| {"id": "AIME25-origin-000", "question": "Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$", "answer": "70"} |
| {"id": "AIME25-origin-001", "question": "In $\\triangle ABC$ points $D$ and $E$ lie on $\\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$.", "answer": "588"} |
| {"id": "AIME25-origin-002", "question": "The $9$ members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$", "answer": "16"} |
| {"id": "AIME25-origin-003", "question": "Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.", "answer": "117"} |
| {"id": "AIME25-origin-004", "question": "There are $8!= 40320$ eight-digit positive integers that use each of the digits $1, 2, 3, 4, 5, 6, 7, 8$ exactly once. Let $N$ be the number of these integers that are divisible by $22$. Find the difference between $N$ and $2025$.$", "answer": "279"} |
| {"id": "AIME25-origin-005", "question": "An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \\neq s$. Find $r^2+s^2$", "answer": "504"} |
| {"id": "AIME25-origin-006", "question": "The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", "answer": "821"} |
| {"id": "AIME25-origin-007", "question": "Let $k$ be a real number such that the system \\begin{align*} &|25 + 20i - z| = 5 \\ &|z - 4 - k| = |z - 3i - k| \\end{align*} has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Here $i = \\sqrt{-1}$.$", "answer": "77"} |
| {"id": "AIME25-origin-008", "question": "The parabola with equation $y = x^2 - 4$ is rotated $60^\\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\\frac{a - \\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.", "answer": "62"} |
| {"id": "AIME25-origin-009", "question": "The $27$ cells of a $3 \\times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \\times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle. [asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label(\"5\",(a,a)); label(\"6\",(1+a,a)); label(\"1\",(2+a,a)); label(\"8\",(3+a,a)); label(\"4\",(4+a,a)); label(\"7\",(5+a,a)); label(\"9\",(6+a,a)); label(\"2\",(7+a,a)); label(\"3\",(8+a,a)); label(\"3\",(a,1+a)); label(\"7\",(1+a,1+a)); label(\"9\",(2+a,1+a)); label(\"5\",(3+a,1+a)); label(\"2\",(4+a,1+a)); label(\"1\",(5+a,1+a)); label(\"6\",(6+a,1+a)); label(\"8\",(7+a,1+a)); label(\"4\",(8+a,1+a)); label(\"4\",(a,2+a)); label(\"2\",(1+a,2+a)); label(\"8\",(2+a,2+a)); label(\"9\",(3+a,2+a)); label(\"6\",(4+a,2+a)); label(\"3\",(5+a,2+a)); label(\"1\",(6+a,2+a)); label(\"7\",(7+a,2+a)); label(\"5\",(8+a,2+a)); [/asy] The number of different ways to fill such a grid can be written as $p^a \\cdot q^b \\cdot r^c \\cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \\cdot a + q \\cdot b + r \\cdot c + s \\cdot d$.", "answer": "81"} |
| {"id": "AIME25-origin-010", "question": "A piecewise linear function is defined by\\[f(x) = \\begin{cases} x & \\operatorname{if} ~ -1 \\leq x < 1 \\ 2 - x & \\operatorname{if} ~ 1 \\leq x < 3\\end{cases}\\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of all these intersection points can be expressed in the form $\\tfrac{a + b\\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$, $b$, $d$ have greatest common divisor equal to $1$, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. Graph [asy] import graph; size(300); Label f; f.p=fontsize(6); yaxis(-2,2,Ticks(f, 2.0)); xaxis(-6.5,6.5,Ticks(f, 2.0)); draw((0, 0)..(1/4,sqrt(1/136))..(1/2,sqrt(1/68))..(0.75,sqrt(0.75/34))..(1, sqrt(1/34))..(2, sqrt(2/34))..(3, sqrt(3/34))..(4, sqrt(4/34))..(5, sqrt(5/34))..(6, sqrt(6/34))..(7, sqrt(7/34))..(8, sqrt(8/34)), red); draw((0, 0)..(1/4,-sqrt(1/136))..(0.5,-sqrt(1/68))..(0.75,-sqrt(0.75/34))..(1, -sqrt(1/34))..(2, -sqrt(2/34))..(3, -sqrt(3/34))..(4, -sqrt(4/34))..(5, -sqrt(5/34))..(6, -sqrt(6/34))..(7, -sqrt(7/34))..(8, -sqrt(8/34)), red); draw((-7,0)--(7,0), black+0.8bp); draw((0,-2.2)--(0,2.2), black+0.8bp); draw((-6,-0.1)--(-6,0.1), black); draw((-4,-0.1)--(-4,0.1), black); draw((-2,-0.1)--(-2,0.1), black); draw((0,-0.1)--(0,0.1), black); draw((2,-0.1)--(2,0.1), black); draw((4,-0.1)--(4,0.1), black); draw((6,-0.1)--(6,0.1), black); draw((-7,1)..(-5,-1), blue); draw((-5,-1)--(-3,1), blue); draw((-3,1)--(-1,-1), blue); draw((-1,-1)--(1,1), blue); draw((1,1)--(3,-1), blue); draw((3,-1)--(5,1), blue); draw((5,1)--(7,-1), blue); [/asy]", "answer": "259"} |
| {"id": "AIME25-origin-011", "question": "The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities\\[x-yz<y-zx<z-xy\\]forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$", "answer": "510"} |
| {"id": "AIME25-origin-012", "question": "Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.", "answer": "204"} |
| {"id": "AIME25-origin-013", "question": "Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\\angle B=\\angle E=60^{\\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$", "answer": "60"} |
| {"id": "AIME25-origin-014", "question": "Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \\leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.", "answer": "735"} |
| {"id": "AIME25-origin-015", "question": "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 $\\triangle BGE.$", "answer": "468"} |
| {"id": "AIME25-origin-016", "question": "Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$.", "answer": "49"} |
| {"id": "AIME25-origin-017", "question": "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 a say that each unit square has $2$ red sides and $2$ 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]", "answer": "82"} |
| {"id": "AIME25-origin-018", "question": "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 $m + n.$", "answer": "106"} |
| {"id": "AIME25-origin-019", "question": "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]", "answer": "336"} |
| {"id": "AIME25-origin-020", "question": "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((0,0), 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]", "answer": "293"} |
| {"id": "AIME25-origin-021", "question": "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 $m+n$.", "answer": "237"} |
| {"id": "AIME25-origin-022", "question": "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.", "answer": "610"} |
| {"id": "AIME25-origin-023", "question": "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 $n+t$.", "answer": "149"} |
| {"id": "AIME25-origin-024", "question": "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 $1000$.", "answer": "907"} |
| {"id": "AIME25-origin-025", "question": "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.", "answer": "113"} |
| {"id": "AIME25-origin-026", "question": "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 $m+n+p+q$.", "answer": "19"} |
| {"id": "AIME25-origin-027", "question": "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 $1000$.", "answer": "248"} |
| {"id": "AIME25-origin-028", "question": "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 $n.$", "answer": "104"} |
| {"id": "AIME25-origin-029", "question": "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 $k$.", "answer": "240"} |
| {"id": "AIME25-fill-001", "question": "A circle has a diameter drawn, splitting it into two semicircular regions. Then 8 chords are drawn, each connecting a random point on the boundary in one semicircle to a random point on the boundary in the other semicircle. Assuming all points are chosen uniformly and independently, what is the expected number of regions into which the interior of the circle is divided?", "answer": "32"} |
| {"id": "AIME25-fill-002", "question": "Alex divides a disk into four equal sectors by drawing two diameters that are perpendicular to each other. He then draws 12 line segments. For each segment, independently of the other segments, he chooses its first endpoint uniformly on the circumference of the disk, then chooses its second endpoint uniformly on the quarter-circle arc in the opposite sector. After drawing the two diameters and the 12 segments, what is the expected number of regions into which the disk is divided? Express your answer as a rational number in simplest form.", "answer": "179/2"} |
| {"id": "AIME25-fill-003", "question": "Find the value of \\(\\displaystyle\\prod_{k=4}^{15} \\frac{\\log_k \\bigl(7^{k^{2}-1}\\bigr)}{\\log_{k+1} \\bigl(7^{k^{2}-4}\\bigr)}\\) expressed as a reduced fraction \\(\\frac{m}{n}\\). What is \\(m+n\\)?", "answer": "87"} |
| {"id": "AIME25-fill-004", "question": "A circle has 20 distinct points chosen uniformly at random on its circumference. These points are randomly paired to form 10 chords, with no two chords sharing an endpoint. Assuming that no three chords intersect at the same interior point, what is the expected number of regions into which the interior of the circle is divided?", "answer": "26"} |
| {"id": "AIME25-fill-005", "question": "Let a_1 = \\frac{7}{2} and for n \\ge 1 define a_{n+1} = \\frac{a_n^2 + 5}{2 a_n}. If a_{12} = \\frac{p}{q} in lowest terms, where p and q are positive integers, find the remainder when p + q is divided by 1000.", "answer": "849"} |
| {"id": "AIME25-fill-006", "question": "Find the number of ordered pairs (x, y) of integers with -100 ≤ x, y ≤ 100 such that x^2 - 5xy + 6y^2 = 0 and x > y.", "answer": "83"} |
| {"id": "AIME25-fill-007", "question": "Let A be the set of all positive divisors of 66. A subset B of A is chosen uniformly at random (the empty set may be chosen). Find the probability that B is nonempty and the least common multiple of the elements of B is 22. Express your answer as a reduced fraction m/n and provide m+n.", "answer": "133"} |
| {"id": "AIME25-fill-008", "question": "Let $k$ be a real number such that the system \\[\\begin{cases} x^2 + y^2 = 7,\\\\ (x - k)^2 + (y - 2)^2 = (x - 5)^2 + (y - k)^2 \\end{cases}\\] has exactly one solution $(x,y)$ in the real numbers. The sum of all possible values of $k$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", "answer": "8"} |
| {"id": "AIME25-fill-009", "question": "Alex draws three diameters that split a circular disk into six equal sectors. He then draws 40 additional line segments as follows: each segment is formed by choosing two points uniformly at random on the circumference, with the condition that the two points lie in opposite sectors (i.e., one point is in a sector and the other point is in the sector directly opposite it). Find the expected number of regions into which the 43 line segments (the three diameters and the 40 random segments) divide the disk.", "answer": "816"} |
| {"id": "AIME25-fill-010", "question": "Let $x_1 = \\frac{3}{2}$ and for $n\\ge 1$ define $x_{n+1}=\\frac{1}{2}\\Bigl(x_n+\\frac{2}{x_n}\\Bigr)$. Write $x_{2025}$ as $\\frac{m}{n}$ in lowest terms. Find the remainder when $m+n$ is divided by $1000$.", "answer": "945"} |
| {"id": "AIME25-fill-011", "question": "Let f be the periodic function with period 4 such that for 0 ≤ t < 2, f(t)=2−t, and for 2 ≤ t < 4, f(t)=t−2, where t = x−4⌊x/4⌋ (the remainder of x upon division by 4). Extend f periodically, so f(x+4)=f(x) for all real x. The graph of y = f(x) is a triangle wave with amplitude 2.\n\nThe parabola x = 3y^2 meets the graph of y = f(x) at finitely many points. Find the sum of the y‑coordinates of all intersection points. Express the sum in the form (a+b√c)/d, where a, b, c, d are positive integers, gcd(a,b,d)=1, and c is square‑free. Compute a+b+c+d.", "answer": "93"} |
| {"id": "AIME25-fill-012", "question": "How many ordered pairs (x, y) of integers satisfy -13 ≤ x, y ≤ 13 and x^2 - 7xy + 12y^2 = 0?", "answer": "15"} |
| {"id": "AIME25-fill-013", "question": "Let k be a real number such that the system\n |z - 3| = 4,\n |z - k| = |z - (5k - 2)|\nhas exactly one complex solution z. The sum of all possible values of k can be written as m/n, where m and n are relatively prime positive integers. Find m+n.", "answer": "11"} |
| {"id": "AIME25-fill-014", "question": "Let x_1 = 13/8 and for k ≥ 1 define x_{k+1} = 1 + 1/x_k. Suppose x_{2026} = m/n where m and n are relatively prime positive integers. Find the remainder when m + n is divided by 1000.", "answer": "453"} |
| {"id": "AIME25-fill-015", "question": "Let $x_1=1$ and for $k\\ge1$ define $x_{k+1}=4+\\frac{1}{x_k}$. If $x_{2025}=\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $1000$.", "answer": "786"} |
| {"id": "AIME25-fill-016", "question": "Two diameters are drawn in a circle, dividing the circle into four quadrants. Eight chords are drawn, each connecting two points chosen independently and uniformly at random on the circumference. Assuming that no three chords intersect at a single interior point (which occurs with probability 1), what is the expected number of regions into which the circle is divided? Express your answer as a reduced fraction.", "answer": "88/3"} |
| {"id": "AIME25-fill-017", "question": "A disk is cut by two perpendicular diameters, dividing it into four quadrants. Then n chords are drawn, each connecting a uniformly random point on the upper semicircle to another uniformly random point on the upper semicircle (both endpoints lie on the same semicircle). All endpoints are chosen independently and uniformly. After drawing the diameters and all chords, the expected number of regions into which the disk is divided can be expressed in lowest terms as (n^2 + p n + q)/r. Find p+q+r.", "answer": "38"} |
| {"id": "AIME25-fill-018", "question": "Six points A, B, C, D, E, and F lie on a straight line in that order. A point G does not lie on the line. The distances AC = 70, BD = 25, CE = 35, DF = 40, AF = 120, CG = 13, and DG = 13 are known. Find the area of triangle BGE.", "answer": "300"} |
| {"id": "AIME25-fill-019", "question": "Alex draws two perpendicular diameters that intersect at the centre of a circle, thereby partitioning it into four equal quadrants. He then randomly draws 40 chords inside the circle by picking a point uniformly at random on the perimeter in one quadrant and another point uniformly at random on the perimeter in the opposite quadrant (i.e. quadrants 180° apart), and joining these two points. All chords are drawn independently. Assuming that no three chords intersect at a single interior point, what is the expected number of regions into which the circle is divided after the two diameters and the 40 chords are drawn?", "answer": "709"} |
| {"id": "AIME25-fill-020", "question": "Let f be the periodic triangle wave defined by f(x) = x for -1 ≤ x < 1, f(x) = 2 - x for 1 ≤ x < 3, and f(x+4) = f(x) for all real x. The parabola x = 2 y^2 meets the graph of y = f(x) at finitely many points. The sum of the y‑coordinates of all intersection points can be written in the form (a + b√c)/d, where a, b, c, d are positive integers, gcd(a,b,d)=1, and c is square‑free. Find a + b + c + d.", "answer": "23"} |
| {"id": "AIME25-fill-021", "question": "Find the value of the product from k = 5 to 24 of (log_k(5^{k^2-1}) / log_{k+1}(5^{k^2-4})). Write the result as a reduced fraction a/b and compute a+b.", "answer": "59"} |
| {"id": "AIME25-fill-022", "question": "Six members of a soccer team go to an ice‑cream shop. Each player orders a single scoop of chocolate, vanilla, or strawberry ice cream. At least one player orders each flavor, and the number of chocolate orders is more than the number of vanilla orders, which is more than the number of strawberry orders. In how many different ways can the six players place their orders?", "answer": "60"} |
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