2016 EGMO

Part of EGMO

Subcontests

(6)

1

Infinitely many numbers of a given form

S

be the set of all positive integers

n

n^4

has a divisor in the range

n^2 +1, n^2 + 2,...,n^2 + 2n

. Prove that there are infinitely many elements of

S

of each of the forms

7m, 7m+1, 7m+2, 7m+5, 7m+6

and no elements of

S

7m+3

7m+4

m

1

Combinatorics Tiles

k

n

be integers such that

k\ge 2

k \le n \le 2k-1

. Place rectangular tiles, each of size

1 \times k

k \times 1

n \times n

chessboard so that each tile covers exactly

k

cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each such

k

n

, determine the minimum number of tiles that such an arrangement may contain.

1

Geometry tangent circles

\omega_1

\omega_2

, of equal radius intersect at different points

X_1

X_2

. Consider a circle

\omega

externally tangent to

\omega_1

T_1

and internally tangent to

\omega_2

T_2

. Prove that lines

X_1T_1

X_2T_2

intersect at a point lying on

\omega

1

Related blue squares in 4m by 4m board

m

be a positive integer. Consider a

4m\times 4m

array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are colored blue, such that every cell is related to at least two blue cells. Determine the minimum number of blue cells.

1

Inequality with odd numbers.

n

be an odd positive integer, and let

x_1,x_2,\cdots ,x_n

be non-negative real numbers. Show that

\min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1})

x_{n+1}=x_1

1

Tangent Line.

ABCD

be a cyclic quadrilateral, and let diagonals

AC

BD

X

C_1,D_1

M

be the midpoints of segments

CX,DX

CD

, respecctively. Lines

AD_1

BC_1

Y

MY

intersects diagonals

AC

BD

at different points

E

F

, respectively. Prove that line

XY

is tangent to the circle through

E,F

X