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 √ n > √ 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 + xx2 + · · · + 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 ) + 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 ≥ (William Lowell Putnam Math Competition) Volume 7(2001) Issue 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· sin α sin β 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 + (A Prymak, O Manziuk, Kyiv) Volume 7(2001) Issue [Problems 160-165: to be found] Volume 7(2001) Issue 166 Let a, b, c be positive real numbers Prove the inequality b2 b6 c6 abc(a + b + c) a6 + + ≥ 2 +c a +c a + b2 (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.) 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 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 dH dN dY √ √ √ √ √ √ + + + + + ≥ 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.) ... 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· sin α sin β sin γ (V Yasinsky, Vinnytsia) 159 Find all the. .. 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. .. 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