2019 IMAR Test

Part of IMAR Test

Subcontests

(4)

1

Graph theory; maximum length of cycle

Show that the length of a cycle that contains every edge of a connected graph is at most the sum between the vertices and nodes of the graph, minus

1.

1

An application of Gauss-Legendre

Consider a natural number

n\equiv 9\pmod {25}.

Prove that there exist three nonnegative integers

a,b,c

having the property that:

n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2}

1

Find a certain polynomial

f_1,f_2,f_3,f_4

be four polynomials with real coefficients, having the property that f_1 (1) =f_2 (0), f_2 (1) =f_3 (0), f_3 (1) =f_4 (0), f_4 (1) =f_1 (0) . Prove that there exists a polynomial

f\in\mathbb{R}[X,Y]

such that f(X,0)=f_1(X), f(1,Y) =f_2(Y) , f(1-X,1) =f_3(X), f(0,1-Y)=f_4(Y) .

1

Fresh geometry problem (prove concurence)

Consider an acute triangle

ABC.

D,E,F

are the feet of the altitudes of

ABC

A,B,C,

M,N,P

are the middlepoints of

BC,CA,AB,

respectively. The circumcircles of

BDP,CDN

X\neq D,

the circumcircles of

CEM,AEP

Y\neq E,

and the circumcircles of

AFN,BFM

Z\neq F.

AX,BY,CZ

are concurrent.