Found problems: 15
2025 AIME, 13
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and
\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]
$x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.
2025 AIME, 7
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2025 AIME, 4
The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
2025 AIME, 2
Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.
2025 AIME, 14
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such \[AK = AL = BK = CL = KL = 14.\] The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$
2025 AIME, 3
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings.
[asy]
size(4cm);
defaultpen(linewidth(1.2));
draw((0, 0) -- (2, 0) -- (2, 1));
draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2));
draw((0, 0) -- (0, 1), dotted);
draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted);
draw((0, 1) -- (0, 2) -- (1, 2), dotted);
[/asy]
2025 AIME, 10
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.
2025 AIME, 5
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees.
[asy]
import olympiad;
size(6cm);
defaultpen(fontsize(10pt));
pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B);
guide circ = circumcircle(D, E, F);
pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0];
draw(B--A--C--cycle);
draw(D--E--F--cycle);
draw(circ);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(J);
label("$A$", A, (0, .8));
label("$B$", B, (-.8, -.8));
label("$C$", C, (.8, -.8));
label("$D$", D, (0, -.8));
label("$E$", E, (.8, .2));
label("$F$", F, (-.8, .2));
label("$G$", G, (0, .8));
label("$H$", H, (-.2, -1));
label("$J$", J, (.2, -.8));
[/asy]
2025 AIME, 6
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle DGF$ and $\triangle CHG$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(5cm);
defaultpen(fontsize(10pt));
pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5));
filldraw(G--H--C--cycle, lightgray);
filldraw(D--G--F--cycle, lightgray);
draw(B--C);
draw(A--D);
draw(E--F--G--H--cycle);
draw(circle(origin, 15));
draw(circle(A, 6));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
label("$A$", A, (.8, -.8));
label("$B$", B, (.8, 0));
label("$C$", C, (-.8, 0));
label("$D$", D, (.4, .8));
label("$E$", E, (.8, -.8));
label("$F$", F, (.8, .8));
label("$G$", G, (-.8, .8));
label("$H$", H, (-.8, -.8));
label("$\omega_1$", (9, -5));
label("$\omega_2$", (-1, -13.5));
[/asy]
2025 AIME, 1
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$
2025 AIME, 9
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$.
2025 AIME, 15
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]
There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
2025 AIME, 12
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that
- The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$,
- $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$,
- The perimeter of $A_1A_2\dots A_{11}$ is $20$.
If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.
2025 AIME, 8
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed $N.$ For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given $N$ if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of $N$ cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of $N$ between $1$ and $1000$ inclusive for which the greedy algorithm succeeds.
2025 AIME, 11
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.