For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that $$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$ Mircea Fianu

Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that $$ n=S(a)=S(b)=S(a+b) , $$where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $ Vasile Zidaru and Mircea Lascu

Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set $$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $ $ \# $ denotes the cardinal. Eugen Păltânea

On the hypotenuse $ BC $ of an isosceles right triangle $ ABC $ let $ M,N $ such that $ BM^2-MN^2+NC^2=0. $ Show that $ \angle MAN= 45^{\circ } . $ Cristinel Mortici

Solve in natural the equation $9^x-3^x=y^4+2y^3+y^2+2y$ _____________________________ Azerbaijan Land of the Fire

In an urban area whose street plan is a grid, a person started walking from an intersection and turned right or left at every intersection he reached until he ended up in the same initial intersection. a) Show that the number of intersections (not necessarily distinct) in which he were is equivalent to $ 1 $ modulo $ 4. $ b) Enunciate and prove a reciprocal statement. Marius Beceanu

Find all real numbers $ a $ such that $ x,y>a\implies x+y+xy>a. $ Gheorghe Iurea

Let be a triangle $ ABC, $ and three points $ A',B',C' $ on the segments $ BC,CA, $ respectively, $ AB, $ such that the lines $ AA',BB',CC' $ are concurent at $ M. $ Name $ a,b,c,x,y,z $ the areas of the triangles $ AB'M,BC'M,CA'M,AC'M,BA'M, $ respectively, $ CB'M. $ Show that: a) $ abc=xyz $ b) $ ab+bc+ca=xy+yz+zx $ Bogdan Enescu and Marcel Chiriță

Let be a natural number $ n\ge 2, n $ real numbers $ b_1,b_2,\ldots ,b_n , $ and $ n-1 $ positive real numbers $ a_1,a_2,\ldots ,a_{n-1} $ such that $ a_1+a_2+\cdots +a_{n-1} =1. $ Prove the inequality $$ b_1^2+\frac{b_2^2}{a_1} +\frac{b_3^2}{a_2} +\cdots +\frac{b_n^2}{a_{n-1}} \ge 2b_1\left( b_2+b_3+\cdots +b_n \right) , $$and specify when equality is attained. Dumitru Acu

Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it. Dan Brânzei

Let $ D,E,F $ be the feet of the interior bisectors from $ A,B, $ respectively $ C, $ and let $ A',B',C' $ be the symmetric points of $ A,B, $ respectively, $ C, $ to $ D,E, $ respectively $ F, $ such that $ A,B,C $ lie on $ B'C',A'C', $ respectively, $ A'B'. $ Show that the $ ABC $ is equilateral. Marius Beceanu

Two identical squares havind a side length of $ 5\text{cm} $ are each divided separately into $ 5 $ regions through intersection with some lines. Show that we can color the regions of the first square with five colors and the regions of the second with the same five colors such that the sum of the areas of the resultant regions that have the same colors at superpositioning the two squares is at least $ 5\text{cm}^2. $