Found problems: 37
2024 JHMT HS, 1
Compute the number of squares of positive area whose vertices all are points on the grid shown below.
[asy]
unitsize(1cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]
2024 Mexico National Olympiad, 5
Let $A$ and $B$ infinite sets of positive real numbers such that:
1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$.
2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$.
Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.
2024 JHMT HS, 14
Let $N_{13}$ be the answer to problem 13, and let $k = \tfrac{1}{N_{13} + 6}$.
Compute the infinite product
\[ (1 - k + k^2)(1 - k^3 + k^6)(1 - k^9 + k^{18})(1 - k^{27} + k^{54})\cdots, \]
where the factors take the form $(1 - k^{3^a} + k^{2\cdot 3^a})$ for all nonnegative integers $a$.
2024 JHMT HS, 2
Let $N_1$ be the answer to problem 1.
On square $JHMT$, point $X$ lies on $\overline{HM}$, and point $Y$ is the intersection point of lines $JM$ and $TX$. Assume that $\tfrac{TY}{XY}=\sqrt5$ and the area of $\triangle{XYM}$ is $N_1$. Compute the area of $\triangle{JYT}$.
2024 JHMT HS, 3
Amelia has $27$ unit cubes. She selects one and paints one of its faces. She then randomly glues all $27$ cubes together to form a $3 \times 3 \times 3$ cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.
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 $\angle AED$.
2024 JHMT HS, 13
Compute the largest nonnegative integer $T \leq 30$ that is the remainder when $T^2 + 4$ is divided by $31$.
2024 JHMT HS, 9
Compute the smallest positive integer $k$ such that the area of the region bounded by
\[ k\min(x,y)+x^2+y^2=0 \]
exceeds $100$.
2024 JHMT HS, 11
Let $N_{10}$ be the answer to problem 10.
Compute the number of ordered pairs of integers $(m,n)$ that satisfy the equation
\[ m^2+n^2=mn+N_{10}. \]
2024 JHMT HS, 14
Compute
\[ \frac{1}{2}\sin\frac{3\pi}{7}+\sin\frac{2\pi}{7}\cos\frac{3\pi}{7}. \]
2024 JHMT HS, 3
Let $N_2$ be the answer to problem 2.
On a number line, Tanya circles the first $\ell$ positive integers. Then, starting with the greatest number in the most recent circle, she circles the next $\ell$ positive integers, so that the two circles have exactly one number in common; she repeats this until $N_2$ is in a circle. Compute the sum of all possible values of $\ell$ for which $N_2$ is the greatest number in a circle.