MathDB
Prove this inequality on cyclic quadrilateral

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020
Let ABCDABCD be a quadrilateral in which A^=C^=90∘\widehat{A}=\widehat{C}=90^{\circ}. Prove that 1BD(AB+BC+CD+DA)+BD2(1AB⋅AD+1CB⋅CD)≥2(2+2)\frac{1}{BD}(AB+BC+CD+DA)+BD^2\left (\frac{1}{AB\cdot AD}+\frac{1}{CB\cdot CD}\right )\geq 2\left (2+\sqrt{2}\right )
geometrycyclic quadrilateralinequalities
Prove this hard inequality

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020
If ai>0a_i >0 (i=1,2,⋯ ,ni=1, 2, \cdots , n) and ∑i=1naik=1\sum \limits_{i=1}^n a_i^k=1, where 1≤k≤n+11\leq k\leq n+1, then ∑i=1nai+1∏i=1nai≥n1−1k+nnk\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}
inequalities
Prove this combinatorial equation

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020
Prove that ∑0≤i<j≤n(i+j)(ni)(nj)=n(22n−1−(2n−1n))\sum \limits_{0\leq i<j\leq n}(i+j) {{n}\choose{i}}{{n}\choose{j}}=n\left (2^{2n-1}-{{2n-1}\choose{n}} \right )
combinatorics
Prove this problem on Banach space and jensen additive mapping

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020
Let θ\theta and p(p<1)p(p<1) ) be nonnegative real numbers.
Suppose that f:X→Yf:X\to Y is mapping with f(0)=0f(0)=0 and ∣∣2f(x+y2)−f(x)−f(y)∣∣Y≤θ(∣∣x∣∣Xp+∣∣y∣∣Xp)\left |\left| 2f\left (\frac{x+y}{2}\right )-f(x)-f(y) \right |\right|_Y \leq \theta\left (\left |\left |x\right |\right |_X^p +\left |\left |y\right |\right |_X^p \right ) for all xx, y∈Zy\in \mathbb{Z} with x⊥yx\perp y where XX is an orthogonality space and YY is a real Banach space.
Prove that there exists a unique orthogonally Jensen additive mapping T:X→YT:X\to Y, namely a mapping TT that satisfies the so-called orthogonally Jensen additive functional equation 2f(x+y2)=f(x)+f(y)2f\left (\frac{x+y}{2}\right )=f(x)+f(y)for all xx, y∈Xy\in \mathbb{X} with x⊥yx\perp y, satisfying the property ∣∣f(x)−T(x)∣∣Y≤2pθ2−2p∣∣x∣∣Xp\left |\left|f(x)-T(x) \right |\right|_Y \leq \frac{2^p\theta}{2-2^p}\left |\left |x\right |\right |_X^p for all x∈Xx\in X
Functional AnalysisBanach Spacesjensen additive mappingorthogonality space
Miklos Schweitzer 1951_9

Source:

10/8/2008
Let {m1,m2,… } \{m_1,m_2,\dots\} be a (finite or infinite) set of positive integers. Consider the system of congruences (1) x\equiv 2m_i^2 \pmod{2m_i\minus{}1} ( i\equal{}1,2,... ). Give a necessary and sufficient condition for the system (1) to be solvable.
modular arithmeticnumber theory proposednumber theory
Prove this inequality holds for all triangle

Source: 2009 Jozsef Wildt International Math Competition

4/27/2020
Prove that for all triangle △ABC\triangle ABC holds the following inequality ∑cyc(1−3tan⁡A2+3tan⁡A2)(1−3tan⁡B2+3tan⁡B2)≥3\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3
inequalitiesgeometrytrigonometry
Miklos Schweitzer 1951_1

Source:

10/8/2008
Choose terms of the harmonic series so that the sum of the chosen terms be finite. Prove that the sequence of these terms is of density zero in the sequence 1,12,13,…,1n,… 1,\frac12,\frac13,\dots,\frac1n,\dots
real analysisreal analysis unsolved
Miklos Schweitzer 1951_2

Source:

10/8/2008
Denote by H \mathcal{H} a set of sequences S\equal{}\{s_n\}_{n\equal{}1}^{\infty} of real numbers having the following properties: (i) If S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}, then S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}; (ii) If S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H} and T\equal{}\{t_n\}_{n\equal{}1}^{\infty}, then S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H} and ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}; (iii) \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}. A real valued function f(S) f(S) defined on H \mathcal{H} is called a quasi-limit of S S if it has the following properties: If S\equal{}{c,c,\dots,c,\dots}, then f(S)\equal{}c; If si≥0 s_i\geq 0, then f(S)≥0 f(S)\geq 0; f(S\plus{}T)\equal{}f(S)\plus{}f(T); f(ST)\equal{}f(S)f(T), f(S')\equal{}f(S) Prove that for every S S, the quasi-limit f(S) f(S) is an accumulation point of S S.
functionalgebrapolynomialreal analysisreal analysis unsolved
Miklos Schweitzer 1951_3

Source:

10/8/2008
Consider the iterated sequence (1) x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots, where f(x) \equal{} 4x \minus{} x^2. Determine the points x0 x_0 of [0,1] [0,1] for which (1) converges and find the limit of (1).
limitreal analysisreal analysis unsolved
Miklos Schweitzer 1951_4

Source:

10/8/2008
Prove that the infinite series 1\minus{}\frac{1}{x(x\plus{}1)}\minus{}\frac{x\minus{}1}{2!x^2(2x\plus{}1)}\minus{}\frac{(x\minus{}1)(2x\minus{}1)}{3!(x^3(3x\plus{}1))}\minus{}\frac{(x\minus{}1)(2x\minus{}1)(3x\minus{}1)}{4!x^4(4x\plus{}1)}\minus{}\cdots is convergent for every positive x x. Denoting its sum by F(x) F(x), find \lim_{x\to \plus{}0}F(x) and lim⁡x→∞F(x) \lim_{x\to \infty}F(x).
limitreal analysisreal analysis unsolved
Miklos Schweitzer 1951_5

Source:

10/8/2008
In a lake there are several sorts of fish, in the following distribution: 18% 18\% catfish, 2% 2\% sturgeon and 80% 80\% other. Of a catch of ten fishes, let x x denote the number of the catfish and y y that of the sturgeons. Find the expectation of \frac {x}{y \plus{} 1}
integrationprobability and stats
Miklos Schweitzer 1951_6

Source:

10/8/2008
In lawn-tennis the player who scores at least four points, while his opponent scores at least two points less, wins a game. The player who wins at least six games, while his opponent wins at least two games less, wins a set. What minimum percentage of all points does the winner have to score in a set?
combinatorics proposedcombinatorics
Miklos Schweitzer 1951_7

Source:

10/8/2008
Let f(x) f(x) be a polynomial with the following properties: (i) f(0)\equal{}0; (ii) \frac{f(a)\minus{}f(b)}{a\minus{}b} is an integer for any two different integers a a and b b. Is there a polynomial which has these properties, although not all of its coefficients are integers?
algebrapolynomialalgebra proposed
Miklos Schweitzer 1951_8

Source:

10/8/2008
Given a positive integer n>3 n>3, prove that the least common multiple of the products x1x2⋯xk x_1x_2\cdots x_k (k≥1 k\geq 1) whose factors xi x_i are positive integers with x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n, is less than n! n!.
number theoryleast common multiplefloor functionnumber theory proposed
Miklos Schweitzer 1951_11

Source:

10/8/2008
Prove that, for every pair n n, rr of positive integers, there can be found a polynomial f(x) f(x) of degree n n with integer coefficients, so that every polynomial g(x) g(x) of degree at most n n, for which the coefficients of the polynomial f(x)\minus{}g(x) are integers with absolute value not greater than r r, is irreducible over the field of rational numbers.
algebrapolynomialabsolute valuealgebra proposed
Miklos Schweitzer 1951_13

Source:

10/8/2008
Of how many terms does the expansion of a determinant of order 2n 2n consist if those and only those elements aik a_{ik} are non-zero for which i\minus{}k is divisible by n n?
combinatorics proposedcombinatorics
Miklos Schweitzer 1951_10

Source:

10/8/2008
Let f(x) f(x) be a polynomial with integer coefficients and let p p be a prime. Denote by z_1,...,z_{p\minus{}1} the (p\minus{}1)th complex roots of unity. Prove that f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}.
algebrapolynomialmodular arithmeticRing Theory
Miklos Schweitzer 1951_12

Source:

10/8/2008
By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions f0(x),f1(x),f2(x),… f_0(x),f_1(x),f_2(x),\dots such that every number theoretical function F(x) F(x) can be uniquely represented in the form F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x), a0,a1,a2,… a_0,a_1,a_2,\dots being integers?
functionnumber theory proposednumber theory
Miklos Schweitzer 1951_15

Source:

10/8/2008
Let the line z\equal{}x, \, y\equal{}0 rotate at a constant speed about the z z-axis; let at the same time the point of intersection of this line with the z z-axis be displaced along the z z-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.
rotationgeometrygeometric transformationadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1951_14

Source:

10/8/2008
For which commutative finite groups is the product of all elements equal to the unit element?
group theoryabstract algebrasuperior algebrasuperior algebra unsolved
Miklos Schweitzer 1951_16

Source:

10/8/2008
Let F \mathcal{F} be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that F \mathcal{F} can be developed (that is, isometrically mapped) into the plane.
Gausscalculusintegrationadvanced fieldsadvanced fields unsolved
Miklos Schweitzer 1951_17

Source:

10/8/2008
Let α \alpha be a projective plane and c c a closed polygon on α \alpha. Prove that α \alpha will be decomposed into two regions by c c if and only if there exists a straight line g g in α \alpha which has an even number of points in common with c c.
advanced fieldsadvanced fields unsolved
Find the limit on fibonacci numbers

Source: 2019 Jozsef Wildt International Math Competition-W. 17

5/18/2020
Let fn=(1+1n)n((2n−1)!Fn)1nf_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}. Find lim⁡n→∞(fn+1−fn)\lim \limits_{n \to \infty}(f_{n+1} - f_n) where FnF_n denotes the nnth Fibonacci number (given by F0=0F_0 = 0, F1=1F_1 = 1, and by Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1} for all n≥1n \geq 1
Fibonacci sequencenumber theory
Find the value of this infinte summation about pell numbers

Source: 2019 Jozsef Wildt International Math Competition-W. 1

4/29/2020
The Pell numbers PnP_n satisfy P0=0P_0 = 0, P1=1P_1 = 1, and Pn=2Pn−1+Pn−2P_n=2P_{n-1}+P_{n-2} for n≥2n\geq 2. Find ∑n=1∞(tan⁡−11P2n+tan⁡−11P2n+2)tan⁡−12P2n+1\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}
pell equationnumber theorysplit method
Prove this inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 2

4/30/2020
If 0<a≤c≤b0<a\leq c\leq b then (b30−a30)(b30−c30)36b10≤(b25−a25)(b25−c25)25≤(b30−a30)(b30−c30)36(ac)10\frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36b^{10}}\leq \frac{(b^{25}-a^{25})(b^{25}-c^{25})}{25}\leq \frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36(ac)^{10}}
inequalities