József Wildt International Mathematical Competition
问题W1-W58的解答须在2024年10月26日之前提交至下面邮箱.
E-mail: benczemihaly@gmail.com and benczemihaly@yahoo.com
W1. Let be an integer number and let be the integer sequence defined by and for ,
Find a close form for .
W2. Let be a polynomial of positive degree with integer coefficients. Show that for every positive integer number , there exists an integer such that the number has at least different prime divisors.
W3. Fill six natural numbers into the following grids, so that the sum of the three numbers in each row, each column, each diagonal all are equal.
W4. Let integer,
Compute
Compute
W5. for Let
for . Find the limit of
when
W6. If such that , prove that
W7. If , with the notations:
Prove the following inequalities:
(a)
(b)
W8. Evaluate
W9. Evaluate
W10. Let be a sequence of positive real numbers such that where is a constant. Prove that if diverges the series
also diverges
W11. If M is a point inside the triangle ABC of sides , area and are the distances from point to sides and respectively, show that:
W12. If M is a point inside the triangle ABC of sides , area and are the distances from point to sides and respectively, show that:
W13. The inscribed circle is tangent to the sides and BC of the triangle ABC , respectively at the points and . Prove that:
W14. Calculate the following sum:
W15. Let the matrix , such that
Show that the matrix is invertible.
W16. Show that
where
W17. Let be then Fibonacci number defined by and for all . Prove that
is an irrational number but not transcendental.
W18. Compute
W19. Find
W20. Find such that:
W21. Find:
where
W22. If , then prove that:
for any positive inyteger n .
W23. Let for all
1). Compute
2). Compute
when and denote the Fibonacci respective Lucas numbers.
W24. Let continuous and bijective function. We assume that the function has a limit for , where . Show that:
where it is differentiable function, with derivate integrable and positive on
W25. Show that:
with .
W26. We consider a prime number , two matrices , , and
such that:
a).
b). , for all
c).
Show that whatever is the natural number and whatever the rational numbers , while
then
W27. Let be the sphere and let be the family of cardioids . Each element of belongs on a plane orthogonal to the axis and its point with the largest abscissa belongs to the sphere. For any cardioid consider the conic surface with vertex in and generatrices the segments joining and the points of the cardioid. Find the maximum in of the volume enclosed by the conic surface
W28. In , I-incenter and the points of contact of the cevians AI, BI, CI with the circle, then the following relationship holds:
W29. In acute -simmetrics of the points to the sides respectively, the following relationship holds:
where area.
W30. In -middle points of the arcs respectively made with the circumscrible circle of the triangle ABC , the following relationship holds:
W31. Let different in pairs such that . If
then are affixses of an equilateral triangle.
W32. In , K-Lemoine's point, F-area of the triangle ABC and the lengths sides, the following relationship holds:
W33. Prove that in any acute angled triangle holds inequality
W34. For any real prove inequality
W35. For any positive such that prove that
W36. a). If are positive real numbers that satisfy inequality
then any of them, let it be , satisfies inequality
b). If are positive real numbers that satisfy
then for any holds inequality
that is be side lengths of some triangle.
W37. Let be a non-obtuse angles triangle with usual notations. Prove that
W38. Let be positive integer and be real number. Find
where and .
W39. Find the following limits:
a).
b). .
c). .
W40. Let be area of a triangle and be angle bisectors, respectively, from vertices . Prove the following double inequality
W41. Prove that
for all
W42. Solve in the equation
W43. Let and be nonegative real numbers such that . Prove that
W44. (a) Solve the following differential equation
(b) Show that the initial value problem
has at least 5 different solutions on any interval around the origin.
W45. Let be and
for all . Prove that:
a).
b).
W46. a). Determine all functions for which is a primitive function of and is a primitive function of for all .
b). Prove that for all holds:
for all
W47. In all acute triangle holds
W48. If then
then
W49. In all triangle ABC holds
W50. Let be the affixes of vertices of a regular hexagon. Prove that
for all
W51. Let such that
for all . Prove that
W52. In all triangle ABC holds
W53. Let be where where denote the fractional part and p is a prime number.
a). Prove that the function is injective
b). Prove that the function f is not surjective
W54. Let be a convex polygon. Prove that
W55. Let be the set . Prove that if then holds:
where denote the fractional part, and denote the integer part.
W56. Find:
a).
b).
W57. Let be an odd integer. Solve in the equation
W58. Let be an odd integer. Solve in the equation , where denotes the transpose of .
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