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Training problems for mathematical contests A. Junior highschool level G176. Let a 1 , a 2 , . . . , a n ∈ R ∗ +, with 1 a 1 + 1 a 2 + . . . + 1 a n = 1. Prove that 1 1 1 a 3 1 + + a2 2 a 3 2 + + . . . + a2 3 a 3 n + a 2 < 1 1 2 . Angela Ţigăeru, Suceava G177. Let k > 0 and a, b, c ∈ [0, +∞) such that a + b + c = 1. Prove that a a 2 + a + k + b b 2 + b + k + c c 2 + c + k ≤ 9 9k + 4 . Titu Zvonaru, Comăneşti G178. If n ∈ N\{0, 1}, show that 1 − n ≤ {nx} − {x} ≤ n − 1 , ∀x ∈ R. n n Gheorghe Iurea, Iaşi G179. Determine the prime numbers a, b, c, d and the number p ∈ N ∗ , so that a 2p + b 2p + c 2p = d 2p + 3. Cosmin Manea and Dragoş Petrică, Piteşti G180. Find the remainder of the division of S = 2010 2009! +2009 2008! +. . .+2 1! +1 0! by 41. Răzvan Ceucă, hight-school student, Iaşi G181. Let k ≥ 1 be a given natural number. Show that there are infinitely many natural numbers n such that n k divides n!. Marian Tetiva, Bârlad G182. The triangle ABC is considered with the points M ∈ [AB], N ∈ [BC], P ∈ [CA] such that MP ∥BC and MN∥AC. Let {Q} = AN ∩ MP and {T } = BP ∩ MN. Prove that A AMN = A P T N + A QP C . Andrei Răzvan Băleanu, hight-school student, Motru G183. Let ABC be an isosceles triangle, with AB = AC şi m(bA) < 30 ◦ . Knowing that the points D ∈ [AB] and E ∈ [AC] exist such that AD = DE = EC = BC, determine the measure of the anglebA. Vasile Chiriac, Bacău G184. The points A ij 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 A ij . Find the minimum length of a path consisting of square sides only, which joins the points A 00 and A 44 . Claudiu Ştefan Popa, Iaşi 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, Iaşi 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 Bîrsan, Iaşi 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 of anglebA and let P and Q be the projections of L on the interior and respectively exterior bisector lines of anglebA. Prove that the lines GK, MN and P Q meet at the same point. Titu Zvonaru, Comăneşti L178. Let ABCD be a rhombus with its side length l = 1 and the points A 1 ∈ (AB), B 1 ∈ (BC), C 1 ∈ (CD), D 1 ∈ (DA). Prove that A 1 B 2 1 +B 1 C 2 1 +C 1 D 2 1 + D 1 A 2 1 ≥ 2 sin 2 A. Neculai Roman, Mirceşti, Iaşi L179. Prove that the following inequality holds in any triangle: 2(9R 2 − p 2 ) 9Rr ≥ cos2 A sin B sin C + cos2 B sin C sin A + cos2 C sin A sin B ≥ 1. I.V. Maftei and Dorel Băiţan, Bucureşti L180. Determine the minimum number of factors in the product P = sin π 4 n · sin 2π 3π · sin 4n 4 n · . . . · sin (22n−1 − 1)π 4 n , n ∈ N ∗ , so that P < 10 −9 . Ionel Tudor, Călugăreni, 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, Iaşi L182. Let (x n ) n≥1 be a sequence of integer numbers with the property that x n+2 − 5x n+1 + x n = 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, Bârlad 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 q n and an integer number b n such that p(X)q n (X) = X 2n + b n X n + 1. Marian Tetiva, Bârlad L184. Show that the function f:N ∗ ×N ∗ →N ∗ , f(x, y)= x2 + y 2 + 2xy − x − 3y + 2 , 2 is bijective. Silviu Boga, Iaşi 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
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Anul XII, Nr. 1 Ianuarie - Iunie 20
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Anul XII, Nr. 1 Ianuarie - Iunie 20
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Mai întâi cu un număr restrâns
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Seminarul Matematic din Iaşi - 100
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Amintiri de la Seminarul Matematic
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doream, să consult şi să împrum
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copiere. Îmi amintesc că, în anu
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Rigla şi compasul Gabriel POPA 1 A
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plus, cum △P AB şi △QAB sunt e
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ezultă că [AM] ≡ [BT ], deci [M
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O inegalitate ponderată cu medii G
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Aplicaţia 2. Fie a, b, p, q ∈ R
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Definiţia 3. Fie I 1 , I 2 ⊂ R d
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3) Considerăm funcţia f : (0, ∞
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Demonstraţie. În (2), luând x 1
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‹ Din faptul că v n+2 v n+1 v n+
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Teorema poate fi demonstrată şi p
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de unde, în urma unor calcule simp
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Aici avem a 2 b 2 + t 2 ≤ 6abt de
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Fie acum A i A j A k A l un trapez
Page 42 and 43: Remarque. detA(x, y, z, t) = x 2
Page 44 and 45: Si b divise v alors b divise u 2 ;
Page 46 and 47: coeficienţi reali şi rădăcini c
Page 48 and 49: În cazul |α| = √ a 2 + b 2 < 1
Page 50 and 51: Deoarece m(ÕBΩA) = 180 ◦ − B
Page 52 and 53: Şcoala Normală ”Vasile Lupu”
Page 54 and 55: în localul şcolii, celelalte fiin
Page 56 and 57: 2. Fie triunghiul △ABC înscris
Page 58 and 59: 3. Fie n ∈ N\0, 1} şi a 1 , a 2
Page 60 and 61: Soluţie. Latura pătratului este d
Page 62 and 63: V.108. Pe tablă sunt scrise numere
Page 64 and 65: Clasa a VII-a VII.102. În urma unu
Page 66 and 67: Soluţie. Se impune ca x ∈ R\{1,
Page 68 and 69: şi, cum tg A ∈ N, rezultă că t
Page 70 and 71: X.97. Fie a, b, c ∈ C ∗ numere
Page 72 and 73: ’ 1 0 . . . 0 0 0 1 . . . 0 Solu
Page 74 and 75: =Ztg n−1 1 x · cos 2 x + 1 ctgn
Page 76 and 77: ) Cum suma şi diferenţa au aceea
Page 78 and 79: maximă când produsul xy este maxi
Page 80 and 81: Notă. Soluţie corectă, bazată p
Page 82 and 83: În cazul în careba =b0, concluzia
Page 84 and 85: Clasele primare Probleme propuse 1
Page 86 and 87: VI.117. Fie I centrul cercului îns
Page 88 and 89: IX.107. Dacă a, b ∈ N ∗ , atun
Page 90 and 91: Probleme pentru pregătirea concurs
Page 94 and 95: Pagina rezolvitorilor CRAIOVA Coleg
Page 96 and 97: IMPORTANT • În scopul unei legă
Page 98: CUPRINS Centenarul SEMINARULUI MATE
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