MathDB
Miklós Schweitzer 1959- Problem 10

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11/8/2015
10. Prove that if a graph with 2n+12n+1 vertices has at least 3n+13n+1 edges, then the graph contains a circuit having an even number of edges. Prove further that this statemente does not hold for 3n3n edges. (By a circuit, we mean a closed line which does not intersect itself.) (C. 5)
college contests
Miklós Schweitzer 1959- Problem 8

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11/8/2015
8. An Oblique lattice-cubs is a lattice-cube of the three-dimensional fundamental lattice no edge of which is perpendicular to any coordinate axis. Prove that for any integer h=8n−1h= 8n-1 (n=1,2,…n= 1, 2, \dots ) there existis an oblique lattice-cube with edges of length hh. Propose a method for finding such a cube. (N. 20)
college contests
Miklós Schweitzer 1959- Problem 9

Source:

11/8/2015
9. Let f(z)=zn+a1zn−1+⋯+anf(z)= z^n +a_1 z^{n-1}+\dots + a_n be a polynomial over the field of the complex numbers and let EfE_f denote the closed (not necessarily connected) domain of complex numbers zz for which ∣f(z)∣≤1\mid f(z) \mid \leq 1. Show that there exists a point z0∈Efz_0 \in E_f such that ∣f′(z0)∣≥n\mid f'(z_0) \mid \geq n. (F. 5)
college contests
Miklós Schweitzer 1959- Problem 7

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11/8/2015
7. Let (zn)n=1∞(z_n)_{n=1}^{\infty} be a sequence of complex numbers tending to zero. Prove that there exists a sequence (ϵn)n=1∞(\epsilon_n)_{n=1}^{\infty} (where ϵn=+1\epsilon_n = +1 or −1-1) such that the series
∑n=1∞ϵnzn\sum_{n=1}^{\infty} \epsilon_n z_n
is convergente. (F. 9)
complex numberscollege contestsreal analysis
Miklos Schweitzer 1964_1

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9/20/2008
Among all possible representations of the positive integer n n as n\equal{}\sum_{i\equal{}1}^k a_i with positive integers k,a1<a2<...<ak k, a_1 < a_2 < ...<a_k, when will the product \prod_{i\equal{}1}^k a_i be maximum?
combinatorics proposedcombinatorics
Miklos Schweitzer 1964_2

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9/20/2008
Let p p be a prime and let l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ . be homogeneous linear polynomials with integral coefficients. Suppose that for every pair (ξ,η) (\xi,\eta) of integers, not both divisible by p p, the values lk(ξ,η),  1≤k≤p2 l_k(\xi,\eta), \;1\leq k\leq p^2 , represent every residue class mod  p \textrm{mod} \;p exactly p p times. Prove that the set of pairs {(ak,bk):1≤k≤p2} \{(a_k,b_k): 1\leq k \leq p^2 \} is identical mod  p \textrm{mod} \;p with the set \{(m,n): 0\leq m,n \leq p\minus{}1 \}.
algebrapolynomialcalculusintegrationadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1964_3

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9/20/2008
Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.
abstract algebrageometric seriessuperior algebrasuperior algebra unsolved
Miklos Schweitzer 1964_4

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9/20/2008
Let A1,A2,...,An A_1,A_2,...,A_n be the vertices of a closed convex n n-gon K K numbered consecutively. Show that at least n\minus{}3 vertices Ai A_i have the property that the reflection of Ai A_i with respect to the midpoint of A_{i\minus{}1}A_{i\plus{}1} is contained in K K. (Indices are meant mod  n . \textrm{mod} \;n\ .)
geometrygeometric transformationreflectionadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1964_5

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9/20/2008
Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?
topologyadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1964_6

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9/20/2008
Let y1(x) y_1(x) be an arbitrary, continuous, positive function on [0,A] [0,A], where A A is an arbitrary positive number. Let yn+1=2∫0xyn(t)dt  (n=1,2,...) . y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ . Prove that the functions yn(x) y_n(x) converge to the function y=x2 y=x^2 uniformly on [0,A] [0,A].
functionintegrationreal analysisreal analysis unsolved
Miklos Schweitzer 1964_8

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9/20/2008
Let F F be a closed set in the n n-dimensional Euclidean space. Construct a function that is 0 0 on F F, positive outside F F , and whose partial derivatives all exist.
functioncalculusderivativetopologyreal analysisreal analysis unsolved
Miklos Schweitzer 1964_9

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9/20/2008
Let E E be the set of all real functions on I\equal{}[0,1]. Prove that one cannot define a topology on E E in which fn→f f_n\rightarrow f holds if and only if fn f_n converges to f f almost everywhere.
functiontopologyreal analysisreal analysis unsolved
Miklos Schweitzer 1964_7

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9/20/2008
Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution f(t) f(t) and any real number c,f(t\plus{}c) is also a solution.
invariantfunctionreal analysisreal analysis unsolved
Miklos Schweitzer 1963_1

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9/19/2008
Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]
geometryperimeter3D geometrytetrahedronadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1964_10

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9/20/2008
Let ε1,ε2,...,ε2n \varepsilon_1,\varepsilon_2,...,\varepsilon_{2n} be independent random variables such that P(\varepsilon_i\equal{}1)\equal{}P(\varepsilon_i\equal{}\minus{}1)\equal{}\frac 12 for all i i, and define S_k\equal{}\sum_{i\equal{}1}^k \varepsilon_i, \;1\leq k \leq 2n. Let N2n N_{2n} denote the number of integers k∈[2,2n] k\in [2,2n] such that either Sk>0 S_k>0, or S_k\equal{}0 and S_{k\minus{}1}>0. Compute the variance of N2n N_{2n}.
integrationprobability and stats
Miklos Schweitzer 1963_2

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9/19/2008
Show that the center of gravity of a convex region in the plane halves at least three chords of the region. [Gy. Hajos]
searchadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1963_3

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9/19/2008
Let R\equal{}R_1\oplus R_2 be the direct sum of the rings R1 R_1 and R2 R_2, and let N2 N_2 be the annihilator ideal of R2 R_2 (in R2 R_2). Prove that R1 R_1 will be an ideal in every ring R~ \widetilde{R} containing R R as an ideal if and only if the only homomorphism from R1 R_1 to N2 N_2 is the zero homomorphism. [Gy. Hajos]
superior algebrasuperior algebra unsolved
Miklos Schweitzer 1963_4

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9/19/2008
Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let f(x) f(x) be a polynomial with f(0)\not\equal{}0 such that f(xn) f(x^n) is positive reducible for some natural number n n. Prove that f(x) f(x) itself is positive reducible. [L. Redei]
algebrapolynomialsuperior algebrasuperior algebra unsolved
Miklos Schweitzer 1963_5

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9/19/2008
Let H H be a set of real numbers that does not consist of 0 0 alone and is closed under addition. Further, let f(x) f(x) be a real-valued function defined on H H and satisfying the following conditions:   f(x)≤f(y) if  x≤y \;f(x)\leq f(y)\ \mathrm{if} \;x \leq y and f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ . Prove that f(x)\equal{}cx on H H, where c c is a nonnegative number. [M. Hosszu, R. Borges]
functionreal analysisreal analysis unsolved
Miklos Schweitzer 1963_6

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9/19/2008
Show that if f(x) f(x) is a real-valued, continuous function on the half-line 0≤x<∞ 0\leq x < \infty, and ∫0∞f2(x)dx<∞ \int_0^{\infty} f^2(x)dx <\infty then the function g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt satisfies \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx. [B. Szokefalvi-Nagy]
functionintegrationreal analysisinequalitieslimitreal analysis unsolved
Miklos Schweitzer 1963_7

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9/19/2008
Prove that for every convex function f(x) f(x) defined on the interval \minus{}1\leq x \leq 1 and having absolute value at most 1 1, there is a linear function h(x) h(x) such that \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}. [L. Fejes-Toth]
functionintegrationabsolute valuereal analysisreal analysis unsolved
Miklos Schweitzer 1963_8

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9/19/2008
Let the Fourier series a02+∑k≥1(akcos⁡kx+bksin⁡kx) \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx) of a function f(x) f(x) be absolutely convergent, and let ak2+bk2≥ak+12+bk+12  (k=1,2,...) . a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ . Show that 1h∫02π(f(x+h)−f(x−h))2dx  (h>0) \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0) is uniformly bounded in h h. [K. Tandori]
trigonometryfunctionintegrationsearchreal analysisreal analysis unsolved
Miklos Schweitzer 1963_9

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9/19/2008
Let f(t) f(t) be a continuous function on the interval 0≤t≤1 0 \leq t \leq 1, and define the two sets of points A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}. Show that the union of all segments AtBt‾ \overline{A_tB_t} is Lebesgue-measurable, and find the minimum of its measure with respect to all functions f f. [A. Csaszar]
functionreal analysisgeometrytrapezoidreal analysis unsolved
Miklos Schweitzer 1966_3

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9/29/2008
Let f(n) f(n) denote the maximum possible number of right triangles determined by n n coplanar points. Show that \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 . P. Erdos
limitadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1963_10

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9/19/2008
Select n n points on a circle independently with uniform distribution. Let Pn P_n be the probability that the center of the circle is in the interior of the convex hull of these n n points. Calculate the probabilities P3 P_3 and P4 P_4. [A. Renyi]
probabilityanalytic geometrygeometrygeometric transformationrotationintegrationcombinatorial geometry