Training problems for mathematical contests

Training problems for mathematical contests A. Junior highschool level G176. Let a1 , a2 , . . . , an ∈ R∗+ , with

1 1 1 + + ... + = 1. Prove that a1 a2 an

1 1 1 1 + 3 + ... + 3 < . a31 + a22 a2 + a23 an + a21 2 Angela T ¸ ig˘ aeru, Suceava G177. Let k > 0 and a, b, c ∈ [0, +∞) such that a + b + c = 1. Prove that b c 9 a + + ≤ . a2 + a + k b2 + b + k c2 + c + k 9k + 4 Titu Zvonaru, Com˘ ane¸sti 1−n n−1 G178. If n ∈ N\{0, 1}, show that ≤ {nx} − {x} ≤ , ∀x ∈ R. n n Gheorghe Iurea, Ia¸si G179. Determine the prime numbers a, b, c, d and the number p ∈ N∗ , so that p p p p a2 + b2 + c2 = d2 + 3. Cosmin Manea and Drago¸s Petric˘ a, Pite¸sti 2009! 2008! G180. Find the remainder of the division of S = 2010 +2009 +. . .+21! +10! by 41. R˘ azvan Ceuc˘ a, hight-school student, Ia¸si G181. Let k ≥ 1 be a given natural number. Show that there are infinitely many natural numbers n such that nk divides n!. Marian Tetiva, Bˆ arlad G182. The triangle ABC is considered with the points M ∈ [AB], N ∈ [BC], P ∈ [CA] such that M P ∥BC and M N ∥AC. Let {Q} = AN ∩ M P and {T } = BP ∩ M N . Prove that AAM N = AP T N + AQP C . Andrei R˘ azvan B˘ aleanu, hight-school student, Motru b < 30◦ . Knowing G183. Let ABC be an isosceles triangle, with AB = AC ¸si m(A) that the points D ∈ [AB] and E ∈ [AC] exist such that AD = DE = EC = BC, b determine the measure of the angle A. Vasile Chiriac, Bac˘ au G184. The points Aij of coordinates (i, j) are considered in the plane xOy, where i, j ∈ {0, 1, 2, 3, 4}. Let P be the set of the squares with their vertices among the considered points Aij . Find the minimum length of a path consisting of square sides only, which joins the points A00 and A44 . Claudiu S ¸ tefan Popa, Ia¸si G185. Show that there exists a coloring of the plane by n colours, where n ≥ 2 is a given natural number, so that any line segment in phe plane contain points colored by each of the n colours. Paul Georgescu and Gabriel Popa, Ia¸si 88

B. Highschool Level L176. Let D, E, F be the projections of the centroid G of the triangle ABC onto the lines BC, CA, and respectively AB. Show that the Cevian lines AD, BE and CF meet at a unique point if and only if the triangle is isosceles. Temistocle Bˆırsan, Ia¸si L177. We consider the triangle ABC with its centroid G and Lemoine′ s point L. Denote by M and N the projections of G onto the interior and exterior bisector lines b and let P and Q be the projections of L on the interior and respectively of angle A b Prove that the lines GK, M N and P Q meet at the exterior bisector lines of angle A. same point. Titu Zvonaru, Com˘ ane¸sti L178. Let ABCD be a rhombus with its side length ℓ = 1 and the points A1 ∈ (AB), B1 ∈ (BC), C1 ∈ (CD), D1 ∈ (DA). Prove that A1 B12 + B1 C12 + C1 D12 + D1 A21 ≥ 2 sin2 A. Neculai Roman, Mirce¸sti, Ia¸si L179. Prove that the following inequality holds in any triangle: 2(9R2 − p2 ) cos2 A cos2 B cos2 C ≥ + + ≥ 1. 9Rr sin B sin C sin C sin A sin A sin B I.V. Maftei and Dorel B˘ ait¸an, Bucure¸sti π L180. Determine the minimum number of factors in the product P = sin n · 4 2π 3π (22n−1 − 1)π ∗ −9 sin n · sin n · . . . · sin , n ∈ N , so that P < 10 . 4 4 4n Ionel Tudor, C˘ alug˘ areni, Giurgiu L181. Let P be a point on the circular boundary of a half-disc, and d the tangent at P to this boundary. Denote by C the rotation body obtained by the rotation of the half-disc around the line d. Study the variation of the volume of C as a function of the position of point P. Paul Georgescu and Gabriel Popa, Ia¸si L182. Let (xn )n≥1 be a sequence of integer numbers with the property that xn+2 − 5xn+1 + xn = 0, ∀n ∈ N∗ . Show that if a term of the sequence is divisible by 22 then infinitely many terms there of have this property. Marian Tetiva, Bˆ arlad L183. Let us consider the numbers a ∈ Z, n ∈ N∗ and the polynomial p(X) = X 2 + a X + 1. Show that there exist a polynomial with integer coefficients qn and an integer number bn such that p(X)qn (X) = X 2n + bn X n + 1. Marian Tetiva, Bˆ arlad 2 2 x + y + 2xy − x − 3y +2 , L184. Show that the function f :N∗ ×N∗ →N∗ , f (x, y)= 2 is bijective. Silviu Boga, Ia¸si L185. Let f : R → R be a function with the property that |f (x + y) − f (x) − f (y)| ≤ |x − y|, ∀x, y ∈ R. Show that lim f (x) = 0 if and only if lim xf (x) = 0. x→0

x→0

Adrian Zahariuc, student, Princeton 89