Found problems: 66
2022 JHMT HS, 7
Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.
2022 JHMT HS, 4
For an integer $a$ and positive integers $n$ and $k$, let $f_k(a, n)$ be the remainder when $a^k$ is divided by $n$. Find the largest composite integer $n\leq 100$ that guarantees the infinite sequence
\[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \]
to be periodic for all integers $a$ (i.e., for each choice of $a$, there is some positive integer $T$ such that $f_k(a,n) = f_{k+T}(a,n)$ for all $k$).
2022 Iranian Geometry Olympiad, 2
We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$.
[i]Proposed by Patrik Bak, Slovakia[/i]
2022 JHMT HS, 3
Triangle $WSE$ has side lengths $WS=13$, $SE=15$, and $WE=14$. Points $J$ and $H$ lie on $\overline{SE}$ such that $SJ=JH=HE=5$. Let the angle bisector of $\angle{WES}$ intersect $\overline{WH}$ and $\overline{WJ}$ at points $M$ and $T$, respectively. Find the area of quadrilateral $JHMT$.
2022 JHMT HS, 1
The graph of $y=C\sin x$, where $C>0$ is a constant, is drawn on the interval $[0,\pi]$. Suppose that there exists a point $P$ on the graph such that the triangle with vertices $(0,0)$, $(\pi,0)$, and $P$ is equilateral. Find $C^2$.
2022 JHMT HS, 4
Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves
\[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \]
partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition.
(The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)
2022 JHMT HS, 3
Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?
2022 JHMT HS, 5
Consider an array of white unit squares arranged in a rectangular grid with $59$ rows of unit squares and $c$ columns of unit squares, for some positive integer $c$. What is the smallest possible value of $c$ such that, if we shade exactly $25$ unit squares in each column black, then there must necessarily be some row with at least $18$ black unit squares?
2022 JHMT HS, 8
An ant is walking on a sidewalk and discovers $12$ sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point $A$ to point $B$ if it can only move
[list]
[*] up, down, or right (along the border of a sidewalk panel), or
[*] up-right (along one of two margin halves of a leaf)
[/list]
and cannot visit any border or margin half more than once (an example path is highlighted in red).
[asy]
unitsize(1cm);
int r = 4;
int c = 5;
for (int i = 0; i < r; ++i) {
for (int j = 0; j < c; ++j) {
pair A = (j,i);
}
}
for (int i = 0; i < r; ++i) {
for (int j = 0; j < c; ++j) {
if (j != c-1) {
draw((j,i)--(j+1,i));
}
if (i != r-1) {
draw((j,i)--(j,i+1));
}
}
}
for (int i = 1; i < r+1; ++i) {
for (int j = 0; j < c-2; ++j) {
fill(arc((i,j),1,90,180)--cycle,deepgreen);
fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen);
draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5));
draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen);
draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen);
draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen);
draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen);
draw(arc((i,j),1,90,180));
draw(arc((i-1,j+1),1,270,360));
}
}
draw((0,3)--(0,1), red+linewidth(1.5));
draw((0,3)--(0,1), red+linewidth(1.5));
draw(arc((1,1),1,90,180), red+linewidth(1.5));
draw((1,2)--(1,1)--(2,1), red+linewidth(1.5));
draw(arc((2,2),1,270,360), red+linewidth(1.5));
draw(arc((4,2),1,90,180), red+linewidth(1.5));
draw((4,3)--(4,0), red+linewidth(1.5));
dot((0,3));
dot((4,0));
label("$A$", (0,3), NW);
label("$B$", (4,0), SE);
[/asy]
2022 JHMT HS, 9
In convex quadrilateral $KALE$, angles $\angle KAL$, $\angle AKL$, and $\angle ELK$ measure $110^\circ$, $50^\circ$, and $10^\circ$, respectively. Given that $KA = LE$ and that $\overline{KL}$ and $\overline{AE}$ intersect at point $X$, compute the value of $\tfrac{KX^2}{AL\cdot EX}$.
2022 JHMT HS, 5
A point $(X, Y, Z)$ is chosen uniformly at random from the ball of radius $4$ centered at the origin (i.e., the set $\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 4^2\}$). Compute the probability that the inequalities $X^2 \leq 1$ and $X^2 + Y^2 + Z^2 \geq 1$ simultaneously hold.
2022 JHMT HS, 2
Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.
2022 JHMT HS, 6
There is a unique choice of positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime and the infinite sums
\[ \sum_{n=0}^{\infty} \left(\left(\frac{a - b\sqrt{c}}{10}\right)^{n-10}\cdot\prod_{k=0}^{9} (n - k)\right) \quad \text{and} \quad \sum_{n=0}^{\infty} \left((a - b\sqrt{c})^{n+1}\cdot\prod_{k=0}^{9} (n - k)\right) \]
are equal (i.e., converging to the same finite value). Compute $a + b + c$.
2022 JHMT HS, 4
Hexagon $ARTSCI$ has side lengths $AR=RT=TS=SC=4\sqrt2$ and $CI=IA=10\sqrt2$. Moreover, the vertices $A$, $R$, $T$, $S$, $C$, and $I$ lie on a circle $\mathcal{K}$. Find the area of $\mathcal{K}$.
2022 ISI Entrance Examination, 8
Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.
2022 JHMT HS, 6
Let $A$ be the number of arrangements of the letters in JOHNS HOPKINS such that no two Os are adjacent, no two Hs are adjacent, no two Ns are adjacent, and no two Ss are adjacent. Find $\frac{A}{8!}$.