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Let $0 \le a, b, c \le 5$ be integers. For how many ordered triples $(a,b,c)$ is $a^2b+b^2c+c^2a-ab^2-bc^2-ca^2 = 0$?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{96} | null | math | null | null | null | null | 0841e827e87038a5f7dbbb01dce6c480 |
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$ , and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{15} | null | math | null | null | null | null | 090ef913673f055a5b2696aa4616a9f9 |
A manufacturer of airplane parts makes a certain engine that has a probability $p$ of failing on any given flight. There are two planes that can be made with this sort of engine, one that has 3 engines and one that has 5. A plane crashes if more than half its engines fail. For what values of $p$ do the two plane models have the same probability of crashing?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{0, \frac{1}{2}, 1} | null | math | null | null | null | null | 09bad46e722455a4f5dce9cfd836dd38 |
从1,2,3,4,5,7,9这7个数字中任取两个数字分别做成对数的底数和真数，则可构成不相等的对数值的数目是.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{27} | null | math | null | null | null | null | 0ab64d93b915db6dcd52487a78a896ca |
Two tourists started from $A$ to $B$ at the same time. The first tourist walked the first half of the time with a speed of $5 \mathrm{km/h}$ and the remaining time with a speed of $4 \mathrm{km/h}$. The second tourist walked the first half of the distance with a speed of $5 \mathrm{km/h}$ and the remaining distance with a speed of $4 \mathrm{km/h}$. Who will arrive at $B$ first?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{First tourist} | null | math | null | null | null | null | 0f0ef510cc05410244e2a7e2589cee75 |
The sum of three positive numbers that form an arithmetic sequence is 15, and these three numbers, when increased by 2, 5, and 13 respectively, form a geometric sequence. Find these three positive numbers.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{3, 5, 7} | null | math | null | null | null | null | 10294a92d4df01ca060d36e836c2f130 |
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{16} | null | math | null | null | null | null | 1a47214b183bf20a19cb5cae1bea3746 |
In the diagram, each of \( \triangle W X Z \) and \( \triangle X Y Z \) is an isosceles right-angled triangle. The length of \( W X \) is \( 6 \sqrt{2} \). The perimeter of quadrilateral \( W X Y Z \) is closest to
Please reason step by step, and put your final answer within \boxed{}. | \boxed{23} | null | math | null | null | null | null | 1c066d626653baf7555f0bb84585ebc8 |
In figure 1, \(BEC\) is a semicircle and \(F\) is a point on the diameter \(BC\). Given that \(BF : FC = 3 : 1\), \(AB = 8\), and \(AE = 4\), find the length of \(EC\).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{4} | null | math | null | null | null | null | 1e2c4acd3e8e286dd9d1cfb44931db80 |
Through the point with coordinates $(2,2)$, lines (including two parallel to the coordinate axes) are drawn, which divide the plane into angles of $18^{\circ}$. Find the sum of the x-coordinates of the points of intersection of these lines with the line $y = 2016 - x$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{10080} | null | math | null | null | null | null | 1ffee597d92c62203bdedd9665242568 |
Determine all pairs of natural integers \(a, b\) such that \(a^{b} = b^{a}\).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{(a, a), (2, 4), (4, 2)} | null | math | null | null | null | null | 218e25f50042ba2c8bfc7ae56435a5cf |
平面α,β所成的锐角为40°，过空间中一个定点P作与α，β所成的角都等于80°的平面，这样的平面有且仅有多少个．
Please reason step by step, and put your final answer within \boxed{}. | \boxed{4 } | null | math | null | null | null | null | 23cf5005b98aaf66633522c3989acee9 |
Several students are competing in a series of three races. A student earns $5$ points for winning a race, $3$ points for finishing second and $1$ point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{13} | null | math | null | null | null | null | 25270361305419ea0df06b8a75549d29 |
Given $V = \mathbb{C}[x]/(x-2) \oplus \mathbb{C}[x]/(x^2)$, find the minimal and characteristic polynomials of the linear transformation $T: V \to V$ where $T$ is multiplication by $x$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{x^2(x - 2)} | null | math | null | null | null | null | 2bd4187dfd79cace1e732c925f9e26ac |
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{6} | null | math | null | null | null | null | 2c3c66c3c537262ec713271c6e8c6a1c |
A train that is initially 160 meters long is moving with a speed of 30 m/s. It starts increasing in length at the rate of 2 m/s while it keeps moving. The train crosses a man standing on the platform in certain amount of time. What is the final speed of the train as it finishes crossing the man, if the acceleration due to increase in length is considered? Assume no wind resistance or other external factors.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{30} | null | math | null | null | null | null | 2f3cff5e9683b9e7d81c621d36ea79aa |
求出所有满足下面要求的不小于 -1 的实数 $t$ ：对任意 $a \in[-2, t]$, 总存在 $b, c \in[-2, t]$, 使得 $a b+c=1$. 请给出满足条件的最小整数。
Please reason step by step, and put your final answer within \boxed{}. | \boxed{-1} | null | math | null | null | null | null | 30c239a6d472e3595d4fa45151d3c4f5 |
Bob plays a game where, for some number $n$, he chooses a random integer between 0 and $n-1$, inclusive. If Bob plays this game for each of the first four prime numbers, what is the probability that the sum of the numbers he gets is greater than 0?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{419} | null | math | null | null | null | null | 32887c542dce7bdb70290b0e93e92c4d |
Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$ , it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set $A$ can have.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{40} | null | math | null | null | null | null | 3389592b4ac751db3b4af14cc1ef8bc2 |
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{$883$} | null | math | null | null | null | null | 352a35b6bf757e100c1673be3993e9a8 |
空间给定9个点，其中任何4点不共面，在9点间连接若干条线段，使图中不存在四面体，问图中最多有多少个三角形？ （1994年中国国家队测验题）
Please reason step by step, and put your final answer within \boxed{}. | \boxed{27} | null | math | null | null | null | null | 37db00dc3d7e8cb741e4929d773fce35 |
In the convex trapezoid $ABCD$, the longer parallel side is fixed. The side $CD$ moves such that neither its length nor the perimeter of the trapezoid changes. What path does the intersection point of the extensions of the non-parallel sides trace?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{an ellipse} | null | math | null | null | null | null | 399413ba024bad956bba3c5b46d0b027 |
Let $S$ be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of $S$ in ascending order.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{0, 1, 3, 4, 5, 6} | null | math | null | null | null | null | 3a007d5f43160158b9971939996878e3 |
Which numbers have the last two digits that match the last two digits of their square?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{00, 01, 25, 76} | null | math | null | null | null | null | 3abbf1694c681281c6a49ab69d3e161a |
Let $P$ be an interior point of triangle $ABC$ . Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$ taking into consideration all possible choices of triangle $ABC$ and of point $P$ .
by Elton Bojaxhiu, Albania
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\frac{2}{\sqrt{3}}} | null | math | null | null | null | null | 3c8d5f4ee37f5ca39e3d6fd46cc56619 |
The height of an isosceles trapezoid is \( h \). The upper base of the trapezoid is viewed from the midpoint of the lower base at an angle of \( 2\alpha \), and the lower base is viewed from the midpoint of the upper base at an angle of \( 2\beta \). Find the area of the trapezoid in general, and calculate it without tables if \( h = 2 \), \( \alpha = 15^\circ \), and \( \beta = 75^\circ \).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{16} | null | math | null | null | null | null | 3d68a2120e2ee26c3361c850c9403989 |
In the following diagram, two sides of a square are tangent to a circle with a diameter of $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ such that the area of the square is $m + \sqrt{n}$. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{536} | null | math | null | null | null | null | 3fd3de622f7e63da417e2af8b7bc0a46 |
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\frac{1}{2}} | null | math | null | null | null | null | 40a5a1ed0834f48ababe38172c6e36e3 |
Find all the solutions to
\[3 \sqrt[3]{3x - 2} = x^3 + 2.\]Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{1,-2} | null | math | null | null | null | null | 40c502ee705f791cedf709d5ae282c6a |
Andrew wants to write the numbers 1, 2, 3, 4, 5, 6, and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. What number should he write in the top circle to satisfy this condition? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{4} | null | math | null | null | null | null | 4189dd094abf49dee5e2b6b53a9e12dd |
Find the least positive integer $n$ such that
$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$
Please reason step by step, and put your final answer within \boxed{}. | \boxed{1} | null | math | null | null | null | null | 420913fd997ea011ebceb82aec7d6be6 |
Find all prime numbers $p$ that satisfy the following condition: For any prime number $q < p$, if $p = k q + r$ with $0 \leq r < q$, there does not exist an integer $a > 1$ such that $a^2$ divides $r$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{2, 3, 5, 7, 13} | null | math | null | null | null | null | 43adf584b238be0333368d1a8847ff4e |
Is it possible to append two digits to the right of the number 277 so that the resulting number is divisible by any number from 2 to 12?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{27720} | null | math | null | null | null | null | 4431f76e6ee235297bcbf62b3dc49012 |
An artist decided to purchase a canvas for a miniature, which should have an area of 72 cm$^{2}$. To stretch the miniature on a frame, there must be strips of blank canvas 4 cm wide on the top and bottom, and 2 cm wide on the sides.
What are the minimum dimensions of the required canvas?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{10 \, \text{cm} \times 20 \, \text{cm}} | null | math | null | null | null | null | 465cbd805661f9a9a873f1717ac7b08e |
Taking a box of matches, I found that I could use them to form any pair of the regular polygons shown in the illustration, using all the matches each time. For example, if I had 11 matches, I could form either a triangle and a pentagon, a pentagon and a hexagon, or a square and a triangle (using only 3 matches for the triangle). However, 11 matches cannot form a triangle with a hexagon, a square with a pentagon, or a square with a hexagon. Of course, each side of the shape must use an equal number of matches.
What is the smallest number of matches that could be in my box?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{36} | null | math | null | null | null | null | 48184f5a049116ee3aa82c4690e055f8 |
(4 points) $f(x)$ is an even function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = 2x + 1$. If $f(m) = 5$, then the value of $m$ is ___.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\pm 2} | null | math | null | null | null | null | 4cc8121b8c65adf1351b9f0c9b7e7c12 |
Let $x, y$ and $z$ be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of $x + y + z$ .
Please reason step by step, and put your final answer within \boxed{}. | \boxed{402} | null | math | null | null | null | null | 52be1fc99ebce7ce9a13328f69a2bec8 |
满足 $A B=1, B C=2, C D=4, D A=3$ 的凸四边形 $A B C D$ 的内切圆半径的取值范围是 $(a, \frac{k \sqrt{m}}{n}]$。请找出 a, k, m, n 的值并计算 a + k + m + n。
Please reason step by step, and put your final answer within \boxed{}. | \boxed{13} | null | math | null | null | null | null | 53236a956aebff4e0bb51264e9116110 |
The product of all the prime numbers between 1 and 100 is equal to $P$. What is the remainder when $P$ is divided by 16?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{6} | null | math | null | null | null | null | 54cb87f26ad175582b34929b1e449487 |
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is
Please reason step by step, and put your final answer within \boxed{}. | \boxed{67.5} | null | math | null | null | null | null | 56282a053ac1cb64e9df37c970562203 |
For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to \[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]
Please reason step by step, and put your final answer within \boxed{}. | \boxed{167} | null | math | null | null | null | null | 57924b2018b36dd76a6d243cd5356d50 |
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{36} | null | math | null | null | null | null | 5b782a5541f9ca499eb5bbbfc15d2f7a |
What is the Fourier transform of the function $\hat{f}(w)$ defined by $\hat{f}(w) = \int_{-\infty}^{+\infty}\max(t-1,0)e^{-i\omega t}dt$, and under what conditions does it converge?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{-\(\frac{e^{-i \omega}}{\omega^2}\)} | null | math | null | null | null | null | 5fcc35f6c59e2059ebcb56faabb9064d |
Misha created a homemade dartboard during the summer at his dacha. The circular board is divided into several sectors by circles, and darts can be thrown at it. Points are awarded according to the number written in the sector, as shown in the illustration.
Misha threw 8 darts three separate times. The second time, he scored 2 times more points than the first time, and the third time, he scored 1.5 times more points than the second time. How many points did he score the second time?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{48} | null | math | null | null | null | null | 607d77e1b7d9ee8f98d8dc6ca4825a13 |
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. Find the coordinates of $p_{2015}$ in the form $(x, y)$, and provide the sum $x + y$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{-9} | null | math | null | null | null | null | 6120764f63f27ccd0d1e0ab668e2b8b3 |
In a math competition with problems $A$, $B$, and $C$, there are 39 participants, each of whom answered at least one question correctly. Among those who answered problem $A$ correctly, the number of participants who answered only problem $A$ is 5 more than those who also answered other problems. Among those who did not answer problem $A$ correctly, the number of participants who answered problem $B$ is twice the number of those who answered problem $C$. It is also given that the number of participants who answered only problem $A$ is equal to the sum of the participants who answered only problem $B$ and only problem $C$. What is the maximum number of participants who answered problem $A$?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{23} | null | math | null | null | null | null | 627d358dadee6f2250361bcad5e72423 |
1、设X是N＊的子集，X的最小元为1，最大元为100，对X中任何一个大于1的数，都可表成X中两个数（可以相同）的和，求|X|的最小值.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{9} | null | math | null | null | null | null | 6323616a3dee647841ea03ff395ff85d |
1. How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{8} | null | math | null | null | null | null | 636ed2fca3115a4724706ff7c10bbdd8 |
Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{20} | null | math | null | null | null | null | 63976e621ac6bd56640e9f42762d0fc2 |
In triangle $ABC$ we have $|AB| \ne |AC|$ . The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$ , respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$ .
Please reason step by step, and put your final answer within \boxed{}. | \boxed{60^\circ} | null | math | null | null | null | null | 651bb7062502a9857bcc62b0cacc55a4 |
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{$\text{Friday}$} | null | math | null | null | null | null | 653c3e64e12ecd03c434892672b67f23 |
The figure shows the map of an (imaginary) country consisting of five states. The goal is to color this map with green, blue, and yellow in such a way that no two neighboring states share the same color. Calculate the exact number of different ways the map can be painted.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{6} | null | math | null | null | null | null | 69d01f5cdb309f82391c294d9f132c6a |
An orchard has 50 apple trees and 30 orange trees. Each apple tree can fill 25 baskets with 18 apples each, while each orange tree can fill 15 baskets with 12 oranges each. How many apples and oranges can you get from the entire orchard?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{5,400} | null | math | null | null | null | null | 6c6d70b880720d5df900b3424f91ea14 |
Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $S.$ Find $M-N.$
Please reason step by step, and put your final answer within \boxed{}. | \boxed{99} | null | math | null | null | null | null | 6cd166934fc231c9faee15d368bbc52a |
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10,444$ and $3,245$, and LeRoy obtains the sum $S = 13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{25} | null | math | null | null | null | null | 6d007b6efb93ecf9bfe55cc55a259627 |
Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha}, \cos{2\alpha}, \ldots, \cos{2^n\alpha}, \ldots$ are negative.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{120} | null | math | null | null | null | null | 6fb3318f141e24679eed3e3a47cef9cf |
Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]
Please reason step by step, and put your final answer within \boxed{}. | \boxed{^3+b^3+^3\geq^3+e^3} | null | math | null | null | null | null | 7336b21cac0ee6f51fbd5d1cb30e6e6e |
设S是由一些不大于15的正整数组成的集合，且S中任何两个不相交的子集，各子集的元素和互不相等，求具有这个性质的S中的元素和的最大值.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{61} | null | math | null | null | null | null | 736cd4834849121a9803ac63b7a15ec2 |
Fifteen square tiles with side 10 units long are arranged as shown. An ant walks along the edges of the tiles, always keeping a black tile on its left. Find the shortest distance that the ant would walk in going from point \( P \) to point \( Q \).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{80} | null | math | null | null | null | null | 784d66c0562f16b483583bdc4e93d1eb |
若等比数列1，a+bi,b+ai,⋯,(a,b∈R且a<0)则其前12项的和与积＝__
Please reason step by step, and put your final answer within \boxed{}. | \boxed{0，-1} | null | math | null | null | null | null | 79697b8cfaf439ca15f9bd8bdc319f91 |
$A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\prime} B C D, A B^{\prime} C D$, $A B C^{\prime} D$, and $A B C D^{\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\frac{125}{27}} | null | math | null | null | null | null | 7a87b649c503e984eaaa3ebaf1a20ed1 |
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{6:\pi} | null | math | null | null | null | null | 7aa5154900ff2248f76ca967030109f8 |
The product of three natural numbers is 600. If one of the factors is decreased by 10, the product decreases by 400. If instead that factor is increased by 5, the product doubles. Which three natural numbers have this property?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{5, 8, 15} | null | math | null | null | null | null | 7f99515faab7a712b07f72ca095bb853 |
Let $f(x) = x^2-3x$. For what values of $x$ is $f(f(x)) = f(x)$? Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{0, 3, -1, 4} | null | math | null | null | null | null | 7ffdb7bc13957be9469fcae91e84d9e1 |
Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$ . Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{9\sqrt{3}} | null | math | null | null | null | null | 82d6c81c5f78edd1a7bf2bd6c65dc8d4 |
Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$, rounded to the nearest integer?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{167} | null | math | null | null | null | null | 84be0f08df118b38f05f004bfcc9eb7f |
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{4} | null | math | null | null | null | null | 870bf8811e983a2ced1abc8bd9556103 |
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{$\dfrac{10}{21}$} | null | math | null | null | null | null | 89c24ee1986a7601e4bb5c2641faad7b |
For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define \[S_n = \sum | A \cap B | ,\] where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3, \cdots , n \right\}$ with $|A| = |B|$. For example, $S_2 = 4$ because the sum is taken over the pairs of subsets \[(A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} ,\] giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$. Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{$245$} | null | math | null | null | null | null | 8d650ca8e19e20f2089026e4e8e1929b |
Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is $1 / 3$. When Ryan swings, the probability that he will hit a home run is $1 / 2$. In one round, what is the probability that Doug will hit more home runs than Ryan hits?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{1/5} | null | math | null | null | null | null | 8e00971795c456eb7ed6bf813473862d |
将4个数1,9,8,8写成一行并进行以下操作：对每一对相邻的数都作一次减法，即用右边的数减左边的数，然后将所得的数写在两数之间，算是完成了一次操作.然后再对这7个数所排成的一行进行同样的操作，如此继续下去，共操作100次，求最后所得到的一行数的和. （第51届莫斯科市奥林匹克试题）
Please reason step by step, and put your final answer within \boxed{}. | \boxed{726} | null | math | null | null | null | null | 8ecbaa3f88de6bf758f82ff395e9d4e5 |
What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[ \text{lcm}(m,n) = 3m \times \gcd(m,n)? \]
Please reason step by step, and put your final answer within \boxed{}. | \boxed{972} | null | math | null | null | null | null | 925ceb5b3dbc7c2f4ec4564238dfa2b4 |
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = x + 5.
\]
Please reason step by step, and put your final answer within \boxed{}. | \boxed{51} | null | math | null | null | null | null | 9407f69081fac592a649cde6f0769172 |
Find the smallest whole number that is larger than the sum \[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\]
Please reason step by step, and put your final answer within \boxed{}. | \boxed{$16$} | null | math | null | null | null | null | 9c0c224403fb45ba25f6f945d25b33c5 |
A regular hexagon \( K L M N O P \) is inscribed in an equilateral triangle \( A B C \) such that the points \( K, M, O \) lie at the midpoints of the sides \( A B, B C, \) and \( A C \), respectively. Calculate the area of the hexagon \( K L M N O P \) given that the area of triangle \( A B C \) is \( 60 \text{ cm}^2 \).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{30} | null | math | null | null | null | null | 9c0d436c0c64c12608122ccf4a5e0891 |
Janet lives in a city built on a grid system. She walks 3 blocks north, then seven times as many blocks west. Then she turns around and walks 8 blocks south and twice as many blocks east in the direction of her home. If Janet can walk 2 blocks/minute, how long will it take her to get home?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{5} | null | math | null | null | null | null | 9efcf13737b1f96f02f61f2d8e7bdad0 |
Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$ .
Please reason step by step, and put your final answer within \boxed{}. | \boxed{2} | null | math | null | null | null | null | 9efef3193042a5fd9bb9a48e4af3b0ca |
A triple of integers \((a, b, c)\) satisfies \(a+b c=2017\) and \(b+c a=8\). Find all possible values of \(c\).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{-6,0,2,8} | null | math | null | null | null | null | a0ac2af217421c5ac7aec30c9dad0865 |
A two-player game is played on a grid of varying sizes (6x7, 6x8, 7x7, 7x8, and 8x8). The game starts with a piece at the bottom-left corner, and players take turns moving the piece up, right, or diagonally up-right. The game ends when a player cannot make a move. How many of these grid sizes allow the first player to guarantee a win? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{4} | null | math | null | null | null | null | a32145c266aade14542e2c7b0997cbec |
Find all integers $n$, not necessarily positive, for which there exist positive integers $a, b, c$ satisfying $a^{n}+b^{n}=c^{n}$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\pm 1, \pm 2} | null | math | null | null | null | null | a321b8f3d96879395b8af9c56da5babd |
A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{13} | null | math | null | null | null | null | a36f5a752c8d8336ab53f55749a8454f |
Determine all real numbers \( x, y, z \) satisfying \( x + y + z = 2 \), \( x^{2} + y^{2} + z^{2} = 6 \), and \( x^{3} + y^{3} + z^{3} = 8 \).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{-1, 1, 2} | null | math | null | null | null | null | a44f8f7f8f5b7ccadd8c8cbe0cc80ec8 |
Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{31} | null | math | null | null | null | null | a4cd18548464685d4a367c336908150e |
Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$ .
Please reason step by step, and put your final answer within \boxed{}. | \boxed{The maximum area of p(C) is \(\sqrt{3}\).} | null | math | null | null | null | null | a5329975f380623fc50abc14f90cca63 |
The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the probability that the Red Sox will win the series?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{1/2} | null | math | null | null | null | null | a635dea740c58351d77fd1aa74f9cb43 |
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$ , then $a_m$ is a divisor of $a_n$ and $a_m<a_n$ . Find the least possible value of $a_{2000}$ .
Please reason step by step, and put your final answer within \boxed{}. | \boxed{128} | null | math | null | null | null | null | aab6ee5a867de381310236be48d376b6 |
We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{233} | null | math | null | null | null | null | ac97f9652c8ab0295ce6e2f53a1e7ccb |
Given the equation:
\[
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
\]
where \([x]\) denotes the greatest integer less than or equal to \(x\), find the smallest value of \(x\) that satisfies this equation.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{11.5} | null | math | null | null | null | null | adb45bff6a94a32c59713167e0728402 |
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{12} | null | math | null | null | null | null | af431629f3d3e299c461f8134dbe7051 |
Is there a pair of rational numbers \(a\) and \(b\) that satisfies the following equality?
$$
\frac{a+b}{a}+\frac{a}{a+b}=b
$$
Please reason step by step, and put your final answer within \boxed{}. | \boxed{(1, -2)} | null | math | null | null | null | null | b21ce888473e92ab155494a80b8c029d |
In the parallelogram $ABCD$ ($AB \parallel CD$), diagonal $BD = a$, and $O$ is the point where the diagonals intersect. Find the area of the parallelogram, given that $\angle DBA = 45^\circ$ and $\angle AOB = 105^\circ$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\dfrac{(\sqrt{3} + 1)a^2}{4}} | null | math | null | null | null | null | b2ae0be5ef6b1edca0ceb2eb13a8e175 |
A secret facility is in the shape of a rectangle measuring $200 \times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths along the external perimeter (while the intruder remained in place). Three guards ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{150} | null | math | null | null | null | null | b2d351f2127ff1cd928f3be9391c248f |
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. Give the answer in the form $r+s\sqrt{t}$ with $r$, $s$, and $t$ positive integers，writing r.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{13} | null | math | null | null | null | null | b59e34cefd485578c53cf6e2a143be85 |
Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.
[asy]
unitsize(0.2inch);
defaultpen(linewidth(0.7));
real f(real y)
{
return (5*sqrt(3)-y)/sqrt(3);
}
draw((-5,0)--(5,0)--(0,5*sqrt(3))--cycle);
draw((-f(5*sqrt(3)/4),5*sqrt(3)/4)--(f(5*sqrt(3)/4),5*sqrt(3)/4));
draw((-f(5*sqrt(3)/2),5*sqrt(3)/2)--(f(5*sqrt(3)/2),5*sqrt(3)/2));
draw((-f(15*sqrt(3)/4),15*sqrt(3)/4)--(f(15*sqrt(3)/4),15*sqrt(3)/4));
label("$A$",(0,5*sqrt(3)),N);
label("$B$",(-f(15*sqrt(3)/4),15*sqrt(3)/4),WNW);
label("$C$",(f(15*sqrt(3)/4),15*sqrt(3)/4),ENE);
label("$D$",(-f(5*sqrt(3)/2),5*sqrt(3)/2),WNW);
label("$E$",(f(5*sqrt(3)/2),5*sqrt(3)/2),ENE);
label("$F$",(-f(5*sqrt(3)/4),5*sqrt(3)/4),WNW);
label("$G$",(f(5*sqrt(3)/4),5*sqrt(3)/4),ENE);
label("$H$",(-5,0),W);
label("$I$",(5,0),E);[/asy]
Please reason step by step, and put your final answer within \boxed{}. | \boxed{\frac{7}{16}} | null | math | null | null | null | null | b5eb73f89dbef7bc6f9ba51d911345bd |
$S$ is the sum of the first 14 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{9} a_{17} > S + 12$ and $a_{11} a_{15} < S + 47$. Indicate all possible values of $a_{1}$.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{-9, -8, -7, -6, -4, -3, -2, -1} | null | math | null | null | null | null | b6e76a12deb5b4d1ffd3f833fe1e5a5b |
For a natural number $n \ge 3$ , we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$ -gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$ ?
Please reason step by step, and put your final answer within \boxed{}. | \boxed{41} | null | math | null | null | null | null | b6e7d70e969e612c2b4d83a752cdc8f4 |
The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$ . Find the greatest integer less than or equal to $ a_{10}$ .
Please reason step by step, and put your final answer within \boxed{}. | \boxed{983} | null | math | null | null | null | null | b8ffec856b668071ceb56aab5c182939 |
The numbers \(a, b, c\) satisfy the equations
\[
a b + a + b = c, \quad b c + b + c = a, \quad c a + c + a = b
\]
Find all possible values of \(a\).
Please reason step by step, and put your final answer within \boxed{}. | \boxed{0, -1, -2} | null | math | null | null | null | null | b9df3978895825d84572a5c5d857ea32 |
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Determine the age that is not one of Mr. Jones's children. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{5} | null | math | null | null | null | null | bb528577bb1085df4ffa4fb1565a7cab |
There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.
Please reason step by step, and put your final answer within \boxed{}. | \boxed{64} | null | math | null | null | null | null | bc54572018f7a6cbdd68cd594c3d585e |
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