determinants: ##\displaystyle{f(2)=\sum_{1000\leq N\leq 9999}\det(N)}.##

**15.**(solved by @Not anonymous ) All squares on a chessboard are labeled from ##1## to ##64## in reading order (from left to right, row by row top-down). Then someone places ##8## rooks on the board such that none threatens any other. Let ##S## be the sum of all squares which carry a rook. List all possible values of ##S.##

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