In the World of Mathematics Problems148159,166?175

In the World of Mathematics
Problems 148-159,166 —175
Volume 7(2001)
148. Prove that the inequality

Issue 1

√
3

1
n > √
3
3 n2
holds for every positive integer n not equal to a cube of an integer. (Here {x} is the fractional
part of the number x, i.e., {x} = x − [x], where [x] is the greatest integer not greater than
x.)
(O. Sarana, Zhytomyr)
149. Let x1 , x2 , . . . , xn and y1 , y2 , . . . , yn be two sets of pairwise different natural numbers for
which the equality
xx1 1 + xx2 2 + · · · + xxnn = y1y1 + y2y2 + · · · + ynyn
holds. Prove that the set y1 , y2 , . . . , yn can be obtained from the set x1 , x2 , . . . , xn by a
permutation.
(A. Prymak, Kyiv)
150. Does there exist a non-constant function f : N2 → N such that
f (x, y) + f (y, x) = f (x2 , y 2 ) + 1
for all positive integers x, y?
(O. Manziuk, A. Prymak, Kyiv)
151. Prove the inequality
α+β+γ ≥α·

sin β
sin γ
sin α
+β·
+γ·
sin α
sin β
sin γ

for any three numbers α, β, γ ∈ (0; π/2).
(V. Yasinsky, Vinnytsa)
152. The triangles ACB and ADE are oriented in the same way. We also have that ∠DEA =
∠ACB = 90◦ , ∠DAE = ∠BAC, E = C. The line l passes through the point D and is
perpendicular to the line EC. Let L be the intersection point of the lines l and AC. Prove
that the points L, E, C, B belong to a common circumference.
(V. Yasinsky, Vinnytsa)
153. Prove that the expression
gcd(m, n) n
n
m
is an integer for all pairs of integers n ≥ m ≥ 1.
(William Lowell Putnam Math. Competition)
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Volume 7(2001)

Issue 2

154. Let ABCD be a trapezoid (BC AD), denote by E the intersection point of its diagonals

and by O the center of the circle circumscribed around the triangle AOD. Let K and L
the points on the segments AC and BD respectively such that BK ⊥ AC and CL ⊥ BD.
Prove that KL ⊥ OE.
(A. Prymak, Kyiv)
155. The sequence {an , n ≥ 1} is defined in the following way:
a1 = 1,

an = an−1 + (an−1

mod n)2 , n ≥ 2,

where a mod b denotes the remainder of division of a by b. What is the maximal possible
number of consecutive equal members of this sequence?
(V. Mazorchuk, Kyiv)
156. Some cities of the Empire are connected by air lines. It is known that for any three cities
there exists a route connecting any two of them and not passing through the third one. The
Empire has 2000 cities. Prove that one can divide the cities between two descendants of the
Emperor so that the descendants get equal number of the cities and any two cities belonging
to one descendant are connected by a route passing only through his cities.
(V. Yasinsky, Vinnytsia)
157. Let A1 , B1 , C1 be the midpoints of the segments BC, AC, AB of the triangle ABC
respectively. Let H1 , H2 , H3 be the intersection points of the altitudes of the triangles
AB1 C1 , BA1 C1 , CA1 B1 . Prove that the lines A1 H1 , B1 H2 , C1 H3 are concurrent.
(M. Kurylo, Lypova Dolyna, Sumska obl.)
158. Let the numbers α, β, γ belong to the interval 0, π2 . Prove the inequality
α+β+γ ≥α·

sin β + sin γ
sin γ + sin α
sin α + sin β

+β·
+γ·
.
2 sin α
2 sin β
2 sin γ
(V. Yasinsky, Vinnytsia)

159. Find all the quadruples (x, y, z, p) of positive integers such that x > 2, the number p is prime
and xy = pz + 1.
(A. Prymak, O. Manziuk, Kyiv)

Volume 7(2001)

Issue 3 [Problems 160-165: to be found]

Volume 7(2001)

Issue 4

166. Let a, b, c be positive real numbers. Prove the inequality
b2

b6
c6
abc(a + b + c)
a6
.
+ 2
+ 2

≥
2
2
+c
a +c
a + b2
2
(M. Kurylo, Lypova Dolyna, Sumska obl.)

167. Solve the equation
(xy)2 + (x + y)2 +

x
= 2001, where x, y are digits.
y
(A. Narovlyansky, Chernigiv.)
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168. Let AA1 , BB1 , CC1 be bisectors in the triangle ABC, let G1 , G2 , G3 be the intersection
points of medians in the triangles AB1 C1 , BA1 C1 and CA1 B1 respectively. Prove that the
straight lines AG1 , BG1 , CG1 intersect in a common point.
(M. Kurylo, Lypova Dolyna, Sumska obl.)
169. In the square with unit side m2 points are located so that no three points lie on one line.
Prove that there exists a triangle with the vertices in these points of area not greater than
1
2(m−1)2 .
(S. Linchuk, Yu. Linchuk, Chernivtsi.)
170. Triangle ABC is circumscribed around a circle of radius r. The circle is tangent to the sides
AB, BC, AC in the points N, Y, H respectively. Denote the distances from the points N, Y

and H to the sides BC, AC and AB by dN , dY and dH respectively. Prove that
√
√
√
1
1
1
dH
1
dN
1
dY
1
2
√
√
√
√
√
√
+
+
+
+
+
≥ .
dN
dH
dH
r

dY + dN dY
dY + dH dY
dN + dH dN
(I. Nagel, Evpatoria.)
171. There are 2001 workers at a factory. Due to results of work there were made two rating
lists of the workers. A list D is composed by the director and a list W is composed by the
workers. The prize f (n, m) for a worker positioned on the n-th place in the list D and on the
m-th place in the list W is calculated by the formula f (n, m) = m · 2001n + m2000 + n2000 .
Suppose we have only the list D and the sum S of all the prizes. Is it possible to pay the
prizes for all the workers correctly?
(I. Bobak, Lutsk.)

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