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{"problem_id": "aime_2026_28", "problem": "Call finite sets of integers $S$ and $T$ cousins if\n- $S$ and $T$ have the same number of elements,\n- $S$ and $T$ are disjoint, and\n- the elements of $S$ can be paired with the elements of $T$ so that the elements in each pair differ by exactly $1$.\nFor example, $\\{1,2,5\... |
{"problem_id": "aime_2026_28-part1", "problem": "Call finite sets of integers $S$ and $T$ cousins if\n- $S$ and $T$ have the same number of elements,\n- $S$ and $T$ are disjoint, and\n- the elements of $S$ can be paired with the elements of $T$ so that the elements in each pair differ by exactly $1$.\nFor example, $\\{... |
{"problem_id": "hmmt_feb_2026_26", "problem": "Let \\(ABC\\) be a triangle, and \\(M\\) be the midpoint of segment \\(\\overline{BC}\\). Points \\(P\\) and \\(Q\\) lie on segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\), respectively, so that \\(\\angle PMB = \\angle QMC = \\frac{1}{2}\\angle BAC\\). Given that \... |
{"problem_id": "hmmt_feb_2026_26-part1", "problem": "Let \\(ABC\\) be a triangle, and \\(M\\) be the midpoint of segment \\(\\overline{BC}\\). Points \\(P\\) and \\(Q\\) lie on segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\), respectively, so that \\(\\angle PMB = \\angle QMC = \\frac{1}{2}\\angle BAC\\). Given ... |
{"problem_id": "hmmt_feb_2026_26-part2", "problem": "Let \\(ABC\\) be a triangle, and \\(M\\) be the midpoint of segment \\(\\overline{BC}\\). Points \\(P\\) and \\(Q\\) lie on segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\), respectively, so that \\(\\angle PMB = \\angle QMC = \\frac{1}{2}\\angle BAC\\). Given ... |
{"problem_id": "hmmt_feb_2026_26-part3", "problem": "Let \\(ABC\\) be a triangle, and \\(M\\) be the midpoint of segment \\(\\overline{BC}\\). Points \\(P\\) and \\(Q\\) lie on segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\), respectively, so that \\(\\angle PMB = \\angle QMC = \\frac{1}{2}\\angle BAC\\). Given ... |
{"problem_id": "hmmt_feb_2026_26-part4", "problem": "Let \\(ABC\\) be a triangle, and \\(M\\) be the midpoint of segment \\(\\overline{BC}\\). Points \\(P\\) and \\(Q\\) lie on segments \\(\\overline{AB}\\) and \\(\\overline{AC}\\), respectively, so that \\(\\angle PMB = \\angle QMC = \\frac{1}{2}\\angle BAC\\). Given ... |
{"problem_id": "hmmt_feb_2026_16", "problem": "Derek currently owes $\\pi$ units of a currency called Money of Indiscrete Type, or MIT for short. Every day, the following happens:\n\n- He flips a fair coin to decide how much of his debt to pay. If he flips heads, he decreases his debt by 1 MIT. If he flips tails, he de... |
{"problem_id": "hmmt_feb_2026_16-part1", "problem": "Derek currently owes $\\pi$ units of a currency called Money of Indiscrete Type, or MIT for short. Every day, the following happens:\n\n- He flips a fair coin to decide how much of his debt to pay. If he flips heads, he decreases his debt by 1 MIT. If he flips tails,... |
{"problem_id": "hmmt_feb_2026_16-part2", "problem": "Derek currently owes $\\pi$ units of a currency called Money of Indiscrete Type, or MIT for short. Every day, the following happens:\n\n- He flips a fair coin to decide how much of his debt to pay. If he flips heads, he decreases his debt by 1 MIT. If he flips tails,... |
{"problem_id": "hmmt_feb_2026_25", "problem": "Three circles of radius 2 are internally tangent to a circle $\\Omega$ centered at $O$ of radius 11, and three chords of $\\Omega$ are each tangent to two of the three circles. Given that $O$ lies inside the triangle formed by the three chords and two of the chords have le... |
{"problem_id": "hmmt_feb_2026_25-part1", "problem": "Three circles of radius 2 are internally tangent to a circle $\\Omega$ centered at $O$ of radius 11, and three chords of $\\Omega$ are each tangent to two of the three circles. Given that $O$ lies inside the triangle formed by the three chords and two of the chords h... |
{"problem_id": "hmmt_feb_2026_14", "problem": "Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes.", "original_problem"... |
{"problem_id": "hmmt_feb_2026_14-part1", "problem": "Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes.", "original_pr... |
{"problem_id": "hmmt_feb_2026_14-part2", "problem": "Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes.", "original_pr... |
{"problem_id": "hmmt_feb_2026_14-part3", "problem": "Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes.", "original_pr... |
{"problem_id": "hmmt_feb_2026_14-part4", "problem": "Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes.", "original_pr... |
{"problem_id": "hmmt_feb_2026_20", "problem": "Let \\(S\\) be the set of all ordered pairs \\((x, y)\\) of nonnegative integers \\(0 \\leq x \\leq 19\\) and \\(0 \\leq y \\leq 2\\). Compute the number of permutations \\((x_1, y_1)\\), \\((x_2, y_2)\\), ..., \\((x_{60}, y_{60})\\) of the elements of \\(S\\) such that\n\... |
{"problem_id": "hmmt_feb_2026_5", "problem": "Compute the largest positive integer \\( n \\) such that\n\n\\[\nn \\text{ divides } (\\lfloor \\sqrt{n} \\rfloor)!^{n!} + 450.\n\\]", "original_problem": "Compute the largest positive integer \\( n \\) such that\n\n\\[\nn \\text{ divides } (\\lfloor \\sqrt{n} \\rfloor)!^{n... |
{"problem_id": "hmmt_feb_2026_5-part1", "problem": "Compute the largest positive integer \\( n \\) such that\n\n\\[\nn \\text{ divides } (\\lfloor \\sqrt{n} \\rfloor)!^{n!} + 450.\n\\]", "original_problem": "Compute the largest positive integer \\( n \\) such that\n\n\\[\nn \\text{ divides } (\\lfloor \\sqrt{n} \\rfloo... |
{"problem_id": "hmmt_feb_2026_5-part2", "problem": "Compute the largest positive integer \\( n \\) such that\n\n\\[\nn \\text{ divides } (\\lfloor \\sqrt{n} \\rfloor)!^{n!} + 450.\n\\]", "original_problem": "Compute the largest positive integer \\( n \\) such that\n\n\\[\nn \\text{ divides } (\\lfloor \\sqrt{n} \\rfloo... |
{"problem_id": "hmmt_feb_2026_9", "problem": "Compute\n\n\\[ \\sum_{k=1}^{\\infty} \\left( 2^{-\\lfloor 101k/1 \\rfloor} + 2^{-\\lfloor 101k/2 \\rfloor} + \\cdots + 2^{-\\lfloor 101k/100 \\rfloor} \\right). \\]", "original_problem": "Compute\n\n\\[ \\sum_{k=1}^{\\infty} \\left( 2^{-\\lfloor 101k/1 \\rfloor} + 2^{-\\lfl... |
{"problem_id": "hmmt_feb_2026_9-part1", "problem": "Compute\n\n\\[ \\sum_{k=1}^{\\infty} \\left( 2^{-\\lfloor 101k/1 \\rfloor} + 2^{-\\lfloor 101k/2 \\rfloor} + \\cdots + 2^{-\\lfloor 101k/100 \\rfloor} \\right). \\]", "original_problem": "Compute\n\n\\[ \\sum_{k=1}^{\\infty} \\left( 2^{-\\lfloor 101k/1 \\rfloor} + 2^{... |
{"problem_id": "hmmt_feb_2026_9-part2", "problem": "Compute\n\n\\[ \\sum_{k=1}^{\\infty} \\left( 2^{-\\lfloor 101k/1 \\rfloor} + 2^{-\\lfloor 101k/2 \\rfloor} + \\cdots + 2^{-\\lfloor 101k/100 \\rfloor} \\right). \\]", "original_problem": "Compute\n\n\\[ \\sum_{k=1}^{\\infty} \\left( 2^{-\\lfloor 101k/1 \\rfloor} + 2^{... |
{"problem_id": "hmmt_feb_2026_15", "problem": "Let $S$ be the set of positive integer divisors of $10^9$. Compute the number of subsets $T$ of $S$ such that\n\n- for every element $s$ of $S$, exactly one of $s$ and $10^9/s$ is in $T$, and\n- for every element $t$ of $T$, all positive integer divisors of $t$ are in $T$.... |
{"problem_id": "hmmt_feb_2026_15-part1", "problem": "Let $S$ be the set of positive integer divisors of $10^9$. Compute the number of subsets $T$ of $S$ such that\n\n- for every element $s$ of $S$, exactly one of $s$ and $10^9/s$ is in $T$, and\n- for every element $t$ of $T$, all positive integer divisors of $t$ are i... |
{"problem_id": "hmmt_feb_2026_15-part2", "problem": "Let $S$ be the set of positive integer divisors of $10^9$. Compute the number of subsets $T$ of $S$ such that\n\n- for every element $s$ of $S$, exactly one of $s$ and $10^9/s$ is in $T$, and\n- for every element $t$ of $T$, all positive integer divisors of $t$ are i... |
{"problem_id": "hmmt_feb_2026_27", "problem": "Let \\( ABC \\) be an isosceles triangle with \\( AB = AC \\). Points \\( P \\) and \\( Q \\) are located inside triangle \\( ABC \\) such that \\( BP = PQ = QC \\). Suppose that \\( \\angle BAP = 20^\\circ \\), \\( \\angle PAQ = 46^\\circ \\), and \\( \\angle QAC = 26^\\c... |
{"problem_id": "hmmt_feb_2026_27-part1", "problem": "Let \\( ABC \\) be an isosceles triangle with \\( AB = AC \\). Points \\( P \\) and \\( Q \\) are located inside triangle \\( ABC \\) such that \\( BP = PQ = QC \\). Suppose that \\( \\angle BAP = 20^\\circ \\), \\( \\angle PAQ = 46^\\circ \\), and \\( \\angle QAC = ... |
{"problem_id": "hmmt_feb_2026_27-part2", "problem": "Let \\( ABC \\) be an isosceles triangle with \\( AB = AC \\). Points \\( P \\) and \\( Q \\) are located inside triangle \\( ABC \\) such that \\( BP = PQ = QC \\). Suppose that \\( \\angle BAP = 20^\\circ \\), \\( \\angle PAQ = 46^\\circ \\), and \\( \\angle QAC = ... |
{"problem_id": "hmmt_feb_2026_27-part3", "problem": "Let \\( ABC \\) be an isosceles triangle with \\( AB = AC \\). Points \\( P \\) and \\( Q \\) are located inside triangle \\( ABC \\) such that \\( BP = PQ = QC \\). Suppose that \\( \\angle BAP = 20^\\circ \\), \\( \\angle PAQ = 46^\\circ \\), and \\( \\angle QAC = ... |
{"problem_id": "hmmt_feb_2026_27-part4", "problem": "Let \\( ABC \\) be an isosceles triangle with \\( AB = AC \\). Points \\( P \\) and \\( Q \\) are located inside triangle \\( ABC \\) such that \\( BP = PQ = QC \\). Suppose that \\( \\angle BAP = 20^\\circ \\), \\( \\angle PAQ = 46^\\circ \\), and \\( \\angle QAC = ... |
{"problem_id": "hmmt_feb_2026_31", "problem": "Jessica the jackrabbit wants to climb down a wall. The wall consists of 2026 horizontal layers stacked vertically. The $n$th layer from the top is partitioned into $2^n - 1$ identical rectangular bricks arranged side by side. Jessica begins in the topmost layer, which cont... |
{"problem_id": "hmmt_feb_2026_31-part1", "problem": "Jessica the jackrabbit wants to climb down a wall. The wall consists of 2026 horizontal layers stacked vertically. The $n$th layer from the top is partitioned into $2^n - 1$ identical rectangular bricks arranged side by side. Jessica begins in the topmost layer, whic... |
{"problem_id": "hmmt_feb_2026_31-part2", "problem": "Jessica the jackrabbit wants to climb down a wall. The wall consists of 2026 horizontal layers stacked vertically. The $n$th layer from the top is partitioned into $2^n - 1$ identical rectangular bricks arranged side by side. Jessica begins in the topmost layer, whic... |
{"problem_id": "hmmt_feb_2026_31-part3", "problem": "Jessica the jackrabbit wants to climb down a wall. The wall consists of 2026 horizontal layers stacked vertically. The $n$th layer from the top is partitioned into $2^n - 1$ identical rectangular bricks arranged side by side. Jessica begins in the topmost layer, whic... |
{"problem_id": "hmmt_feb_2026_8", "problem": "Let \\( a_0, a_1, a_2, \\ldots \\) be the unique sequence of nonnegative integers less than 397 with \\( a_0 = 1 \\) and\n\n\\[ a_{n+1}(a_n + 1)^2 \\equiv a_n \\pmod{397} \\]\n\nfor all nonnegative integers \\( n \\). Given that \\( a_{2026} = 9 \\), compute the remainder w... |
{"problem_id": "hmmt_feb_2026_8-part1", "problem": "Let \\( a_0, a_1, a_2, \\ldots \\) be the unique sequence of nonnegative integers less than 397 with \\( a_0 = 1 \\) and\n\n\\[ a_{n+1}(a_n + 1)^2 \\equiv a_n \\pmod{397} \\]\n\nfor all nonnegative integers \\( n \\). Given that \\( a_{2026} = 9 \\), compute the remai... |
{"problem_id": "hmmt_feb_2026_8-part2", "problem": "Let \\( a_0, a_1, a_2, \\ldots \\) be the unique sequence of nonnegative integers less than 397 with \\( a_0 = 1 \\) and\n\n\\[ a_{n+1}(a_n + 1)^2 \\equiv a_n \\pmod{397} \\]\n\nfor all nonnegative integers \\( n \\). Given that \\( a_{2026} = 9 \\), compute the remai... |
{"problem_id": "hmmt_feb_2026_7", "problem": "Positive real numbers \\( x, y, \\) and \\( z \\) satisfy the following equations:\n\n\\[\nxyz = 3,\n\\]\n\\[\n(x-y)(y-z)(z-x) = 4,\n\\]\n\\[\n(x+y)(y+z)(z+x) = 40.\n\\]\n\nCompute the minimum possible value for \\( x \\).\n\nGive the minimum value of x as an exact expressi... |
{"problem_id": "hmmt_feb_2026_7-part1", "problem": "Positive real numbers \\( x, y, \\) and \\( z \\) satisfy the following equations:\n\n\\[\nxyz = 3,\n\\]\n\\[\n(x-y)(y-z)(z-x) = 4,\n\\]\n\\[\n(x+y)(y+z)(z+x) = 40.\n\\]\n\nCompute the minimum possible value for \\( x \\).\n\nGive the minimum value of x as an exact ex... |
{"problem_id": "hmmt_feb_2026_7-part2", "problem": "Positive real numbers \\( x, y, \\) and \\( z \\) satisfy the following equations:\n\n\\[\nxyz = 3,\n\\]\n\\[\n(x-y)(y-z)(z-x) = 4,\n\\]\n\\[\n(x+y)(y+z)(z+x) = 40.\n\\]\n\nCompute the minimum possible value for \\( x \\).\n\nGive the minimum value of x as an exact ex... |
{"problem_id": "hmmt_feb_2026_7-part3", "problem": "Positive real numbers \\( x, y, \\) and \\( z \\) satisfy the following equations:\n\n\\[\nxyz = 3,\n\\]\n\\[\n(x-y)(y-z)(z-x) = 4,\n\\]\n\\[\n(x+y)(y+z)(z+x) = 40.\n\\]\n\nCompute the minimum possible value for \\( x \\).\n\nGive the minimum value of x as an exact ex... |
{"problem_id": "hmmt_feb_2026_7-part4", "problem": "Positive real numbers \\( x, y, \\) and \\( z \\) satisfy the following equations:\n\n\\[\nxyz = 3,\n\\]\n\\[\n(x-y)(y-z)(z-x) = 4,\n\\]\n\\[\n(x+y)(y+z)(z+x) = 40.\n\\]\n\nCompute the minimum possible value for \\( x \\).\n\nGive the minimum value of x as an exact ex... |
{"problem_id": "hmmt_feb_2026_28", "problem": "Let $ABC$ be a triangle with orthocenter $H$. The internal angle bisector of $\\angle BAC$ meets the circumcircles of triangles $ABH$, $ACH$, and $ABC$ again at points $P$, $Q$, and $M$, respectively. Suppose that points $A$, $P$, $Q$, and $M$ are distinct and lie on the i... |
{"problem_id": "hmmt_feb_2026_28-part1", "problem": "Let $ABC$ be a triangle with orthocenter $H$. The internal angle bisector of $\\angle BAC$ meets the circumcircles of triangles $ABH$, $ACH$, and $ABC$ again at points $P$, $Q$, and $M$, respectively. Suppose that points $A$, $P$, $Q$, and $M$ are distinct and lie on... |
{"problem_id": "hmmt_feb_2026_28-part2", "problem": "Let $ABC$ be a triangle with orthocenter $H$. The internal angle bisector of $\\angle BAC$ meets the circumcircles of triangles $ABH$, $ACH$, and $ABC$ again at points $P$, $Q$, and $M$, respectively. Suppose that points $A$, $P$, $Q$, and $M$ are distinct and lie on... |
{"problem_id": "hmmt_feb_2026_28-part3", "problem": "Let $ABC$ be a triangle with orthocenter $H$. The internal angle bisector of $\\angle BAC$ meets the circumcircles of triangles $ABH$, $ACH$, and $ABC$ again at points $P$, $Q$, and $M$, respectively. Suppose that points $A$, $P$, $Q$, and $M$ are distinct and lie on... |
{"problem_id": "hmmt_feb_2026_28-part4", "problem": "Let $ABC$ be a triangle with orthocenter $H$. The internal angle bisector of $\\angle BAC$ meets the circumcircles of triangles $ABH$, $ACH$, and $ABC$ again at points $P$, $Q$, and $M$, respectively. Suppose that points $A$, $P$, $Q$, and $M$ are distinct and lie on... |
{"problem_id": "hmmt_feb_2026_28-part5", "problem": "Let $ABC$ be a triangle with orthocenter $H$. The internal angle bisector of $\\angle BAC$ meets the circumcircles of triangles $ABH$, $ACH$, and $ABC$ again at points $P$, $Q$, and $M$, respectively. Suppose that points $A$, $P$, $Q$, and $M$ are distinct and lie on... |
{"problem_id": "hmmt_feb_2026_18", "problem": "A regular hexagon with side length 4 is subdivided into a lattice of 96 equilateral triangles of side length 1. Let $S$ be the set of all vertices of this lattice. Compute the number of nondegenerate triangles with vertices in $S$ that contain the center of the hexagon str... |
{"problem_id": "hmmt_feb_2026_4", "problem": "Let \\( a, b, \\) and \\( c \\) be pairwise distinct complex numbers such that\n\n\\[\na^2 + ab + b^2 = 3(a + b),\n\\]\n\\[\na^2 + ac + c^2 = 3(a + c),\n\\]\n\\[\nb^2 + bc + c^2 = 5(b + c) + 1.\n\\]\n\nCompute \\( a \\).", "original_problem": "Let \\( a, b, \\) and \\( c \\... |
{"problem_id": "hmmt_feb_2026_4-part1", "problem": "Let \\( a, b, \\) and \\( c \\) be pairwise distinct complex numbers such that\n\n\\[\na^2 + ab + b^2 = 3(a + b),\n\\]\n\\[\na^2 + ac + c^2 = 3(a + c),\n\\]\n\\[\nb^2 + bc + c^2 = 5(b + c) + 1.\n\\]\n\nCompute \\( a \\).", "original_problem": "Let \\( a, b, \\) and \\... |
{"problem_id": "hmmt_feb_2026_4-part2", "problem": "Let \\( a, b, \\) and \\( c \\) be pairwise distinct complex numbers such that\n\n\\[\na^2 + ab + b^2 = 3(a + b),\n\\]\n\\[\na^2 + ac + c^2 = 3(a + c),\n\\]\n\\[\nb^2 + bc + c^2 = 5(b + c) + 1.\n\\]\n\nCompute \\( a \\).", "original_problem": "Let \\( a, b, \\) and \\... |
{"problem_id": "hmmt_feb_2026_4-part3", "problem": "Let \\( a, b, \\) and \\( c \\) be pairwise distinct complex numbers such that\n\n\\[\na^2 + ab + b^2 = 3(a + b),\n\\]\n\\[\na^2 + ac + c^2 = 3(a + c),\n\\]\n\\[\nb^2 + bc + c^2 = 5(b + c) + 1.\n\\]\n\nCompute \\( a \\).", "original_problem": "Let \\( a, b, \\) and \\... |
{"problem_id": "hmmt_feb_2026_10", "problem": "Let\n\n\\[ S = \\sum_{k=0}^{2026} k \\binom{2k}{k} 2^k. \\]\n\nCompute the remainder when \\( S \\) is divided by 2027. (Note that 2027 is prime.)", "original_problem": "Let\n\n\\[ S = \\sum_{k=0}^{2026} k \\binom{2k}{k} 2^k. \\]\n\nCompute the remainder when \\( S \\) is ... |
{"problem_id": "hmmt_feb_2026_10-part1", "problem": "Let\n\n\\[ S = \\sum_{k=0}^{2026} k \\binom{2k}{k} 2^k. \\]\n\nCompute the remainder when \\( S \\) is divided by 2027. (Note that 2027 is prime.)", "original_problem": "Let\n\n\\[ S = \\sum_{k=0}^{2026} k \\binom{2k}{k} 2^k. \\]\n\nCompute the remainder when \\( S \... |
{"problem_id": "hmmt_feb_2026_17", "problem": "Let $S$ be the set of vertices of a right prism whose bases are regular decagons $A_1A_2 \\ldots A_{10}$ and $B_1B_2 \\ldots B_{10}$. A plane, not passing through any vertex of $S$, partitions the vertices of $S$ into two sets, one of which is $M$. Compute the number of po... |
{"problem_id": "hmmt_feb_2026_17-part1", "problem": "Let $S$ be the set of vertices of a right prism whose bases are regular decagons $A_1A_2 \\ldots A_{10}$ and $B_1B_2 \\ldots B_{10}$. A plane, not passing through any vertex of $S$, partitions the vertices of $S$ into two sets, one of which is $M$. Compute the number... |
{"problem_id": "hmmt_feb_2026_17-part2", "problem": "Let $S$ be the set of vertices of a right prism whose bases are regular decagons $A_1A_2 \\ldots A_{10}$ and $B_1B_2 \\ldots B_{10}$. A plane, not passing through any vertex of $S$, partitions the vertices of $S$ into two sets, one of which is $M$. Compute the number... |
{"problem_id": "hmmt_feb_2026_29", "problem": "Let $ABC$ be triangle with incenter $I$ and incircle $\\omega$. The circumcircle of triangle $BIC$ intersects $\\omega$ at points $E$ and $F$. Suppose that $\\Gamma \\neq \\omega$ is a circle passing through $E$ and $F$ and tangent to lines $AB$ and $AC$. If $AB = 5$, $AC ... |
{"problem_id": "hmmt_feb_2026_29-part1", "problem": "Let $ABC$ be triangle with incenter $I$ and incircle $\\omega$. The circumcircle of triangle $BIC$ intersects $\\omega$ at points $E$ and $F$. Suppose that $\\Gamma \\neq \\omega$ is a circle passing through $E$ and $F$ and tangent to lines $AB$ and $AC$. If $AB = 5$... |
{"problem_id": "hmmt_feb_2026_29-part2", "problem": "Let $ABC$ be triangle with incenter $I$ and incircle $\\omega$. The circumcircle of triangle $BIC$ intersects $\\omega$ at points $E$ and $F$. Suppose that $\\Gamma \\neq \\omega$ is a circle passing through $E$ and $F$ and tangent to lines $AB$ and $AC$. If $AB = 5$... |
{"problem_id": "hmmt_feb_2026_29-part3", "problem": "Let $ABC$ be triangle with incenter $I$ and incircle $\\omega$. The circumcircle of triangle $BIC$ intersects $\\omega$ at points $E$ and $F$. Suppose that $\\Gamma \\neq \\omega$ is a circle passing through $E$ and $F$ and tangent to lines $AB$ and $AC$. If $AB = 5$... |
{"problem_id": "hmmt_feb_2026_30", "problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppose that the orthocenter of triangle $AGO$ lies on line $BC$. Given that $AB = 11$ and $AC = 13$, compute $BC$.", "original_problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppos... |
{"problem_id": "hmmt_feb_2026_30-part1", "problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppose that the orthocenter of triangle $AGO$ lies on line $BC$. Given that $AB = 11$ and $AC = 13$, compute $BC$.", "original_problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. ... |
{"problem_id": "hmmt_feb_2026_30-part2", "problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppose that the orthocenter of triangle $AGO$ lies on line $BC$. Given that $AB = 11$ and $AC = 13$, compute $BC$.", "original_problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. ... |
{"problem_id": "hmmt_feb_2026_30-part3", "problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppose that the orthocenter of triangle $AGO$ lies on line $BC$. Given that $AB = 11$ and $AC = 13$, compute $BC$.", "original_problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. ... |
{"problem_id": "hmmt_feb_2026_30-part4", "problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppose that the orthocenter of triangle $AGO$ lies on line $BC$. Given that $AB = 11$ and $AC = 13$, compute $BC$.", "original_problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. ... |
{"problem_id": "hmmt_feb_2026_30-part5", "problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Suppose that the orthocenter of triangle $AGO$ lies on line $BC$. Given that $AB = 11$ and $AC = 13$, compute $BC$.", "original_problem": "Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. ... |
{"problem_id": "hmmt_feb_2026_24", "problem": "Let \\( ABC \\) be a triangle with \\( \\angle BAC = 90^\\circ \\). Points \\( X \\) and \\( Y \\) are such that \\( B, X, Y, \\) and \\( C \\) lie on segment \\( \\overline{BC} \\) in that order, \\( BX = 4, XY = 5, \\) and \\( YC = 3 \\). Let \\( T \\) be a point lying o... |
{"problem_id": "hmmt_feb_2026_24-part1", "problem": "Let \\( ABC \\) be a triangle with \\( \\angle BAC = 90^\\circ \\). Points \\( X \\) and \\( Y \\) are such that \\( B, X, Y, \\) and \\( C \\) lie on segment \\( \\overline{BC} \\) in that order, \\( BX = 4, XY = 5, \\) and \\( YC = 3 \\). Let \\( T \\) be a point l... |
{"problem_id": "hmmt_feb_2026_24-part2", "problem": "Let \\( ABC \\) be a triangle with \\( \\angle BAC = 90^\\circ \\). Points \\( X \\) and \\( Y \\) are such that \\( B, X, Y, \\) and \\( C \\) lie on segment \\( \\overline{BC} \\) in that order, \\( BX = 4, XY = 5, \\) and \\( YC = 3 \\). Let \\( T \\) be a point l... |
{"problem_id": "hmmt_feb_2026_24-part3", "problem": "Let \\( ABC \\) be a triangle with \\( \\angle BAC = 90^\\circ \\). Points \\( X \\) and \\( Y \\) are such that \\( B, X, Y, \\) and \\( C \\) lie on segment \\( \\overline{BC} \\) in that order, \\( BX = 4, XY = 5, \\) and \\( YC = 3 \\). Let \\( T \\) be a point l... |
{"problem_id": "hmmt_feb_2026_32", "problem": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha\\beta + \\alpha + \\beta + 100 = 0$. Suppose that $|\\alpha| = |\\beta| = M$ for some nonnegative real number $M$. Determine, with proof, all possible values of $M$.\n\nOutput your answer as a comma-separated... |
{"problem_id": "hmmt_feb_2026_32-part1", "problem": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha\\beta + \\alpha + \\beta + 100 = 0$. Suppose that $|\\alpha| = |\\beta| = M$ for some nonnegative real number $M$. Determine, with proof, all possible values of $M$.\n\nOutput your answer as a comma-sep... |
{"problem_id": "hmmt_feb_2026_32-part2", "problem": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha\\beta + \\alpha + \\beta + 100 = 0$. Suppose that $|\\alpha| = |\\beta| = M$ for some nonnegative real number $M$. Determine, with proof, all possible values of $M$.\n\nOutput your answer as a comma-sep... |
{"problem_id": "hmmt_feb_2026_32-part3", "problem": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha\\beta + \\alpha + \\beta + 100 = 0$. Suppose that $|\\alpha| = |\\beta| = M$ for some nonnegative real number $M$. Determine, with proof, all possible values of $M$.\n\nOutput your answer as a comma-sep... |
{"problem_id": "hmmt_feb_2026_32-part4", "problem": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha\\beta + \\alpha + \\beta + 100 = 0$. Suppose that $|\\alpha| = |\\beta| = M$ for some nonnegative real number $M$. Determine, with proof, all possible values of $M$.\n\nOutput your answer as a comma-sep... |
{"problem_id": "hmmt_feb_2026_32-part5", "problem": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha\\beta + \\alpha + \\beta + 100 = 0$. Suppose that $|\\alpha| = |\\beta| = M$ for some nonnegative real number $M$. Determine, with proof, all possible values of $M$.\n\nOutput your answer as a comma-sep... |
{"problem_id": "hmmt_feb_2026_6", "problem": "The numbers 1, 2, ..., 2100 are written on a board. Every second, Mark takes two numbers on the board, \\( a \\) and \\( b \\), erases them, and replaces them with \\( \\gcd(a, b) \\) and \\( \\operatorname{lcm}(a, b) \\). Mark stops once any move he makes will not change t... |
{"problem_id": "hmmt_feb_2026_6-part1", "problem": "The numbers 1, 2, ..., 2100 are written on a board. Every second, Mark takes two numbers on the board, \\( a \\) and \\( b \\), erases them, and replaces them with \\( \\gcd(a, b) \\) and \\( \\operatorname{lcm}(a, b) \\). Mark stops once any move he makes will not ch... |
{"problem_id": "hmmt_feb_2026_6-part2", "problem": "The numbers 1, 2, ..., 2100 are written on a board. Every second, Mark takes two numbers on the board, \\( a \\) and \\( b \\), erases them, and replaces them with \\( \\gcd(a, b) \\) and \\( \\operatorname{lcm}(a, b) \\). Mark stops once any move he makes will not ch... |
{"problem_id": "hmmt_feb_2026_19", "problem": "Let \\( A_1, A_2, A_3, \\ldots \\) be a sequence of finite nonempty sets of positive integers. Given that \\( |A_i \\cap A_j| = \\gcd(i, j) \\) for all (not necessarily distinct) positive integers \\( i \\) and \\( j \\), compute the minimum possible value of\n\n\\[\n\\sum... |
{"problem_id": "hmmt_feb_2026_19-part1", "problem": "Let \\( A_1, A_2, A_3, \\ldots \\) be a sequence of finite nonempty sets of positive integers. Given that \\( |A_i \\cap A_j| = \\gcd(i, j) \\) for all (not necessarily distinct) positive integers \\( i \\) and \\( j \\), compute the minimum possible value of\n\n\\[\... |
{"problem_id": "hmmt_feb_2026_19-part2", "problem": "Let \\( A_1, A_2, A_3, \\ldots \\) be a sequence of finite nonempty sets of positive integers. Given that \\( |A_i \\cap A_j| = \\gcd(i, j) \\) for all (not necessarily distinct) positive integers \\( i \\) and \\( j \\), compute the minimum possible value of\n\n\\[\... |
{"problem_id": "hmmt_feb_2026_19-part3", "problem": "Let \\( A_1, A_2, A_3, \\ldots \\) be a sequence of finite nonempty sets of positive integers. Given that \\( |A_i \\cap A_j| = \\gcd(i, j) \\) for all (not necessarily distinct) positive integers \\( i \\) and \\( j \\), compute the minimum possible value of\n\n\\[\... |
{"problem_id": "hmmt_feb_2026_33", "problem": "The numbers 1, 2, ..., 2026 are written on a blackboard. An operation consists of replacing any number on the blackboard with the positive difference between the largest and smallest numbers currently on the blackboard. Determine, with proof, the least number of operations... |
{"problem_id": "hmmt_feb_2026_33-part1", "problem": "The numbers 1, 2, ..., 2026 are written on a blackboard. An operation consists of replacing any number on the blackboard with the positive difference between the largest and smallest numbers currently on the blackboard. Determine, with proof, the least number of oper... |
{"problem_id": "hmmt_feb_2026_33-part2", "problem": "The numbers 1, 2, ..., 2026 are written on a blackboard. An operation consists of replacing any number on the blackboard with the positive difference between the largest and smallest numbers currently on the blackboard. Determine, with proof, the least number of oper... |
{"problem_id": "apex_2025_5", "problem": "Hannah has a $2024 \\times 2025$ rectangle in the coordinate plane, with sides parallel to the axes. She makes a cut from one side to another side which only goes down and/or right along grid lines. Then she puts the two pieces together, possibly with rotations and/or reflectio... |
{"problem_id": "apex_2025_9", "problem": "Let $R$ be the region in the complex plane enclosed by the curve $f(\\theta)=e^{i \\theta}+e^{2 i \\theta}+\\frac{1}{3} e^{3 i \\theta}$ for $0 \\leq$ $\\theta \\leq 2 \\pi$. Compute the perimeter of $R$.", "original_problem": "Let $R$ be the region in the complex plane enclose... |
{"problem_id": "apex_2025_9-part1", "problem": "Let $R$ be the region in the complex plane enclosed by the curve $f(\\theta)=e^{i \\theta}+e^{2 i \\theta}+\\frac{1}{3} e^{3 i \\theta}$ for $0 \\leq$ $\\theta \\leq 2 \\pi$. Compute the perimeter of $R$.", "original_problem": "Let $R$ be the region in the complex plane e... |
{"problem_id": "apex_2025_8", "problem": "Anika draws a $4$ by $6$ rectangle. How many ways can she completely tile this rectangle with L-shaped triominoes (each forming a $2\\times 2$ square missing one corner) and color each triomino red, green, or blue, such that any two neighboring triominoes are different colors? ... |
{"problem_id": "apex_2025_8-part1", "problem": "Anika draws a $4$ by $6$ rectangle. How many ways can she completely tile this rectangle with L-shaped triominoes (each forming a $2\\times 2$ square missing one corner) and color each triomino red, green, or blue, such that any two neighboring triominoes are different co... |
{"problem_id": "apex_2025_3", "problem": "Let $\\triangle A B C$ be an acute triangle with circumcircle $\\omega_{1}$, and let $D$ be a point on segment $B C$. Circle $\\omega_{2}$ is tangent to segment $A D$, segment $B D$, and $\\omega_{1}$. Circle $\\omega_{3}$ is tangent to segment $A D$, segment $C D$, and $\\omeg... |
{"problem_id": "apex_2025_10", "problem": "Let $S$ be the set of all strings of length $15$ formed from five $1$s, $2$s, and $3$s. Say a string in $S$ is threnodic if:\n\n\\begin{itemize}\n\\item No two adjacent characters are the same, and\n\\item Through a sequence of removals of contiguous substrings $123,231$, and ... |
{"problem_id": "shortlist_2025_3", "problem": "A list of positive integers satisfies the following properties:\n(A) The mean of the list is $8$.\n(2) The median of the list is $13$.\n(D) The mode of the list is $15$.\n\nMoreover, the range of the list is $27$. What is the fewest possible number of elements that could b... |
{"problem_id": "shortlist_2025_45", "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 roads. Let... |
{"problem_id": "shortlist_2025_45-part1", "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... |
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