Found problems: 9
2025 Kosovo EGMO Team Selection Test, P2
Find all natural numbers $m$ and $n$ such that $3^m+n!-1$ is the square of a natural number.
India EGMO 2024 TST, 6
Let $ABC$ 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 $ABC$.
[i]Proposed by Pranjal Srivastava[/i]
India EGMO 2024 TST, 1
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 $\angle AOG = 90^{\circ}$.
[i]Proposed by Pranjal Srivastava[/i]
India EGMO 2024 TST, 5
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]
[i]Proposed by Muralidharan Somasundaran[/i]
India EGMO 2024 TST, 2
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 $a_{11} = a_1$.
[i]Proposed by Sutanay Bhattacharya[/i]
India EGMO 2024 TST, 3
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 \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\]
[i]Proposed by Sutanay Bhattacharya[/i]
2024 Turkey EGMO TST, 1
Let $ABC$ be a triangle and its circumcircle be $\omega$. Let $I$ be the incentre of the $ABC$. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. The line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet again at point $K$. Prove that the lines $BN$, $CM$, $AK$ are concurrent.
2024 Kosovo EGMO Team Selection Test, P3
Let $\triangle ABC$ be a right triangle at the vertex $A$ such that the side $AB$ is shorter than the side $AC$.
Let $D$ be the foot of the altitude from $A$ to $BC$ and $M$ the midpoint of $BC$. Let $E$ be a point on the ray $AB$, outside of the segment $AB$. Line $ED$ intersects the segment $AM$ at the point $F$. Point $H$ is on the side $AC$ such that $\angle EFH=90^{\circ}$. Suppose that $ED=FH$. Find the measure of the angle $\angle AED$.
India EGMO 2024 TST, 4
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.
[i]Proposed by Sutanay Bhattacharya[/i]