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{"problem_id": "shortlist_2025_45-part2", "problem": "The country of ELMOpia has $n \\ge 4$ cities, where some pairs of cities are connected by a road. An astute traveler notices that for any $2$ cities $A$ and $B$, there exist (distinct) cities $C$ and $D$ such that $A-C$, $C-B$, $B-D$, $D-A$ are all connected by road... |
{"problem_id": "shortlist_2025_45-part3", "problem": "The country of ELMOpia has $n \\ge 4$ cities, where some pairs of cities are connected by a road. An astute traveler notices that for any $2$ cities $A$ and $B$, there exist (distinct) cities $C$ and $D$ such that $A-C$, $C-B$, $B-D$, $D-A$ are all connected by road... |
{"problem_id": "shortlist_2025_2", "problem": "The intersection between a plane and a cube is a convex pentagon $ABCDE$ satisfying $AB = BC = 10$, $CD = AE = 8$, and $DE = 3$. What is the surface area of the cube?", "original_problem": "The intersection between a plane and a cube is a convex pentagon $ABCDE$ satisfying... |
{"problem_id": "hmmt_feb_2025_10", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \\LaTeX... |
{"problem_id": "hmmt_feb_2025_10-part1", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part2", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part3", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part4", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part5", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part6", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part7", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_10-part8", "problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \... |
{"problem_id": "hmmt_feb_2025_14", "problem": "Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$. If Sophie is at $(x, y)$, in a single step she can move to one of $(x+1, y),(x, y+1),(x-1, y+1)$, or $(x+1, y-1)$. She cannot revisit any points along her path, and neither her $x$-coordinate nor h... |
{"problem_id": "hmmt_feb_2025_14-part1", "problem": "Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$. If Sophie is at $(x, y)$, in a single step she can move to one of $(x+1, y),(x, y+1),(x-1, y+1)$, or $(x+1, y-1)$. She cannot revisit any points along her path, and neither her $x$-coordinate... |
{"problem_id": "hmmt_feb_2025_14-part2", "problem": "Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$. If Sophie is at $(x, y)$, in a single step she can move to one of $(x+1, y),(x, y+1),(x-1, y+1)$, or $(x+1, y-1)$. She cannot revisit any points along her path, and neither her $x$-coordinate... |
{"problem_id": "hmmt_feb_2025_15", "problem": "In an $11 \\times 11$ grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in ... |
{"problem_id": "hmmt_feb_2025_16", "problem": "Compute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen do... |
{"problem_id": "hmmt_feb_2025_16-part1", "problem": "Compute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are cho... |
{"problem_id": "hmmt_feb_2025_16-part2", "problem": "Compute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are cho... |
{"problem_id": "hmmt_feb_2025_16-part3", "problem": "Compute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are cho... |
{"problem_id": "hmmt_feb_2025_17", "problem": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number of $x_{... |
{"problem_id": "hmmt_feb_2025_17-part1", "problem": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number o... |
{"problem_id": "hmmt_feb_2025_17-part2", "problem": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number o... |
{"problem_id": "hmmt_feb_2025_17-part3", "problem": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number o... |
{"problem_id": "hmmt_feb_2025_17-part4", "problem": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number o... |
{"problem_id": "hmmt_feb_2025_18", "problem": "Albert writes $2025$ numbers $a_{1}, \\ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1]$. Then, each second, he \\emph{simultaneously} replaces $a_{i}$ with $\\max \\left(... |
{"problem_id": "hmmt_feb_2025_19", "problem": "Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.", "original_problem": "Two points are selected independently a... |
{"problem_id": "hmmt_feb_2025_20", "problem": "The circumference of a circle is divided into $45$ arcs, each of length $1$. Initially, there are $15$ snakes, each of length $1$, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\\frac{1}... |
{"problem_id": "hmmt_feb_2025_24", "problem": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, which i... |
{"problem_id": "hmmt_feb_2025_24-part1", "problem": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, w... |
{"problem_id": "hmmt_feb_2025_24-part2", "problem": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, w... |
{"problem_id": "hmmt_feb_2025_24-part3", "problem": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, w... |
{"problem_id": "hmmt_feb_2025_24-part4", "problem": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, w... |
{"problem_id": "hmmt_feb_2025_25", "problem": "Let $\\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\\triangle A B C$ such that $\\angle B P C=120^{\\circ}$. The circle with diameter $\\overline{A P}$ meets the circumcircle of $\\triangle A B C$ again at $X \\neq A... |
{"problem_id": "hmmt_feb_2025_25-part1", "problem": "Let $\\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\\triangle A B C$ such that $\\angle B P C=120^{\\circ}$. The circle with diameter $\\overline{A P}$ meets the circumcircle of $\\triangle A B C$ again at $X \... |
{"problem_id": "hmmt_feb_2025_25-part2", "problem": "Let $\\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\\triangle A B C$ such that $\\angle B P C=120^{\\circ}$. The circle with diameter $\\overline{A P}$ meets the circumcircle of $\\triangle A B C$ again at $X \... |
{"problem_id": "hmmt_feb_2025_25-part3", "problem": "Let $\\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\\triangle A B C$ such that $\\angle B P C=120^{\\circ}$. The circle with diameter $\\overline{A P}$ meets the circumcircle of $\\triangle A B C$ again at $X \... |
{"problem_id": "hmmt_feb_2025_26", "problem": "Trapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.", "origina... |
{"problem_id": "hmmt_feb_2025_26-part1", "problem": "Trapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.", "o... |
{"problem_id": "hmmt_feb_2025_26-part2", "problem": "Trapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.", "o... |
{"problem_id": "hmmt_feb_2025_26-part3", "problem": "Trapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.", "o... |
{"problem_id": "hmmt_feb_2025_27", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original_probl... |
{"problem_id": "hmmt_feb_2025_27-part1", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_27-part2", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_27-part3", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_27-part4", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_27-part5", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_27-part6", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_27-part7", "problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).", "original... |
{"problem_id": "hmmt_feb_2025_28", "problem": "Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, comput... |
{"problem_id": "hmmt_feb_2025_28-part1", "problem": "Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, ... |
{"problem_id": "hmmt_feb_2025_28-part2", "problem": "Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, ... |
{"problem_id": "hmmt_feb_2025_28-part3", "problem": "Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, ... |
{"problem_id": "hmmt_feb_2025_29", "problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.", "original... |
{"problem_id": "hmmt_feb_2025_29-part1", "problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.", "or... |
{"problem_id": "hmmt_feb_2025_29-part2", "problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.", "or... |
{"problem_id": "hmmt_feb_2025_29-part3", "problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.", "or... |
{"problem_id": "hmmt_feb_2025_29-part4", "problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.", "or... |
{"problem_id": "hmmt_feb_2025_29-part5", "problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.", "or... |
{"problem_id": "hmmt_feb_2025_30", "problem": "A plane $\\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths $45,66,63,55,54$, and 77, in that order. Compute the distance from the center of the rectangular prism to $\\mathcal{P}$.", "original_problem": "A plane $\\mathcal{P}$ intersects a re... |
{"problem_id": "hmmt_feb_2025_30-part1", "problem": "A plane $\\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths $45,66,63,55,54$, and 77, in that order. Compute the distance from the center of the rectangular prism to $\\mathcal{P}$.", "original_problem": "A plane $\\mathcal{P}$ intersect... |
{"problem_id": "hmmt_feb_2025_4", "problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. Compute\n\n$$\n\\sum_{j=-1000}^{1000}\\left\\lfloor\\frac{2025}{j+0.5}\\right\\rfloor\n$$", "original_problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. C... |
{"problem_id": "hmmt_feb_2025_4-part1", "problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. Compute\n\n$$\n\\sum_{j=-1000}^{1000}\\left\\lfloor\\frac{2025}{j+0.5}\\right\\rfloor\n$$", "original_problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to ... |
{"problem_id": "hmmt_feb_2025_4-part2", "problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. Compute\n\n$$\n\\sum_{j=-1000}^{1000}\\left\\lfloor\\frac{2025}{j+0.5}\\right\\rfloor\n$$", "original_problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to ... |
{"problem_id": "hmmt_feb_2025_4-part3", "problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. Compute\n\n$$\n\\sum_{j=-1000}^{1000}\\left\\lfloor\\frac{2025}{j+0.5}\\right\\rfloor\n$$", "original_problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to ... |
{"problem_id": "hmmt_feb_2025_5", "problem": "Let $\\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P(\\sqrt{3}+\\sqrt{2})=$ $P(\\sqrt{3}-\\sqrt{2})$. If $Q$ is an element of $\\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0)$.", "or... |
{"problem_id": "hmmt_feb_2025_5-part1", "problem": "Let $\\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P(\\sqrt{3}+\\sqrt{2})=$ $P(\\sqrt{3}-\\sqrt{2})$. If $Q$ is an element of $\\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0)$.... |
{"problem_id": "hmmt_feb_2025_5-part2", "problem": "Let $\\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P(\\sqrt{3}+\\sqrt{2})=$ $P(\\sqrt{3}-\\sqrt{2})$. If $Q$ is an element of $\\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0)$.... |
{"problem_id": "hmmt_feb_2025_6", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)",... |
{"problem_id": "hmmt_feb_2025_6-part1", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is pri... |
{"problem_id": "hmmt_feb_2025_6-part2", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is pri... |
{"problem_id": "hmmt_feb_2025_6-part3", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is pri... |
{"problem_id": "hmmt_feb_2025_6-part4", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is pri... |
{"problem_id": "hmmt_feb_2025_6-part5", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is pri... |
{"problem_id": "hmmt_feb_2025_6-part6", "problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)", "original_problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is pri... |
{"problem_id": "hmmt_feb_2025_7", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There exists... |
{"problem_id": "hmmt_feb_2025_7-part1", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There ... |
{"problem_id": "hmmt_feb_2025_7-part2", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There ... |
{"problem_id": "hmmt_feb_2025_7-part3", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There ... |
{"problem_id": "hmmt_feb_2025_7-part4", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There ... |
{"problem_id": "hmmt_feb_2025_7-part5", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There ... |
{"problem_id": "hmmt_feb_2025_7-part6", "problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.", "original_problem": "There ... |
{"problem_id": "hmmt_feb_2025_8", "problem": "Define $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radians.)",... |
{"problem_id": "hmmt_feb_2025_8-part1", "problem": "Define $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radia... |
{"problem_id": "hmmt_feb_2025_8-part2", "problem": "Define $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radia... |
{"problem_id": "hmmt_feb_2025_8-part3", "problem": "Define $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radia... |
{"problem_id": "hmmt_feb_2025_9", "problem": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is the co... |
{"problem_id": "hmmt_feb_2025_9-part1", "problem": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is ... |
{"problem_id": "hmmt_feb_2025_9-part2", "problem": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is ... |
{"problem_id": "hmmt_feb_2025_9-part3", "problem": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is ... |
{"problem_id": "hmmt_feb_2025_9-part4", "problem": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is ... |
{"problem_id": "imo-bench_number_theory-013", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used multi... |
{"problem_id": "imo-bench_number_theory-013-part1", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part2", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part3", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part4", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part5", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part6", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part7", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
{"problem_id": "imo-bench_number_theory-013-part8", "problem": " For a positive integer $n \\geq 2$, let the set $C_n$ be the set of integers $2^n - 2^i$ for integers $i$ such that $0 \\leq i < n$. Find the largest positive integer that cannot be expressed as a sum of numbers in $C_n$ (where the same number can be used... |
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