the solar cell in terms of $A, R_{\\odot}, d_{\\odot}, T_{\\mathrm{s}}, v$ and the fundamental constants $c, h, k_{\\mathrm{B}}$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: ee38c137525dddde\n", "## source_type: AI-MO/NuminaMath-CoT/olympiads\n", "## Question statement\n", "\n", "To determine the parameters \\( a_{1}, a_{2}, \\) and \\( a_{3} \\) in the formula \\( y=a_{1} x^{2}+a_{2} x+a_{3} \\), values of \\( y \\) were measured at various values of \\( x \\). The sample obtained is:\n", "\n", "\\[\n", "\\begin{array}{rrrrrrrrrr}\n", "x_{k} & -1 & -0.75 & -0.5 & -0.25 & 0 & 0.25 & 0.5 & 0.75 & 1 \\\\\n", "y_{k} & 6.01 & 5.07 & 4.30 & 3.56 & 3.07 & 2.87 & 2.18 & 2.00 & 2.14\n", "\\end{array}\n", "\\]\n", "\n", "Using these data, determine the parameters \\( a_{1}, a_{2} \\), and \\( a_{3} \\) by the method of least squares. Measurement errors of \\( x_{1}, \\ldots, x_{9} \\) and computational inaccuracies can be neglected.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 2b8218fdf3bda4e5\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Tetrahedron $ABCD$ has $AD=BC=28$ , $AC=BD=44$ , and $AB=CD=52$ . For any point $X$ in space, define $f(X)=AX+BX+CX+DX$ . The least possible value of $f(X)$ can be expressed as $m\\sqrt{n}$ , where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: f86027628b14dafe\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer. For how many integer Fahrenheit temperatures between $32$ and $1000$ inclusive does the original temperature equal the final temperature?\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 2213a3433344dec5\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\\angle O_1 P O_2 = 120^\\circ$ , then $AP=\\sqrt{a}+\\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 6c48f6ffa1ce97a3\n", "## source_type: qfq/openaimath/Precalculus\n", "## Question statement\n", "\n", "In a right triangle, one of the acute angles $\\alpha$ satisfies\n", "\\[\\tan \\frac{\\alpha}{2} = \\frac{1}{\\sqrt[3]{2}}.\\]Let $\\theta$ be the angle between the median and the angle bisector drawn from this acute angle. Find $\\tan \\theta.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 6ab3b964c7e4c75a\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "The Fibonacci numbers $F_1, F_2, F_3, \\ldots$ are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for each integer $n \\ge 1$ . Let $P$ be the unique polynomial of least degree for which $P(n) = F_n$ for all integers $1 \\le n \\le 10$ . Compute the integer $m$ for which \\[P(100) - \\sum_{k=11}^{98} P(k) = \\frac{m}{10} \\dbinom{98}{9} + 144.\\]\n", "\n", "*Proposed by Michael Tang*\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 7a6816402d6e5607\n", "## source_type: qfq/openaimath/Precalculus\n", "## Question statement\n", "\n", "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ $\\mathbf{d}$ be four distinct unit vectors in space such that\n", "\\[\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a} \\cdot \\mathbf{c} = \\mathbf{b} \\cdot \\mathbf{c} =\\mathbf{b} \\cdot \\mathbf{d} = \\mathbf{c} \\cdot \\mathbf{d} = -\\frac{1}{11}.\\]Find $\\mathbf{a} \\cdot \\mathbf{d}.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 950d9b30fa76fbb2\n", "## source_type: qfq/openaimath/Precalculus\n", "## Question statement\n", "\n", "In triangle $ABC,$ let $D,$ $E,$ $F$ be the midpoints of $\\overline{BC},$ $\\overline{AC},$ $\\overline{AB},$ respectively. Let $P,$ $Q,$ $R$ be the midpoints of $\\overline{AD},$ $\\overline{BE},$ $\\overline{CF},$ respectively. Compute\n", "\\[\\frac{AQ^2 + AR^ 2+ BP^2 + BR^2 + CP^2 + CQ^2}{AB^2 + AC^2 + BC^2}.\\]\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 9f7f1563a23537aa\n", "## source_type: baber/agieval/math_agieval\n", "## Question statement\n", "\n", "Let $\\lambda$ be a constant, $0 \\le \\lambda \\le 4,$ and let $f : [0,1] \\to [0,1]$ be defined by\n", "\\[f(x) = \\lambda x(1 - x).\\]Find the values of $\\lambda,$ $0 \\le \\lambda \\le 4,$ for which there exists an $x \\in [0,1]$ such that $f(x) \\neq x$ but $f(f(x)) = x.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 37f719cc9e67e64b\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1,$ what is the least possible total for the number of bananas?\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 5cdb4c6d0c15130a\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$ , and $(0, 3, 0)$ , and whose apex is the point $(1, 1, 3)$ . Let $Q$ be a square pyramid whose base is the same as the base of $P$ , and whose apex is the point $(2, 2, 3)$ . Find the volume of the intersection of the interiors of $P$ and $Q$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 30a96a6f8e988a3e\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \\leq a \\leq 100$ and $b \\geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 58d8003c0f2a224b\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Determine all positive integers $n\\geq 2$ for which there exist integers $x_1,x_2,\\ldots ,x_{n-1}$ satisfying the condition that if $0
value.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 08f3a7a834d4be1b\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "A compound with the molecular formula C8H6O2 gives a positive DNP test, a positive silver mirror test, and also reacts with KMnO4 to give benzoic acid. Which one of the following proton NMR data corresponds to the given compound?\n", "Answer Choices:\n", "(A) 1H NMR (ppm): 9.88 (s, 2H), 8.08 (s, 4H).\n", "(B) 1H NMR (ppm): 9.88 (s, 2H), 8.17 (d, 2H), 7.72 (t, 1H), and 7.63 (s, 1H).\n", "(C) 1H NMR (ppm): 10.36 (s, 2H), 8.08 (d, 2H), 7.92 (d, 2H)\n", "(D) 1H NMR (ppm): 9.48 (s, 1H), 7.89 (d, 2H), 7.64 (m, 2H), 7.73 (m, 1H)\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: d7a6393dad8c3a16\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: d9edf8131036e885\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is $m - n\\sqrt [3]{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m + n + p$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 444b959c049d1311\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Given a triangle with acute angle $ \\angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \\angle BME \\equal{} \\angle ECA.$ Denote by $ \\theta$ the measure of angle $ \\angle BEC.$ Evaluate $ \\frac{MC}{AB}$ in terms of $ \\theta.$ \n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 760635ba833444fd\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Circle $C$ with radius 2 has diameter $\\overline{AB}$ . Circle $D$ is internally tangent to circle $C$ at $A$ . Circle $E$ is internally tangent to circle $C$ , externally tangent to circle $D$ , and tangent to $\\overline{AB}$ . The radius of circle $D$ is three times the radius of circle $E$ , and can be written in the form $\\sqrt{m}-n$ , where $m$ and $n$ are positive integers. Find $m+n$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 51ac4d1c6147a8fa\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Given integer $m\\geq2$ , $x_1,...,x_m$ are non-negative real numbers,prove that: $$ (m-1)^{m-1}(x_1^m+...+x_m^m)\\geq(x_1+...+x_m)^m-m^mx_1...x_m $$ and please find out when the equality holds.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: c44d2bf7affbe631\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Given a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \\ldots, n_{70}$ such that $k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}\\rfloor = \\cdots = \\lfloor\\sqrt[3]{n_{70}}\\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \\leq i \\leq 70.$ Find the maximum value of $\\frac{n_{i}}{k}$ for $1\\leq i \\leq 70.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: ffc2777ae9e0cf20\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "On a hot summer day (30 degrees Celsius), a chemist received a set of samples of three metals that were similar in appearance (color) and found a way to distinguish them. To do this, he exposed the samples to the action of a concentrated solution of acids and sodium hydroxide solution. The results of his research can be presented in the form of the following table, where the “+” sign means cases when the metal dissolves, and the “-” sign means cases when the metal does not react with the acid or alkali solution:\n", "Reagent/ Metal №1 Metal №2 Metal №3\n", "Hydrochloric Acid/ - + +\n", "Nitric Acid/ + - +\n", "Sodium Hydroxide solution/ - + +\n", "Note. Three separate samples of each metal were tested separately with three different reagents.\n", "\n", "Add the atomic weights of all three metals and indicate the range in which they fall.\n", "Answer Choices:\n", "(A) from 130 to 150\n", "(B) from 220 to 240\n", "(C) from 190 to 210\n", "(D) from 160 to 180\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 914dd9e8da28a36d\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Find the eigenvalues and their multiplicities of the Laplace operator $\\Delta = \\text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 0b6d429d407a2a60\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Circles $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$ , and the product of the radii is $68$ . The x-axis and the line $y = mx$ , where $m > 0$ , are tangent to both circles. It is given that $m$ can be written in the form $a\\sqrt {b}/c$ , where $a$ , $b$ , and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a + b + c$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 903e6c9abc4fad43\n", "## source_type: Hothan/OlympiadBench/Open-ended/Physics\n", "## Question statement\n", "\n", "In the far future, the Earth received an enormous amount of charge as a result of Mad Scientist ecilA's nefarious experiments. Specifically, the total charge on Earth is $Q=1.0 \\times 10^{11} \\mathrm{C}$. (compare this with the current $5 \\times 10^{5} \\mathrm{C}$ ).\n", "Estimate the maximum height of a \"mountain\" on Earth that has a circular base with diameter $w=1.0$ $\\mathrm{km}$, if it has the shape of a spherical sector. You may assume that $h_{\\max } \\ll w$. The tensile strength of rock is $10 \\mathrm{MPa}$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: aafe1efafccd0b50\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: e7dc378a65fe121c\n", "## source_type: Hothan/OlympiadBench/Open-ended/Physics\n", "## Question statement\n", "\n", "(b) Any real inductor has undesired, or parasitic, resistance. We can model the real inductor as an ideal inductor $L$ in series with a parasitic resistance $R$.\n", "\n", "Due to the thermal noise $\\frac{d\\left\\langle V^{2}\\right\\rangle}{d f}=4 k T R$ of its parasitic resistance, this (real) inductor will support a nonzero per-frequency mean-squared current, $\\frac{d\\left\\langle I^{2}\\right\\rangle}{d f}$, even when both sides of the inductor are grounded. Compute $\\frac{d\\left\\langle I^{2}\\right\\rangle}{d f}$ as a function of $f, L, T$ and $R$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 349047e567587b90\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$ , and $\\left(P(2)\\right)^2 = P(3)$ . Then $P(\\tfrac72)=\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 7cc28a494d0a4538\n", "## source_type: qfq/openaimath/Counting & Probability\n", "## Question statement\n", "\n", "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\\{1, 2, 3, 6,9\\}$ is $9-6+3-2+1=5$ and for $\\{5\\}$ it is simply $5$. Find the sum of all such alternating sums for $n=7$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: a1148db9187b8b62\n", "## source_type: daman1209arora/jeebench/chem\n", "## Question statement\n", "\n", "The boiling point of water in a 0.1 molal silver nitrate solution (solution $\\mathbf{A}$ ) is $\\mathbf{x}^{\\circ} \\mathrm{C}$. To this solution $\\mathbf{A}$, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution $\\mathbf{B}$. The difference in the boiling points of water in the two solutions $\\mathbf{A}$ and $\\mathbf{B}$ is $\\mathbf{y} \\times 10^{-2}{ }^{\\circ} \\mathrm{C}$.\n", "(Assume: Densities of the solutions $\\mathbf{A}$ and $\\mathbf{B}$ are the same as that of water and the soluble salts dissociate completely.\n", "Use: Molal elevation constant (Ebullioscopic Constant), $K_b=0.5 \\mathrm{~K} \\mathrm{~kg} \\mathrm{~mol}^{-1}$; Boiling point of pure water as $100^{\\circ} \\mathrm{C}$.) What is the value of $|\\mathbf{y}|$?\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: d73758dd2431d32f\n", "## source_type: KbsdJames/Omni-MATH\n", "## Question statement\n", "\n", "Svitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations: - if $n$ is even, she can replace $n$ with $\\frac{n}{2}$; - if $n$ is odd, she can replace $n$ with $\\frac{n+255}{2}$; and - if $n \\geq 64$, she can replace $n$ with $n-64$. Compute the number of possible values that Svitlana can obtain by doing zero or more operations.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 3af4dd9ea22a4efa\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "You would like to use PCR to amplify the following sequence (written from 5’ to 3’ direction) for subcloning into an expression vector. In order to clone it in the correct orientation for subsequent expression, you decide to clone it using two different restriction enzymes.\n", "\n", "ATGAATCCGCGGCAGGGGTATTCCCTCAGCGGATACTACACCCATCCATTTCAAG\n", "GCTATGAGCACAGACAGCTCAGGTACCAGCAGCCTGGGCCAGGATCTTCCCCCAG\n", "TAGTTTCCTGCTTAAGCAAATAGAATTTCTCAAGGGGCAGCTCCCAGAAGCACCGG\n", "TGATTGGAAAGCAGACACCGTCACTGCCACCTTCCCTCCCAGGACTCCGGCCAAG\n", "GTTTCCAGTACTACTTGCCTCCAGTACCAGAGGCAGGCAAGTGGACATCAGGGGT\n", "GTCCCCAGGGGCGTGCATCTCGGAAGTCAGGGGCTCCAGAGAGGGTTCCAGCAT\n", "CCTTCACCACGTGGCAGGAGTCTGCCACAGAGAGGTGTTGATTGCCTTTCCTCAC\n", "ATTTCCAGGAACTGAGTATCTACCAAGATCAGGAACAAAGGATCTTAAAGTTCCTG\n", "GAAGAGCTTGGGGAAGGGAAGGCCACCACAGCACATGATCTGTCTGGGAAACTTG\n", "GGACTCCGAAGAAAGAAATCAATCGAGTTTTATACTCCCTGGCAAAGAAGGGCAAG\n", "CTACAGAAAGAGGCAGGAACACCCCCTTTGTGGAAAATCGCGGTCTC\n", "\n", "You have the enzymes ApaI, BamHI, BstNI, HindIII, KpnI, and ScaI in your freezer.\n", "The order of features on your expression vector looks like this (the cleavage sites in MCS are shown within brackets):\n", "\n", "--CMV promoter---[SmaI--HindIII--BglII--KpnI---PstI--BamHI--EcoRI]-------\n", "\n", "Using the information you have been given, choose a suitable pair of primers for PCR (shown in the 5’ to 3’ direction). Ensure your primers will make it possible to clone the sequence given above into the expression vector in a directional manner.\n", "Answer Choices:\n", "(A) aaaaggatccATGAATCCGCGGCAGGG; aaaaagcttGAGACCGCGATTTTCCACAAAGG\n", "(B) aaaaaagcttATGAATCCGCGGCAGGG; aaaggtaccGAGACCGCGATTTTCCACAAAGG\n", "(C) aaaaaagcttATGAATCCGCGGCAGGG; aaaggatccGAGACCGCGATTTTCCACAAAGG\n", "(D) aaaacccgggATGAATCCGCGGCAGGG; aaaggatccGAGACCGCGATTTTCCACAAAGG\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 84454560a79a90e4\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Let $ ABC$ be an equilateral triangle of center $ O$ , and $ M\\in BC$ . Let $ K,L$ be projections of $ M$ onto the sides $ AB$ and $ AC$ respectively. Prove that line $ OM$ passes through the midpoint of the segment $ KL$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 4396b7ccc09ddf68\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "(a)\tDecide whether there exist two decimal digits $a$ and $b$ , such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$ .\n", "(b)\tDecide whether there exist two decimal digits $c$ and $d$ , such that every integer with decimal representation $cd222... 231$ is divisible by $79$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 13eb9d92b2babbf6\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "The points $A$ , $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$ . It is given that $AB=13$ , $BC=14$ , $CA=15$ , and that the distance from $O$ to triangle $ABC$ is $\\frac{m\\sqrt{n}}k$ , where $m$ , $n$ , and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: e6301ec8f6c7d312\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "A collimated beam emerges from an analyzing spectrometer with E=4GeV. The beam is composed of particle X, with m{X}=1.48GeV and \\tau{0}=8.7\\times10^{-5}s. At what distance is the flux reduced to 25%?\n", "\n", "PS: Use a LaTeX online editor for the maths.\n", "Answer Choices:\n", "(A) 90954 m\n", "(B) 48635 m\n", "(C) 64218 m\n", "(D) 17586 m\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 5d5fbc08bbed4af6\n", "## source_type: Hothan/OlympiadBench/Open-ended/Physics\n", "## Question statement\n", "\n", "1.2.1. Write down the expression giving the atmospheric pressure $p$ as a function of the altitude $Z$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 0f7b165fa4540e6e\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of $500$ points. Alpha scored $160$ points out of $300$ points attempted on the first day, and scored $140$ points out of $200$ points attempted on the second day. Beta who did not attempt $300$ points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\\frac{300}{500} = \\frac{3}{5}$ . The largest possible two-day success ratio that Beta could achieve is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 2958d2d518405ee2\n", "## source_type: qfq/openaimath/Precalculus\n", "## Question statement\n", "\n", "In triangle $ABC$, $AB = 13$, $BC = 15$, and $CA = 14$. Point $D$ is on $\\overline{BC}$ with $CD = 6$. Point $E$ is on $\\overline{BC}$ such that $\\angle BAE = \\angle CAD$. Find $BE.$\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 434924513f213b98\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "A hexagon that is inscribed in a circle has side lengths $22$ , $22$ , $20$ , $22$ , $22$ , and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$ , where $p$ and $q$ are positive integers. Find $p+q$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 3dff6dca15751dea\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "A set $\\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\\mathcal{S}$ is an integer. Given that 1 belongs to $\\mathcal{S}$ and that 2002 is the largest element of $\\mathcal{S},$ what is the greatest number of elements that $\\mathcal{S}$ can have?\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: dc5e70c8f6617cf8\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "Enya and John are of normal phenotype but they have a son with a genetic condition that is caused by a mutation in the DMD gene and is a sex-linked recessive disorder. What are the chances of having a carrier and diseased child with each pregnancy if the sex of the child is unknown and there were no epigenetic changes involved in the progression of the disease?\n", "Answer Choices:\n", "(A) 50% carrier and 50% diseased\n", "(B) 100% carrier and 50% diseased\n", "(C) 25% carrier and 25% diseased\n", "(D) 50% carrier and 25% diseased\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: ec5d9654107ab668\n", "## source_type: KbsdJames/Omni-MATH\n", "## Question statement\n", "\n", "For a continuous and absolutely integrable complex-valued function $f(x)$ on $\\mathbb{R}$, define a function $(S f)(x)$ on $\\mathbb{R}$ by $(S f)(x)=\\int_{-\\infty}^{+\\infty} e^{2 \\pi \\mathrm{i} u x} f(u) \\mathrm{d} u$. Find explicit forms of $S\\left(\\frac{1}{1+x^{2}}\\right)$ and $S\\left(\\frac{1}{\\left(1+x^{2}\\right)^{2}}\\right)$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: f2025557f2cbd011\n", "## source_type: qfq/openaimath/Intermediate Algebra\n", "## Question statement\n", "\n", "Let a sequence be defined as follows: $a_1 = 3,$ $a_2 = 3,$ and for $n \\ge 2,$\n", "\\[a_{n + 1} a_{n - 1} = a_n^2 + 2007.\\]Find the largest integer less than or equal to $\\frac{a_{2007}^2+a_{2006}^2}{a_{2007}a_{2006}}$.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 74170d1655e58eed\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Let $\\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\\tau (n) + \\tau (n+1) = 7$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 21eed436c6e75084\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Suppose that $a$ , $b$ , and $c$ are positive real numbers such that $a^{\\log_3 7} = 27$ , $b^{\\log_7 11} = 49$ , and $c^{\\log_{11}25} = \\sqrt{11}$ . Find \\[a^{(\\log_3 7)^2} + b^{(\\log_7 11)^2} + c^{(\\log_{11} 25)^2}.\\]\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 9025d78ae2c60023\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "In triangle $ABC$ , $AB = 125$ , $AC = 117$ , and $BC = 120$ . The angle bisector of angle $A$ intersects $\\overline{BC}$ at point $L$ , and the angle bisector of angle $B$ intersects $\\overline{AC}$ at point $K$ . Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\\overline{BK}$ and $\\overline{AL}$ , respectively. Find $MN$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: fd6d3f9a996b040f\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "Arrange given compounds (1. Acetophenone, 2. propane-2,2-diyldibenzene, 3. Styrene, 4. 1-oxaspiro[4.4]nonane) in increasing oxidation state of central carbon atom (A). Also, select the proper sequence of reaction when an ester is converted first into an alcohol and then into an acid (B).\n", "\n", "1. Oxidizing reagent followed by reducing reagent\n", "2. Reducing reagent followed by oxidizing reagent\n", "Answer Choices:\n", "(A) A = 2, 3, 1, and 4, B = 2\n", "(B) A = 3, 2, 4, and 1, B = 2\n", "(C) A = 2, 3, 1, and 4, B = 1\n", "(D) A = 3, 2, 4, and 1, B = 1\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: a2259bec7a70b7e4\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "The real root of the equation $8x^3-3x^2-3x-1=0$ can be written in the form $\\frac{\\sqrt[3]{a}+\\sqrt[3]{b}+1}{c}$ , where $a$ , $b$ , and $c$ are positive integers. Find $a+b+c$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: c4e68a57fbacd618\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences. \n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 4832c8a27909b8c7\n", "## source_type: KbsdJames/Omni-MATH\n", "## Question statement\n", "\n", "Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on $[0,1]$. To generate this number, Chebby uses the factorial base: the number $0 . A_{1} A_{2} A_{3} A_{4} \\ldots$ stands for $\\sum_{n=0}^{\\infty} \\frac{A_{n}}{(n+1)!}$, where each $A_{n}$ is an integer between 0 and $n$, inclusive. Chebby has an infinite number of cards labeled $0, 1, 2, \\ldots$. He begins by putting cards $0$ and $1$ into a hat and drawing randomly to determine $A_{1}$. The card assigned $A_{1}$ does not get reused. Chebby then adds in card 2 and draws for $A_{2}$, and continues in this manner to determine the random number. At each step, he only draws one card from two in the hat. Unfortunately, this method does not result in a uniform distribution. What is the expected value of Chebby's final number?\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: f7010ac204354ee8\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Given a positive integer $k$ , find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties:\n", "\\[|S_j|=k \\text{ for } j=1, \\cdots , 5 , \\quad |\\bigcup_{j=1}^{5} S_j | = n_k ;\\]\n", "\\[|S_i \\cap S_{i+1}| = 0 = |S_5 \\cap S_1|, \\quad \\text{for } i=1,\\cdots ,4 \\]\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: c2fc8504c94d6a75\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \\[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\\] is divisible by $3$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 5f5c112a268e5ed3\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "For the integer $n>1$ , define $D(n)=\\{ a-b\\mid ab=n, a>b>0, a,b\\in\\mathbb{N} \\}$ . Prove that for any integer $k>1$ , there exists pairwise distinct positive integers $n_1,n_2,\\ldots,n_k$ such that $n_1,\\ldots,n_k>1$ and $|D(n_1)\\cap D(n_2)\\cap\\cdots\\cap D(n_k)|\\geq 2$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 4dc52341c1bd27f4\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\\theta = \\arg \\left(\\tfrac{w-z}{z}\\right)$ . The maximum possible value of $\\tan^2 \\theta$ can be written as $\\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . (Note that $\\arg(w)$ , for $w \\neq 0$ , denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.)\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 59696d411cd25bd4\n", "## source_type: AI-MO/NuminaMath-CoT/aops_forum\n", "## Question statement\n", "\n", "Let $ABCDEF$ be a regular hexagon and $P$ be a point on the shorter arc $EF$ of its circumcircle. Prove that the value of $$ \\frac{AP+BP+CP+DP}{EP+FP} $$ is constant and find its value.\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 9fdd953f846e56d0\n", "## source_type: qfq/openaimath/Precalculus\n", "## Question statement\n", "\n", "We define two sequences of vectors $(\\mathbf{v}_n)$ and $(\\mathbf{w}_n)$ as follows: First, $\\mathbf{v}_0 = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix},$ $\\mathbf{w}_0 = \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}.$ Then for all $n \\ge 1,$ $\\mathbf{v}_n$ is the projection of $\\mathbf{w}_{n - 1}$ onto $\\mathbf{v}_0,$ and $\\mathbf{w}_n$ is the projection of $\\mathbf{v}_n$ onto $\\mathbf{w}_0.$ Find\n", "\\[\\mathbf{v}_1 + \\mathbf{v}_2 + \\mathbf{v}_3 + \\dotsb.\\]\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: f8f43679754aa59e\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "For a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$ , let $s(p)$ denote the sum of the three $3$ -digit numbers $a_1a_2a_3$ , $a_4a_5a_6$ , and $a_7a_8a_9$ . Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$ . Let $n$ denote the number of permutations $p$ with $s(p) = m$ . Find $|m - n|$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 78ec502828fb712d\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Let $p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3$ . Suppose that $p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0$ . There is a point $\\left(\\frac {a}{c},\\frac {b}{c}\\right)$ for which $p\\left(\\frac {a}{c},\\frac {b}{c}\\right) = 0$ for all such polynomials, where $a$ , $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$ . Find $a + b + c$ .\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 2ef1d89f465f8145\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "Astronomers are studying a star with a 1.4 solar radius and 1.2 solar masses. When the star's surface is not covered by dark spots, its Teff is 6000K. However, when 40% of its surface is covered by spots, the overall photospheric effective temperature decreases to 5500 K. In the stellar photosphere, astronomers have observed that the ratio of ionized to neutral atoms of Mg decreases when the star has spots. What is the factor by which this ratio changes when the star does not have spots compared to when it has spots? Assume the energy of ionization corresponds to a wavelength of approximately 1448 Å, and that the stellar photosphere is in LTE.\n", "Answer Choices:\n", "(A) ~ 3.9\n", "(B) ~ 5.2\n", "(C) ~ 8.2\n", "(D) ~ 6.3\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 3303ceee5a75c570\n", "## source_type: Idavidrein/gpqa\n", "## Question statement\n", "\n", "A spin-half particle is in a mixed ensemble. The spin half system is in state |\\psi1 \\rangle with probabilty (1/3) and in state in state |\\psi2 \\rangle with probability (2/3).\n", " Here |\\psi1 \\rangle is a state in linear superposition (1/2)|\\uparrow\\rangle+(\\sqrt(3)/2) |\\downarrow\\rangle of its spin-up and spin-down states and \n", "|\\psi2 \\rangle is linear superposition (1/\\sqrt(2)|\\uparrow\\rangle+(1/\\sqrt(2)) |\\downarrow\\rangle of its spin-up and spin-down states. \n", "If |\\uparrow\\rangle and |\\downarrow\\rangle are the eigenstates of \\sigma{z} , then what is the expectation value up to two decimal place, of the operator 10\\sigma{z}+5\\sigma_{x} ? Here, symbols have their usual meanings\n", "Answer Choices:\n", "(A) 3.86\n", "(B) 1.24\n", "(C) 4.62\n", "(D) 5.28\n", "\n", "\n", "----------------------------------------------------------------------------------------------------\n", "## Question hash: 39e92361c025735f\n", "## source_type: qq8933/AIME_1983_2024\n", "## Question statement\n", "\n", "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$ , $B(0,12)$ , $C(16,0)$ , $A'(24,18)$ , $B'(36,18)$ , $C'(24,2)$ . A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0
