MathDB

1

Interesting conditional geo in orthocentre configuration

ABC

be an acute angled triangle with orthocentre

H

. Let

E = BH \cap AC

and

F= CH \cap AB

. Let

D, M, N

denote the midpoints of segments

AH, BD, CD

respectively, and

T = FM \cap EN

. Suppose

D, E, T, F

are concylic. Prove that

DT

passes through the circumcentre of

ABC

.
Proposed by Pranjal Srivastava

5

1

Everyone loves tiling

1. Can a

7 \times 7~

square be tiled with the two types of tiles shown in the figure? (Tiles can be rotated and reflected but cannot overlap or be broken)
2. Find the least number

N

of tiles of type

A

that must be used in the tiling of a

1011 \times 1011

square. Give an example of a tiling that contains exactly

N

tiles of type

A

. [asy] size(4cm, 0); pair a = (-10,0), b = (0, 0), c = (10, 0), d = (20, 0), e = (20, 10), f = (10, 10), g = (0, 10), h = (0, 20), ii = (-10, 20), j = (-10, 10); draw(a--b--c--f--g--h--ii--cycle); draw(g--b); draw(j--g); draw(f--c); draw((30, 0)--(30, 20)--(50,20)--(50,0)--cycle); draw((40,20)--(40,0)); draw((30,10)--(50,10)); label((0,0), "

(A)

", S); label((40,0), "

(B)

", S); [/asy] Proposed by Muralidharan Somasundaran

4

1

I am always bigger than my index double

Let

N \geq 3

be an integer, and let

a_0, \dots, a_{N-1}

be pairwise distinct reals so that

a_i \geq a_{2i}

for all

i

(indices are taken

\bmod~ N

). Find all possible

N

for which this is possible.
Proposed by Sutanay Bhattacharya

3

1

If P has an integer root, so does f(P)

Find all functions

f: \mathbb{N} \mapsto \mathbb{N}

so that for any positive integer

n

and finite sequence of positive integers

a_0, \dots, a_n

, whenever the polynomial

a_0+a_1x+\dots+a_nx^n

has at least one integer root, so does

f(a_0)+f(a_1)x+\dots+f(a_n)x^n.

Proposed by Sutanay Bhattacharya

2

1

What are those floors doing here?

Given that

a_1, a_2, \dots, a_{10}

are positive real numbers, determine the smallest possible value of

\sum \limits_{i = 1}^{10} \left\lfloor \frac{7a_i}{a_i+a_{i+1}}\right\rfloor

where we define

a_{11} = a_1

.
Proposed by Sutanay Bhattacharya

1

Euler Line parallel to A-tangent?

Let

ABC

be a triangle with circumcentre

O

and centroid

G

. Let

M

be the midpoint of

BC

and

N

the reflection of

M

across

O

. Prove that

NO = NA

if and only if

\angle AOG = 90^{\circ}

.
Proposed by Pranjal Srivastava

India EGMO 2024 TST

Subcontests

Interesting conditional geo in orthocentre configuration

Everyone loves tiling

I am always bigger than my index double

If P has an integer root, so does f(P)

What are those floors doing here?

Euler Line parallel to A-tangent?