text stringlengths 499 13M | id stringlengths 1 126 | source stringclasses 19
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<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction.</title>
<body>
To determine the coefficient of \(x^2y^6\) in the expansion of \(\left(\frac{3}{5}x - \frac{... | AI-MO/NuminaMath-TIR:0 | AI-MO/NuminaMath-TIR | 815 | 1,923 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f′(0) > 0$, and $f(x)\geqslant 0$ for any real number $x$, find the minimum value of $\frac{f(1)}{f′(0)}$.</title>
<body>
To solve the given problem, we need to f... | AI-MO/NuminaMath-TIR:1 | AI-MO/NuminaMath-TIR | 2,445 | 6,509 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>If $n\left(n\neq 0\right)$ is a root of the equation $x^{2}+mx+3n=0$ with respect to $x$, then the value of $m+n$ is ______.</title>
<body>
To solve the problem, we need to determine the value of \(m + n\) given that \(n \, (n \neq 0)\) is a root of... | AI-MO/NuminaMath-TIR:2 | AI-MO/NuminaMath-TIR | 799 | 2,117 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$</title>
<body>
To solve this problem, we need to find the smallest positive integer \( a \) that is a multiple of both 4 and 14. This means \( a \) has ... | AI-MO/NuminaMath-TIR:3 | AI-MO/NuminaMath-TIR | 367 | 1,042 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The product of 8 and 40 is \_\_\_\_\_\_, and 5 times $\frac {1}{6}$ is \_\_\_\_\_\_. The sum of 6 instances of 500 ends with \_\_\_\_\_\_ zeros, and the sum is \_\_\_\_\_\_.</title>
<body>
To solve this problem, we need to break it down into smaller... | AI-MO/NuminaMath-TIR:4 | AI-MO/NuminaMath-TIR | 915 | 2,342 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let $S$ be a set of real numbers with mean $M$ . If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$ , respectively, then how many elements does $S$ have?</title>
<body>
To solve this problem, let's denot... | AI-MO/NuminaMath-TIR:5 | AI-MO/NuminaMath-TIR | 925 | 2,349 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given positive numbers $m$ and $n$ satisfying $2m+n=1$, find the minimum value of $\frac {1}{m}+ \frac {2}{n}$.</title>
<body>
To solve this problem, we need to find the minimum value of the expression \(\frac{1}{m} + \frac{2}{n}\) subject to the co... | AI-MO/NuminaMath-TIR:6 | AI-MO/NuminaMath-TIR | 623 | 1,838 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Petya picked four distinct digits, none of which are zero. Then he formed all possible four-digit numbers without repeating any of the digits. The sum of all these numbers turned out to be 73,326. What four digits did Petya pick?</title>
<body>
To s... | AI-MO/NuminaMath-TIR:7 | AI-MO/NuminaMath-TIR | 924 | 2,885 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find all real values of $x$ which satisfy
\[\frac{1}{x + 1} + \frac{6}{x + 5} \ge 1.\]</title>
<body>
To solve the inequality \(\frac{1}{x + 1} + \frac{6}{x + 5} \ge 1\), we will break down the solution into several clear steps with the aid of Pytho... | AI-MO/NuminaMath-TIR:8 | AI-MO/NuminaMath-TIR | 1,839 | 4,217 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>On Tuesday, Max's mom gave him $8 dollars for a hot dog at the Grand Boulevard park. On Wednesday, his mom gave him 5 times as much money as she gave him on Tuesday. On Thursday, his mom gave him $9 more in money than she gave him on Wednesday. How ... | AI-MO/NuminaMath-TIR:9 | AI-MO/NuminaMath-TIR | 387 | 1,448 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given that the function $y=f\left(x\right)$ is an odd function, defined on $R$, and satisfies $f\left(x+4\right)=f\left(x\right)$. When $x\in \left(0,2\right)$, $f\left(x\right)=2^{x}+\log _{3}(x+2)$, find $f\left(2023\right)$.</title>
<body>
To sol... | AI-MO/NuminaMath-TIR:10 | AI-MO/NuminaMath-TIR | 1,518 | 3,749 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$ , $P_{0}P_{3}P_{6}$ , $P_{0}P_{6}P_{7}$ , $P_{0}P_{7}P_{8}$ , $P_{1}P_{2}P_{3}$ , $... | AI-MO/NuminaMath-TIR:11 | AI-MO/NuminaMath-TIR | 1,232 | 3,108 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>If the complex number $z = (x^2 - 3x + 2) + (x - 1)i$ is a pure imaginary number, then the real number $x = \boxed{2}$.</title>
<body>
To determine the value of \( x \) for which the given complex number \( z = (x^2 - 3x + 2) + (x - 1)i \) is a pure... | AI-MO/NuminaMath-TIR:12 | AI-MO/NuminaMath-TIR | 606 | 1,767 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>By the first of September, Vlad bought several ballpoint and gel pens. He noticed that if all the purchased pens were gel pens, he would have paid 4 times more than he actually did. Conversely, if all the pens were ballpoint pens, the purchase would... | AI-MO/NuminaMath-TIR:13 | AI-MO/NuminaMath-TIR | 1,322 | 3,569 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find the least integer value of $x$ for which $2|x| + 7 < 17$.</title>
<body>
To find the least integer value of \( x \) for which \( 2|x| + 7 < 17 \), let's reason through the inequality step by step.
First, isolate the term involving the ab... | AI-MO/NuminaMath-TIR:14 | AI-MO/NuminaMath-TIR | 433 | 1,206 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A psychologist believes one in every four people has a disorder he calls ZYX syndrome. The clinic where he works has recently doubled its previous number of 26 patients. How many patients is the psychologist likely to diagnose with ZYX syndrome?</ti... | AI-MO/NuminaMath-TIR:15 | AI-MO/NuminaMath-TIR | 408 | 1,545 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$ . Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]</title>
<body>
To find the limit of the se... | AI-MO/NuminaMath-TIR:16 | AI-MO/NuminaMath-TIR | 1,298 | 2,702 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A positive integer is called a "lucky number" if it satisfies the following conditions:
(1) It consists of only the digits "6" and "8";
(2) It has at least two consecutive 8's.
For example, 886 is a three-digit "lucky number", 8866 is a four-digit "... | AI-MO/NuminaMath-TIR:17 | AI-MO/NuminaMath-TIR | 456 | 1,558 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>In the expansion of $(2x^{3}+\frac{1}{\sqrt{x}})^{6}$, the coefficient of $x^{4}$ is ______.</title>
<body>
To solve the problem, we need to find the coefficient of \(x^4\) in the expansion of \((2x^3 + \frac{1}{\sqrt{x}})^6\). We can use the Binomi... | AI-MO/NuminaMath-TIR:18 | AI-MO/NuminaMath-TIR | 500 | 1,305 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>There are $4$ different gifts. If they are all to be distributed to two people, A and B, with each person receiving at least one gift, the total number of different ways to distribute the gifts is ______.</title>
<body>
To solve this problem using P... | AI-MO/NuminaMath-TIR:19 | AI-MO/NuminaMath-TIR | 613 | 2,024 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Linda owns a store. She sells jeans at 11 dollars each and tees at 8 dollars each. If she sold 7 tees and 4 jeans in a day, how much money, in dollars, did she have at the end of the day?</title>
<body>
To determine how much money Linda made at the ... | AI-MO/NuminaMath-TIR:20 | AI-MO/NuminaMath-TIR | 377 | 1,234 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
$\textbf{(A) }44\qquad \textbf{(B) }55\qquad \te... | AI-MO/NuminaMath-TIR:21 | AI-MO/NuminaMath-TIR | 953 | 2,487 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Dave breaks 2 guitar strings per night when playing live. If he performs 6 shows a week for 12 weeks, how many guitar strings will he need to replace?</title>
<body>
To determine the total number of guitar strings Dave will need to replace, we need... | AI-MO/NuminaMath-TIR:22 | AI-MO/NuminaMath-TIR | 324 | 1,148 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>John has to restring 3 basses with 4 strings each, twice as many guitars which have 6 strings each, and 3 fewer 8 string guitars than the normal guitars. How many strings does he need?</title>
<body>
To solve this problem, let's break it down step ... | AI-MO/NuminaMath-TIR:23 | AI-MO/NuminaMath-TIR | 695 | 2,218 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The real-coefficient polynomials \( f(x) \) and \( g(x) \) satisfy \( (1+\mathrm{i} x)^{2001}=f(x)+\mathrm{i} g(x) \). Then the sum of the coefficients of \( f(x) \) is _______.</title>
<body>
To solve this problem, we need to carefully analyze the ... | AI-MO/NuminaMath-TIR:24 | AI-MO/NuminaMath-TIR | 1,296 | 3,179 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find the number of subsets \( S \) of \(\{1,2, \ldots 6\}\) satisfying the following conditions:
- \( S \) is non-empty.
- No subset of \( S \) has the property that the sum of its elements is 10.</title>
<body>
To solve this problem, we need to fin... | AI-MO/NuminaMath-TIR:25 | AI-MO/NuminaMath-TIR | 654 | 1,921 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>( Fixed points and limits) Find all functions from $\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}^{*}$ such that for all $x, y > 0$, \( f(x f(y)) = y f(x) \) and \( \lim _{x \rightarrow+\infty} f(x) = 0 \).</title>
<body>
To solve this problem, we'll fi... | AI-MO/NuminaMath-TIR:26 | AI-MO/NuminaMath-TIR | 1,470 | 4,006 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>In how many ways can all ten digits be arranged in a row such that the digits $2, 5, 8$ are placed next to each other?</title>
<body>
To determine the number of ways to arrange all ten digits such that the digits \(2, 5, 8\) are placed next to each ... | AI-MO/NuminaMath-TIR:27 | AI-MO/NuminaMath-TIR | 519 | 1,585 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$. It is known that $a_1 + 2a_2 + 3a_3 + \ldots + na_n = (n-1)S_n + 2n$ ($n \in \mathbb{N}^*$).
1. Find the values of $a_2$ and $a_3$.
2. Prove that the sequence $\{S_n + 2\... | AI-MO/NuminaMath-TIR:28 | AI-MO/NuminaMath-TIR | 1,418 | 2,726 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A semicircle is inscribed in triangle \( ABC \) such that its diameter lies on side \( BC \), and the arc touches sides \( AB \) and \( AC \) at points \( C_1 \) and \( B_1 \) respectively. Prove that
$$
\frac{A C_{1}}{C_{1} B} \cdot \frac{B H}{H C... | AI-MO/NuminaMath-TIR:29 | AI-MO/NuminaMath-TIR | 1,562 | 4,886 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Two farmers agree that pigs are worth $\$300$ and that goats are worth $\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\$390$ debt... | AI-MO/NuminaMath-TIR:30 | AI-MO/NuminaMath-TIR | 471 | 1,620 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find the value of \(10 \cdot \operatorname{ctg}(\operatorname{arcctg} 3 + \operatorname{arcctg} 7 + \operatorname{arcctg} 13 + \operatorname{arcctg} 21)\).</title>
<body>
To solve the given problem \(10 \cdot \operatorname{ctg}(\operatorname{arcctg}... | AI-MO/NuminaMath-TIR:31 | AI-MO/NuminaMath-TIR | 775 | 2,007 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given two sets $A=\{1, 2, 3\}$ and $B=\{2, 3\}$. Define the operation "$*$" between sets $A$ and $B$ as $A*B=\{x | x=x_1+x_2, x_1 \in A, x_2 \in B\}$. Determine the largest element in the set $A*B$ and the number of all proper subsets of $A*B$.</tit... | AI-MO/NuminaMath-TIR:32 | AI-MO/NuminaMath-TIR | 885 | 2,408 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The head of a fish weighs as much as the tail and half of the body, the body weighs as much as the head and the tail together. The tail weighs 1 kg. How much does the fish weigh?</title>
<body>
To solve this problem, let's define variables and set u... | AI-MO/NuminaMath-TIR:33 | AI-MO/NuminaMath-TIR | 514 | 1,531 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Prove that for any four points \( A, B, C, D \) the following equality holds: \(\overline{A B} \cdot \overline{C D}+\overline{A C} \cdot \overline{D B}+\overline{A D} \cdot \overline{B C}=0\).</title>
<body>
To prove the given statement, we'll utili... | AI-MO/NuminaMath-TIR:34 | AI-MO/NuminaMath-TIR | 980 | 2,581 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>If \(2^{x}=15\) and \(15^{y}=32\), the value of \(xy\) is:
(A) 5
(B) 8
(C) 16
(D) 6
(E) 4</title>
<body>
To find the value of \(xy\), we first need to solve for \(x\) and \(y\).
Given:
\[2^x = 15\]
\[15^y = 32\]
We can take the logarithm of both ... | AI-MO/NuminaMath-TIR:35 | AI-MO/NuminaMath-TIR | 655 | 1,433 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>In an arithmetic sequence $\{a_n\}$, it is known that $a_1 = 3$ and $a_9 = 11$. Find the sum of the first 9 terms, $S_9$.</title>
<body>
To solve this problem, we'll first determine the common difference of the arithmetic sequence, and then use it t... | AI-MO/NuminaMath-TIR:36 | AI-MO/NuminaMath-TIR | 635 | 1,644 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find the largest 12-digit number \( N \) that satisfies the following two conditions:
- The decimal representation of \( N \) contains six digits "4" and six digits "7";
- In the decimal representation of \( N \), no four consecutive digits form th... | AI-MO/NuminaMath-TIR:37 | AI-MO/NuminaMath-TIR | 1,125 | 4,285 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>In a lake, there is a patch of water lilies. The area of the water lilies doubles every day. If it takes 48 days for the water lilies to cover the entire lake, how many days does it take for them to cover half of the lake?</title>
<body>
To solve th... | AI-MO/NuminaMath-TIR:38 | AI-MO/NuminaMath-TIR | 487 | 1,639 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The function $y=\sin \frac{\pi }{3}x$ takes at least $2$ maximum values in the interval $[0,t]$. The minimum value of the positive integer $t$ is ______.</title>
<body>
To solve the problem, we need to find the minimum value of the positive integer ... | AI-MO/NuminaMath-TIR:39 | AI-MO/NuminaMath-TIR | 1,020 | 3,136 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A hospital has 5 internists and 6 surgeons. They need to dispatch 4 doctors to join a disaster relief medical team. If both internists and surgeons are required to be part of the team, how many ways are there to choose the doctors (answer with a num... | AI-MO/NuminaMath-TIR:40 | AI-MO/NuminaMath-TIR | 589 | 2,066 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>In the arithmetic sequence $\{a_n\}$, it is known that $a_1=1$ and $a_5=9$. Find the sum of the first $5$ terms, $S_n=$ ______.</title>
<body>
To solve the problem, we need to find the sum of the first 5 terms of the arithmetic sequence given the in... | AI-MO/NuminaMath-TIR:41 | AI-MO/NuminaMath-TIR | 690 | 1,845 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A chessboard of size $8 \times 8$ is considered.
How many ways are there to place 6 rooks such that no two rooks are ever on the same row or column?</title>
<body>
To solve the problem of placing 6 rooks on an 8x8 chessboard such that no two rooks ... | AI-MO/NuminaMath-TIR:42 | AI-MO/NuminaMath-TIR | 601 | 1,839 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The base of the pyramid is an isosceles triangle \( A B C \) with a base \( AC = 2 \) and a lateral side \( \sqrt{7} \). The face \( A C D \) is perpendicular to the base plane and is an equilateral triangle. Find the edge \( B D \), as well as the ... | AI-MO/NuminaMath-TIR:43 | AI-MO/NuminaMath-TIR | 1,621 | 4,503 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>If $a,b,c$ are integers from the set of positive integers less than $7$ such that \begin{align*}
abc&\equiv 1\pmod 7,\\
5c&\equiv 2\pmod 7,\\
6b&\equiv 3+b\pmod 7,
\end{align*}then what is the remainder when $a+b+c$ is divided by $7$?</t... | AI-MO/NuminaMath-TIR:44 | AI-MO/NuminaMath-TIR | 817 | 1,974 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Each turtle lays a clutch of 20 eggs. If 40% of the eggs successfully hatch, how many hatchlings do 6 turtles produce?</title>
<body>
To solve this problem, we need to follow these steps:
1. Calculate the total number of eggs laid by 6 turtles.
2. D... | AI-MO/NuminaMath-TIR:45 | AI-MO/NuminaMath-TIR | 373 | 1,178 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Convert the binary number $101_{(2)}$ to a decimal number.</title>
<body>
To solve the problem of converting the binary number \(101_{(2)}\) to a decimal number, we can follow these steps:
1. Understand binary to decimal conversion: Each digit in a... | AI-MO/NuminaMath-TIR:46 | AI-MO/NuminaMath-TIR | 341 | 1,056 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have?</title>
<body>
To solve the problem, we need to use the following property of regular polygons:
- The sum of the exterior angles of any polygo... | AI-MO/NuminaMath-TIR:47 | AI-MO/NuminaMath-TIR | 324 | 1,077 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>If \(x \%\) of 60 is 12, what is \(15\%\) of \(x\)?
(A) \(\frac{3}{4}\)
(B) \(\frac{1}{3}\)
(C) 4
(D) 3
(E) 9</title>
<body>
To solve this problem step-by-step, we will follow these procedures:
1. Convert the word problem into mathematica... | AI-MO/NuminaMath-TIR:48 | AI-MO/NuminaMath-TIR | 568 | 1,252 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The Absent-minded Scientist constructed a device consisting of a sensor and a transmitter. The average lifespan (expected value) of the sensor is 3 years, and the average lifespan of the transmitter is 5 years. Knowing the distribution of the lifesp... | AI-MO/NuminaMath-TIR:49 | AI-MO/NuminaMath-TIR | 1,001 | 3,513 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction.</... | AI-MO/NuminaMath-TIR:50 | AI-MO/NuminaMath-TIR | 1,129 | 2,977 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Kai will celebrate his 25th birthday in March 2020. In what year was Kai born?
(A) 1975
(B) 1990
(C) 1995
(D) 2000
(E) 1955</title>
<body>
To determine the year Kai was born, we need to calculate the year by subtracting his age (25 years) from the y... | AI-MO/NuminaMath-TIR:51 | AI-MO/NuminaMath-TIR | 372 | 1,045 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let \( a = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \) and \( b = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \). Find the value of \( a^4 + b^4 + (a+b)^4 \).</title>
<body>
To solve this problem using Python, we'll go through a series of... | AI-MO/NuminaMath-TIR:52 | AI-MO/NuminaMath-TIR | 912 | 2,032 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The numbers \(1, 2, \cdots, 9\) are randomly placed into a \(3 \times 3\) grid such that each cell contains exactly one number and each number is used exactly once. What is the probability that the sum of the numbers in each row and each column is o... | AI-MO/NuminaMath-TIR:53 | AI-MO/NuminaMath-TIR | 1,135 | 3,700 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let $S = \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004 \}$. Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$?</title>
<body>
To solve the problem of finding how many elements of the set \( S ... | AI-MO/NuminaMath-TIR:54 | AI-MO/NuminaMath-TIR | 750 | 1,798 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let \( B \) be a point on the circle centered at \( O \) with diameter \( AC \) and let \( D \) and \( E \) be the circumcenters of the triangles \( OAB \) and \( OBC \) respectively. Given that \( \sin \angle BOC = \frac{4}{5} \) and \( AC = 24 \),... | AI-MO/NuminaMath-TIR:55 | AI-MO/NuminaMath-TIR | 1,692 | 4,372 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given quadratic polynomials \( f \) and \( g \) with the same leading coefficients, it is known that the sum of the four roots of these polynomials is \( p \). Find the sum of the roots of the polynomial \( f+g \), provided that it has two roots.</t... | AI-MO/NuminaMath-TIR:56 | AI-MO/NuminaMath-TIR | 860 | 2,372 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Kolya and Vitya play the following game on an infinite grid paper. Starting with Kolya, they take turns marking grid nodes, which are the points of intersection of vertical and horizontal lines. Each player must mark a node in such a way that after ... | AI-MO/NuminaMath-TIR:57 | AI-MO/NuminaMath-TIR | 1,294 | 5,427 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Janet has 10 coins consisting of nickels, dimes, and quarters. Seven of the coins are either dimes or quarters, and eight of the coins are either dimes or nickels. How many dimes does Janet have?</title>
<body>
To solve this problem, we need to set ... | AI-MO/NuminaMath-TIR:58 | AI-MO/NuminaMath-TIR | 702 | 1,936 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A positive integer will be called "sparkly" if its smallest positive divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers \(2, 3, \ldots, 2003\) are sparkly?</title>
<body>
To determine how many numbers b... | AI-MO/NuminaMath-TIR:59 | AI-MO/NuminaMath-TIR | 589 | 1,868 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given the parabola $x^{2}=ay$, find the value of $a$ if the equation of its directrix is $y=- \frac {1}{4}$.</title>
<body>
To solve this problem, we need to understand the relationship between the parabola \(x^2 = ay\) and its directrix.
The stan... | AI-MO/NuminaMath-TIR:60 | AI-MO/NuminaMath-TIR | 646 | 1,945 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given that \( x \geq 1 \), \( y \geq 1 \), and \(\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}\), find the maximum value of \( u=\lg x y \).</title>
<body>
To solve the given problem, we need to carefully analyze the equation \(\lg^2 x + \lg^2 y =... | AI-MO/NuminaMath-TIR:61 | AI-MO/NuminaMath-TIR | 1,436 | 3,674 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Dongfang Elementary School organized a tree-planting activity. The fifth grade planted 114 trees, which was twice as many as the third grade. The fourth grade planted 32 more trees than the third grade. How many trees did the fourth grade plant?</ti... | AI-MO/NuminaMath-TIR:62 | AI-MO/NuminaMath-TIR | 616 | 1,797 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>What is the domain of the function $f(x)=\log_2(\log_3(\log_4(\log_5x)))$?</title>
<body>
To solve for the domain of the function \( f(x) = \log_2(\log_3(\log_4(\log_5x))) \), we need to ensure that the arguments in each of the logarithmic functions... | AI-MO/NuminaMath-TIR:63 | AI-MO/NuminaMath-TIR | 1,120 | 2,455 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A factory needs to build a warehouse. According to market research analysis, shipping costs are directly proportional to the distance between the factory and the warehouse, while storage costs are inversely proportional to this distance. When the di... | AI-MO/NuminaMath-TIR:64 | AI-MO/NuminaMath-TIR | 1,029 | 3,021 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given that $(2x-1)^{6}=a\_0+a\_1x+a\_2x^{2}+a\_3x^{3}+...+a\_6x^{6}.$, find the value of $a\_1+2a\_2+3a\_3+...+6a\_6 =$ ___.</title>
<body>
To solve the problem, we need to expand the expression \((2x - 1)^6\) and then find the coefficients \(a_1, a... | AI-MO/NuminaMath-TIR:65 | AI-MO/NuminaMath-TIR | 584 | 1,406 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>A polygon is drawn on graph paper with a perimeter of 36, and its sides follow the grid lines. What is the maximum possible area it can have?</title>
<body>
To find the maximum possible area of a polygon with a given perimeter that follows grid line... | AI-MO/NuminaMath-TIR:66 | AI-MO/NuminaMath-TIR | 740 | 2,543 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Find the smallest integer $k$ for which the conditions
(1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
(3) $a_9=k$
are satisfied by more than one sequence.</title>
<body>
To solve t... | AI-MO/NuminaMath-TIR:67 | AI-MO/NuminaMath-TIR | 867 | 2,554 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The insurance contract provides an insured amount of 500,000 rubles and a deductible of 1%. The actual damage amounted to 4,000 rubles. What will be the amount of insurance compensation?</title>
<body>
To determine the amount of insurance compensati... | AI-MO/NuminaMath-TIR:68 | AI-MO/NuminaMath-TIR | 423 | 1,443 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}.\]</title>
<body>
Given the constraints \( x, y, z > 0 \) and \( x + y + z = 1 \), we want to find the ... | AI-MO/NuminaMath-TIR:69 | AI-MO/NuminaMath-TIR | 958 | 2,412 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>The company "Horns and Hooves" split into the company "Horns" and the company "Hooves" with a different number of employees. The director of the company "Horns" receives the same salary as the director of the company "Hooves", and the average salary... | AI-MO/NuminaMath-TIR:70 | AI-MO/NuminaMath-TIR | 1,011 | 3,257 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Given that $a$ and $b$ are two consecutive integers, and $a < \sqrt{13} < b$, find the value of $a + b$.</title>
<body>
To solve this problem, we need to identify the two consecutive integers \( a \) and \( b \) such that \( a < \sqrt{13} &... | AI-MO/NuminaMath-TIR:71 | AI-MO/NuminaMath-TIR | 408 | 1,157 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \... | AI-MO/NuminaMath-TIR:72 | AI-MO/NuminaMath-TIR | 1,817 | 5,065 |
<doc>
<meta source="AI-MO/NuminaMath-TIR" domain="education" />
<title>John buys 5 notebooks that have 40 pages each. He uses 4 pages per day. How many days do the notebooks last?</title>
<body>
To solve this problem, we need to determine the total number of pages John has and then divide that by the number of pages ... | AI-MO/NuminaMath-TIR:73 | AI-MO/NuminaMath-TIR | 382 | 1,209 |
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