MathDB
Miklos Schweitzer 1950_1

Source: first round of 1950

10/2/2008
Let \{k_n\}_{n \equal{} 1}^{\infty} be a sequence of real numbers having the properties k1>1 k_1 > 1 and k_1 \plus{} k_2 \plus{} \cdots \plus{} k_n < 2k_n for n \equal{} 1,2,.... Prove that there exists a number q>1 q > 1 such that kn>qn k_n > q^n for every positive integer n n.
algebra proposedalgebra
Miklos Schweitzer 1950_2

Source: first round of 1950

10/2/2008
Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.
quadraticsgeometry proposedgeometry
Miklos Schweitzer 1950_7

Source: first round of 1950

10/2/2008
Let x x be an arbitrary real number in (0,1) (0,1). For every positive integer k k, let fk(x) f_k(x) be the number of points mx\in [k,k \plus{} 1) m \equal{} 1,2,... Show that the sequence f1(x)f2(x)⋯fn(x)n \sqrt [n]{f_1(x)f_2(x)\cdots f_n(x)} is convergent and find its limit.
floor functionceiling functionalgebra proposedalgebra
Miklos Schweitzer 1950_3

Source: first round of 1950

10/2/2008
Let E E be a system of n^2 \plus{} 1 closed intervals of the real line. Show that E E has either a subsystem consisting of n \plus{} 1 elements which are monotonically ordered with respect to inclusion or a subsystem consisting of n \plus{} 1 elements none of which contains another element of the subsystem.
combinatorics proposedcombinatorics
Miklos Schweitzer 1950_4

Source: first round of 1950

10/2/2008
Find the polynomials f(x) f(x) having the following properties: (i) f(0) \equal{} 1, f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0 (ii) f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0
algebrapolynomialalgebra proposed
Miklos Schweitzer 1950_5

Source: first round of 1950

10/2/2008
Prove that for every positive integer k k there exists a sequence of k k consecutive positive integers none of which can be represented as the sum of two squares.
number theory proposednumber theory
Miklos Schweitzer 1950_6

Source: first round of 1950

10/2/2008
Prove the following identity for determinants: |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}
linear algebralinear algebra unsolved
Miklos Schweitzer 1950_3

Source: second part of 1950

10/3/2008
For any system x1,x2,...,xn x_1,x_2,...,x_n of positive real numbers, let f_1(x_1,x_2,...,x_n) \equal{} x_1, and f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}} for \nu \equal{} 2,3,...,n. Show that for any ϵ>0 \epsilon > 0, a positive integer n0<n0(ϵ) n_0 < n_0(\epsilon) can be found such that for every n>n0 n > n_0 there exists a system x1′,x2′,...,xn′ x_1',x_2',...,x_n' of positive real numbers with x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1 and fν(x1′,x2′,...,xn′)≤ϵ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon for \nu \equal{} 1,2,...,n .
algebra proposedalgebra
Miklos Schweitzer 1950_9

Source: first round of 1950

10/2/2008
Find the sum of the series x\plus{}\frac{x^3}{1\cdot 3}\plus{}\frac{x^5}{1\cdot 3\cdot 5}\plus{}\cdots\plus{}\frac{x^{2n\plus{}1}}{1\cdot 3\cdot 5\cdot \cdots \cdot (2n\plus{}1)}\plus{}\cdots
calculusreal analysisreal analysis unsolved
Miklos Schweitzer 1950_8

Source: first round of 1950

10/2/2008
Let A \equal{} (a_{ik}) be an n×n n\times n matrix with nonnegative elements such that \sum_{k \equal{} 1}^n a_{ik} \equal{} 1 for i \equal{} 1,...,n. Show that, for every eigenvalue λ \lambda of A A, either ∣λ∣<1 |\lambda| < 1 or there exists a positive integer k k such that \lambda^k \equal{} 1
linear algebramatrixsearchlinear algebra unsolved
Miklos Schweitzer 1950_10

Source: first part of 1950

10/2/2008
Consider an arc of a planar curve such that the total curvature of the arc is less than π \pi. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.
calculusderivativeadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1950_1

Source: second part of 1950

10/2/2008
Let a>0 a>0, d>0 d>0 and put f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots Give a closed form for f(x) f(x).
real analysisreal analysis unsolved
Miklos Schweitzer 1950_2

Source: second part of 1950

10/2/2008
Show that there exists a positive constant c c with the following property: To every positive irrational α \alpha, there can be found infinitely many fractions pq \frac{p}{q} with (p,q)\equal{}1 satisfying \left|\alpha\minus{}\frac{p}{q}\right|\le \frac{c}{q^2}
continued fractionnumber theory proposednumber theory
Miklos Schweitzer 1950_6

Source: second part of 1950

10/3/2008
Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than π2 \frac{\pi}{2}. Let P1,P2,P3,P4,P5 P_1,P_2,P_3,P_4,P_5 and P6 P_6 be any points on this arc, subject to the only condition that the radius of curvature at Pk P_k is greater than at Pj P_j if j<k j<k. Prove that the radius of the circle passing through the points P1,P3 P_1,P_3 and P5 P_5 is less than the radius of the circle through P2,P4 P_2,P_4 and P6 P_6
functioncalculusderivativeadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1950_4

Source: second part of 1950

10/3/2008
Put M\equal{}\begin{pmatrix}p&q&r\\ r&p&q\\q&r&p\end{pmatrix} where p,q,r>0 p,q,r>0 and p\plus{}q\plus{}r\equal{}1. Prove that \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\ \frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}
linear algebramatrixlimitvectorlinear algebra unsolved
Miklos Schweitzer 1950_5

Source: second part of 1950

10/3/2008
Let 1≤a1<a2<⋯<am≤N 1\le a_1<a_2<\cdots<a_m\le N be a sequence of integers such that the least common multiple of any two of its elements is not greater than N N. Show that m≤2[N] m\le 2\left[\sqrt{N}\right], where [N] \left[\sqrt{N}\right] denotes the greatest integer ≤N \le \sqrt{N}
least common multiplenumber theory proposednumber theory
Miklos Schweitzer 1950_8

Source: second part of 1950

10/3/2008
A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of 60 60 kilometers an hour. Let the probability of a hit be 0.75x^{ \minus{} 2}, where x x denotes the distance (in kilometers) between the cruiser and the coast (x≥1 x\geq 1), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after n n hits be 1 \minus{} \frac {1}{4^n} ( n \equal{} 0,1,...). Show that the probability of the cruiser escaping is 223π \frac {2\sqrt {2}}{3\pi}
probabilityprobability and stats
Miklos Schweitzer 1950_7

Source: second part of 1950

10/3/2008
Examine the behavior of the expression \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n as n→∞ n\rightarrow \infty
logarithmsintegrationcalculusEulerfunctionreal analysisreal analysis unsolved
Miklos Schweitzer 1950_9

Source: second part of 1950

10/3/2008
Find the necessary and sufficient conditions for two conics that every tangent to one of them contains a real point of the other.
conicsreal analysisreal analysis unsolved
Miklos Schweitzer 1952_5

Source:

10/12/2008
Let G G be anon-commutative group. Consider all the one-to-one mappings a→a′ a\rightarrow a' of G G onto itself such that (ab)'\equal{}b'a' (i.e. the anti-automorphisms of G G). Prove that this mappings together with the automorphisms of G G constitute a group which contains the group of the automorphisms of G G as direct factor.
group theorysuperior algebrasuperior algebra unsolved
Miklos Schweitzer 1952_1

Source:

10/12/2008
Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).
geometry3D geometrypyramidtetrahedronconvex
Miklos Schweitzer 1952_2

Source:

10/12/2008
Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?
conicsgeometry proposedgeometry
Miklos Schweitzer 1952_4

Source:

10/12/2008
Let K K be a finite field of p p elements, where p p is a prime. For every polynomial f(x)\equal{}\sum_{i\equal{}0}^na_ix^i (∈K[x] \in K[x]) put \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}. Prove that for any pair of polynomials f(x),g(x)∈K[x] f(x),g(x)\in K[x], f(x)‾∣g(x)‾ \overline{f(x)}|\overline{g(x)} if and only if f(x)∣g(x) f(x)|g(x).
algebrapolynomialsuperior algebrasuperior algebra unsolved
Miklos Schweitzer 1952_3

Source:

10/12/2008
Prove:If a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n} is a perfect number, then 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4 ; if moreover, a a is odd, then the upper bound 4 4 may be reduced to 223 2\sqrt[3]{2}.
inequalitiesnumber theory proposednumber theory
Miklos Schweitzer 1952_6

Source:

10/12/2008
Let 2n 2n distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join the couples by chords. Determine the probability that no two of these n n chords intersect. (All possible arrangement into pairs are supposed to have the same probability.)
probabilityratioprobability and stats