x - Recreaţii Matematice

Training problems for mathematical contests
A. Junior highschool level
G176. Let a1, a2, . . . , an ∈ R ∗ +, with 1
a1
1
a3 1 + a2 1
+
2 a3 2 + a2 1
+ . . . +
3 a3 n + a2 1
+ 1
+ . . . +
a2
1
= 1. Prove that
an
< 1
2 .
Angela T¸ igăeru, 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, Comăne¸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
a2p + b2p + c2p = d2p + 3.
Cosmin Manea and Drago¸s Petrică, Pite¸sti
G180. Find the remainder of the division of S = 20102009! +20092008! +. . .+21! +10! by 41.
Răzvan Ceucă, 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, 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 AAMN = AP T N + AQP C.
Andrei Răzvan Băleanu, hight-school student, Motru
G183. Let ABC be an isosceles triangle, with AB = AC ¸si m(A) < 30◦ . Knowing
that the points D ∈ [AB] and E ∈ [AC] exist such that AD = DE = EC = BC,
determine the measure of the angleA.
Vasile Chiriac, Bacău
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
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