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Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? $ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)... | math | 91 | 0.8 | openr1_math_220k | 475f73ab201777369b5272221a02833cce3ca9398bef32348f3cd8242544ec81 | 24,895 | 20260928_094616 |
6. The line $y=2 x$ intersects the curve $$ (8 \sin \theta+\cos \theta+1) y=2(2 \sin \theta-\cos \theta+3) x^{2} $$ to form a chord of maximum length of $\qquad$ . | math | 40 | 0.8 | openr1_math_220k | 3405b1e15ebcbcc34233671cdb16b221bf41e3927cc0051ec8317e98c8ac84c0 | 45,338 | 20260928_094616 |
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions : 1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$ 2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$. | math | 56 | 0.8 | openr1_math_220k | 06b5dd6b993543ffe23b4f50232342817e1ae9622e966d8b16f6aa267c7aa720 | 62,814 | 20260928_094616 |
Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For exampl... | math | 95 | 0.8 | openr1_math_220k | 92a571d2e4a1e0cd1bb388cdccd7bd9c0bedbbaeb07a77e2a1f666a9757b5209 | 61,039 | 20260928_094616 |
Example 4 If $3 x-y-1=0$, find $$ z=\left|\sqrt{x^{2}+y^{2}-8 x-2 y+17}-\sqrt{x^{2}+y^{2}-8 y+16}\right| $$ the maximum value. | math | 31 | 0.65 | openr1_math_220k | 764471cb47f6774e57c09f5109ff76beb3fae8b51ce22357e3872a5801b92d7e | 58,826 | 20260928_094616 |
What relationship exists between the numbers $a, b, c$, if there is a pair of numbers $x, y$ for which the following equalities hold: (1) $x+y=a$, (2) $x^{2}+y^{2}=b$, (3) $x^{3}+y^{3}=c$. | math | 47 | 0.8 | openr1_math_220k | abffd2f79f0d4a8afbf1cdcd6ab6b61c5c7fbae0637b7ef2d540bc8f2f69bbc3 | 40,607 | 20260928_094616 |
8. Let the sequence $\left\{a_{n}\right\}$ satisfy: $$ \begin{array}{l} a_{1}=\frac{1}{4}, a_{n+1}=a_{n}+a_{n}^{2}\left(n \in \mathbf{Z}_{+}\right) . \\ \text {Let } T_{2020}=\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{2020}+1} . \end{array} $$ If the value of $T_{2020}$ lies in the interval $(k, k+1)$, then... | math | 87 | 0.65 | openr1_math_220k | 3b391a036f38e0ac3a7ba79dce7fb63a3ba2d3e97a198ff81384e369dd3b64d7 | 25,788 | 20260928_094616 |
3. The numbers $1,2,3,4,5,6,7,8,9$ are written into the cells of a $3 \times 3$ table. After that, all possible sums of numbers standing in adjacent (by side) cells are written down in a notebook. What is the smallest number of different numbers that could have been written in the notebook? | math | 72 | 0.8 | openr1_math_220k | 0cce8ef47c52865c3c3e9477e8dbda54cb92e97e7fae5c7c0328ff3784a1a0f4 | 7,753 | 20260928_094616 |
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the... | math | 498 | 0.9 | openr1_math_220k | 0c35475c177f2282927172bfb8251a4be540e96067add0992e0ac334b708d12a | 14,735 | 20260928_094616 |
1.77 If $x=\sqrt{7}+\sqrt{6}$, then $\left(x+\frac{1}{x}\right):\left(x-\frac{1}{x}\right)$ equals (A) $\sqrt{7}: \sqrt{6}$. (B) $7: 6$. (C) $x^{2}: 1$. (D) $x$. (1st "Five Sheep Cup" Junior High School Mathematics Competition, 1989) | math | 58 | 0.75 | openr1_math_220k | 7e3f8c7ffb6fd7e8823fd22100c8245fc751dbfccd42ad132cd79890064016ff | 17,924 | 20260928_094616 |
11. (Mathematics for Middle School, 1994, Issue 4, Olympiad Training Problem) Find the real-coefficient polynomial $f(x)$ that satisfies the following conditions: (1) For any real number $a$, $f(a+1)=f(a)+f(1)$; (2) There exists a real number $k_{1} \neq 0$ such that $f\left(k_{1}\right)=k_{2}, f\left(k_{2}\right)=k_{3... | math | 105 | 0.8 | openr1_math_220k | 31f0960711d79928815327766ce54bc083bb434cb643a0f7703846455807b2fc | 32,301 | 20260928_094616 |
Let $A,B,C$ be angles of a triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} \end{align*} There are positive integers $p$, $q$, $r$, and $s$ for which \[\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \fra... | math | 112 | 0.85 | openr1_math_220k | ee4150651a642ecdde8525c3dd4afef3a24e7376d20f8ed005aeb942f4a52957 | 15,388 | 20260928_094616 |
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill? $\textbf{(A)}\ \text{\textdollar}120\qquad\textbf{(B)}\ \text{\textdollar}128\qquad\textbf{(C)}\ \text{\te... | math | 103 | 0.8 | openr1_math_220k | 75eea70ed279862dccb49a176a25239685ccb32625398e0a43517099d0af6749 | 24,572 | 20260928_094616 |
(15) 1 Represent the set using the roster method: $A=\left\{x \left\lvert\, x=\frac{a}{|a|}+\frac{|b|}{b}+\frac{|c|}{c}+\frac{a b c}{|a b c|}\right., a, b\right.$, $c \in \mathbf{R}, a b c \neq 0\}$ is . $\qquad$ | math | 53 | 0.75 | openr1_math_220k | a7ec4823e5d1f98e184342d9b7b1c88a5e6f0f056f0d9e47c21d1c461d3ac2d3 | 34,631 | 20260928_094616 |
10 guests came to visit and each left a pair of galoshes in the hallway. All pairs of galoshes are of different sizes. The guests began to leave one by one, putting on any pair of galoshes that they could fit into (i.e., each guest could put on a pair of galoshes not smaller than their own). At some point, it was disco... | math | 115 | 0.75 | openr1_math_220k | 04f2ca06a593750c749c521239de2a4621d75ebadfdd5e68939da631653fab4e | 47,165 | 20260928_094616 |
## Task 1 - 220831 On a day in 1981, Cathrin asks her grandfather about his birth year. The grandfather, a friend of puzzle questions, replied: "I am older than 65 years, but younger than 100 years. The year of my birth is not divisible by 2, 3, or 5. The remainder when this year is divided by 60 is not a prime number.... | math | 137 | 0.85 | openr1_math_220k | bb6b48ee56f312b27025dfe3049e4075bee31804309b23d79c367761c43a2a29 | 29,769 | 20260928_094616 |
2. (China Mathematical Olympiad 1993) Given $k \in \mathbf{N}$ and a real number $a>0$, let $k_{1}, k_{2}, \cdots, k_{r}$ satisfy the following conditions $k_{1}+k_{2}+\cdots+$ $k_{r}=k, k_{i} \in \mathbf{N}, 1 \leqslant r \leqslant k$, find the maximum value of $a^{k_{1}}+a^{k_{2}}+\cdots+a^{k_{r}}$. | math | 75 | 0.75 | openr1_math_220k | cd8f3a7e347570022b0bbf9611022b49b33ffc228a3679eeff8c7f7693bb1734 | 50,038 | 20260928_094616 |
## 1. task At exactly 9 o'clock, Petar lit two candles simultaneously, one 29 cm long and the other 34 cm long. It is known that 2 mm of the shorter candle burns in 4 minutes, and 2 mm of the longer candle in 3 minutes. After some time, Petar noticed that one candle was 2 cm longer than the other. At what time could th... | math | 84 | 0.8 | openr1_math_220k | 2943e678e466899f8110eafd8d2f6a3a61ab6afdfd9ec6f1b45f5e0206b3ea00 | 30,716 | 20260928_094616 |
9,10 Someone has arranged a 10-volume collection of works in a random order. Let's call a disorder a pair of volumes where the volume with the higher number is to the left. For this arrangement of volumes, the number $S$ of all disorders has been calculated. What values can $S$ take? | math | 71 | 0.8 | openr1_math_220k | b7bd6ae9eb94d914b0f7b1b0559608e1fafda64801f38e53645b8957d79f430e | 4,365 | 20260928_094616 |
## Task 2 - 310732 A person answers the question about their birthday: "In the year 1989, I was $a$ years old. I was born on the $t$-th day of the $m$-th month of the year $(1900+j)$. The numbers $a, j, m, t$ are natural numbers; for them, $a \cdot j \cdot m \cdot t=105792.$" Determine whether the numbers $a, j, m, t$ ... | math | 102 | 0.8 | openr1_math_220k | 1a76b1153b2f75191702dc5bd54e661ef3669ce023033fac257a74648e866c37 | 43,523 | 20260928_094616 |
Example 4. Solve the equation $(x+y+1) d x+(2 x+2 y-1) d y=0$. The above text has been translated into English, preserving the original text's line breaks and format. | math | 41 | 0.8 | openr1_math_220k | 928d20c85f9130a1de67f6d15f3b6931fc15506d3f7943f7130a0cbf3a439de0 | 1,633 | 20260928_094616 |
4. Given that the three sides $a$, $b$, and $c$ of $\triangle A B C$ satisfy $\frac{3}{a}=\frac{2}{b}+\frac{1}{c}$. Then $\angle A()$. (A) is acute (B) is right (C) is obtuse (D) is not a right angle | math | 49 | 0.8 | openr1_math_220k | 14884c456a71539f14f9a600375aecd3ef1d3c466664b9730baa2399057e4388 | 473 | 20260928_094616 |
Example 2. Find the solution of the equation $$ y^{\prime \prime}+y=0 $$ satisfying the initial conditions $$ \left.y\right|_{x=0}=1,\left.\quad y^{\prime}\right|_{x=0}=0 $$ | math | 43 | 0.8 | openr1_math_220k | 1395d5d387466e400c491480ef44fc40553ed738598ce0522ab6f50ebc0927d9 | 23,080 | 20260928_094616 |
9. Let $x, y \in \mathbf{R}$ satisfy $$ x-6 \sqrt{y}-4 \sqrt{x-y}+12=0 \text {. } $$ Then the range of values for $x$ is $\qquad$ | math | 32 | 0.8 | openr1_math_220k | 9e4581a7692dd49d0766f392cbbc5d45ef0dde71765be6faa7095d8c92df3baa | 43,230 | 20260928_094616 |
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$? | math | 40 | 0.65 | openr1_math_220k | 6fe5867331d37c4d6fecf232efbc905fc78a73db2bbb6c3206de47686ae425bc | 40,900 | 20260928_094616 |
B2. The Smith family went to a restaurant and bought two Pizzas, three Chillies and four Pastas. They paid $£ 53$ in total. The Patel family went to the same restaurant and bought five of the same Pizzas, six of the same Chillies and seven of the same Pastas. They paid $£ 107$ in total. How much more does a Pizza cost ... | math | 83 | 0.8 | openr1_math_220k | f3f101dd7057e7e76f48c7f7e794297eb82fe303826fd024f6acff26d14a78f8 | 9,832 | 20260928_094616 |
2. In the interval $0 \leq x \leq \pi$ find the solutions to the equation $$ \frac{1}{\sin x}-\frac{1}{\cos x}=2 \sqrt{2} $$ | math | 31 | 0.8 | openr1_math_220k | 5c8f419afb8ee30b007ee1fe89a7f8660bd2e068c603721facb0c8648d08e215 | 57,017 | 20260928_094616 |
Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | math | 59 | 0.8 | openr1_math_220k | f062081d281f606b2c581ad2a3d6c0a88ecbd4af783dc78e1d1729369d8b40e3 | 4,463 | 20260928_094616 |
3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions. | math | 53 | 0.8 | openr1_math_220k | 393f1e927c31ccbba8e94a6a72cd43b797f2c2f13c68e01db8dc85c44014cfd0 | 51,255 | 20260928_094616 |
9. Let $[x]$ be the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1, \quad n \in N^{*}$, find $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$. | math | 63 | 0.75 | openr1_math_220k | 60c79b3b65f35abc5da3934235353e480e8e135d5ccbf4913d02501281c4d92d | 58,964 | 20260928_094616 |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $\textbf{(A)}\ 22 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 119 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 231$ | math | 68 | 0.8 | openr1_math_220k | 9d859576590ef71960d57d1c39d20e8b64927b49dfcb06ddcfe6ccaee20d891b | 44,742 | 20260928_094616 |
1. Which whole numbers from 1 to $4 \cdot 10^{25}$ (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | math | 42 | 0.8 | openr1_math_220k | 4ffdb04cdd8a8af8d89ae282b004e7a0724f8d39274e8554af6b722f845ae7b2 | 24,581 | 20260928_094616 |
Example 1 Find all functions $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$ such that for all $m, n \in \mathbf{Z}_{+}$, we have $$ \begin{array}{l} f(m n)=f(m) f(n), \\ (m+n) \mid(f(m)+f(n)) .{ }^{[1]} \end{array} $$ (2016, Turkey National Team Selection Exam) | math | 65 | 0.75 | openr1_math_220k | 24210508938e4e2c3bfdf9dc3fc59aab493e1ce0af165cccf9c2f0c5eea02186 | 33,416 | 20260928_094616 |
18. Master Li made 8 identical rabbit lanterns in 3 days, making at least 1 lantern each day. Master Li has ( ) different ways to do this. | math | 34 | 0.8 | openr1_math_220k | a655a2333173b11b2a41d5d03e05e98b211b71ac447ce9a8da26e6431824f573 | 19,623 | 20260928_094616 |
77. As shown in the figure, 6 identical squares are embedded in rectangle $ABCD$. Given that $AB=22$ cm, $BC=$ 20 cm, the area of each square is $\qquad$ square cm. | math | 41 | 0.8 | openr1_math_220k | 81bda6106dff1d8eb13a3cad4fae2c9a1da6d44e8397e3ce9e1f5c9782cec1ca | 23,318 | 20260928_094616 |
$6 \cdot 38$ If $y=\log _{5} 6 \cdot \log _{6} 7 \cdot \log _{7} 8 \cdot \log _{8} 9 \cdot \log _{9} 10$, then (A) $y \in(0,1)$. (B) $y=1$. (C) $y \in(1,2)$. (D) $y=2$. (E) $y \in(2,3)$. (Shanghai High School Mathematics Competition, 1982) | math | 59 | 0.65 | openr1_math_220k | 6f889a13529d2d312caf9eed4efe6669e1ce75056b41cfa8c3b9e33b7f74e2ea | 30,991 | 20260928_094616 |
4. The sequence $\left\{x_{n}\right\}$ is defined as follows: $$ x_{1}=\frac{2}{3}, x_{n+1}=\frac{x_{n}}{2(2 n+1) x_{n}+1}\left(n \in \mathbf{Z}_{+}\right) \text {. } $$ Then $x_{1}+x_{2}+\cdots+x_{2014}=$ $\qquad$ | math | 53 | 0.65 | openr1_math_220k | c7b1bbfa21d20e43f17cd80e95d88009ca7501958dd0247eeb85f685faa10091 | 60,799 | 20260928_094616 |
3. The diagonal $AC$ of the inscribed quadrilateral $ABCD$ is the diameter of the circumscribed circle $\omega$ around it. A line perpendicular to the segment $BC$ was drawn from point $D$, intersecting the circle $\omega$ again at point $E$. Find the ratio of the areas of triangle $BCD$ and quadrilateral $ABEC$. | math | 78 | 0.75 | openr1_math_220k | 38ad780e345dccd8393954992b509ce3c4cdf6245b3a1215f64aaf345b664f60 | 28,725 | 20260928_094616 |
3. In the field of real numbers, solve the system of equations $$ x^{2}=\frac{1}{y}+\frac{1}{z}, \quad y^{2}=\frac{1}{z}+\frac{1}{x}, \quad z^{2}=\frac{1}{x}+\frac{1}{y} . $$ | math | 43 | 0.75 | openr1_math_220k | e6a23f539b1ae7ca6ebe1c65d5ae238b953e1ffdd0a43610c882c6e36be92c00 | 57,890 | 20260928_094616 |
3. a) Determine the remainder of the division of a natural number by 42, knowing that when divided by 6 it gives a remainder of 5 and when divided by 7 it gives a remainder of 3. b) Show that the number $21^{33} \cdot 33^{77} \cdot 77^{21}$ is a perfect square. | math | 65 | 0.8 | openr1_math_220k | f4fd5314b5e919d7997f075742afcc328e267717e76d786b936dacf84d718416 | 40,337 | 20260928_094616 |
How many two-digit positive integers have at least one $7$ as a digit? $\mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 30$ | math | 48 | 0.8 | openr1_math_220k | 0c53d04d965a16e2e68376685412913407c688c8a88f757ef5e6c2fc62ee45fc | 39,002 | 20260928_094616 |
Given $f: k \rightarrow R$, for all $x, y \in \mathbf{R}$, it satisfies $$ f\left(x^{2}-y^{2}\right)=x f(x)-y f(y) . $$ Find $f(x)$. | math | 33 | 0.75 | openr1_math_220k | f008e6df3915f104cd42a334b5720950bb07ac97cec42966b6f499400d4da50c | 19,679 | 20260928_094616 |
2. Four of the following points are vertices of the same square. Which point is not a vertex of this square? A $(-1,3)$ B $(0,-4)$ C $(-2,-1)$ $\mathrm{D}(1,1)$ $\mathrm{E}(3,-2)$ | math | 44 | 0.8 | openr1_math_220k | c03fb3e5ffdbe649644c8a41b6dcea814d1579799816333b98ccbb4ab795c567 | 11,827 | 20260928_094616 |
The encryption games involved 168 players in 50 teams, which had two to five members. The most were four-member teams, there were 20 three-member teams, and the games were attended by at least one five-member team. How many two-member, four-member, and five-member teams were there? (M. Mach) | math | 73 | 0.8 | openr1_math_220k | 0cfe094d22d3465b6bfa2e13201ea5fe9609040e9ac50cf3d5fbcc58e16ba618 | 8,088 | 20260928_094616 |
## Problem Statement Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$. $A(-10 ; 0 ; 9)$ $B(12 ; 4 ; 11)$ $C(8 ; 5 ; 15)$ | math | 45 | 0.8 | openr1_math_220k | f91af3675cfd53bd41f44a677f28eecb21d153136080709ab1a240eb330d6d2f | 16,475 | 20260928_094616 |
4. A two-digit number $N$ was multiplied by 2, the digits of the result were swapped, and then the number was divided by 2. The result was the same number $N$. How many such numbers $N$ exist? Answers: A) none (-) B) exactly 4 (-) C) at least 10 (+) D) at least 14 (+) E) at least 15 (-) | math | 71 | 0.8 | openr1_math_220k | fc8a7e449f316d36604840edb9f5e64f9f2c9cd71f2863aff9e84ccfe2d4a632 | 44,881 | 20260928_094616 |
[b]p1.[/b] Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly a) Three other points? b) Four other points? [b]p2.[/b] Martian bank notes can have denomination of $1, 3, 5, 25$ marts. Is it possible to change a note of $25$ marts... | math | 164 | 0.85 | openr1_math_220k | 39b55c009b3aa78dd0803f29b3cbe9646385a126f6a3d5f94d1c3f61d9fe1bf8 | 18,802 | 20260928_094616 |
A computer generates even integers half of the time and another computer generates even integers a third of the time. If $a_i$ and $b_i$ are the integers generated by the computers, respectively, at time $i$, what is the probability that $a_1b_1 +a_2b_2 +\cdots + a_kb_k$ is an even integer. | math | 72 | 0.8 | openr1_math_220k | 7373447c2251b77c02f4452237919171a741de75250fc7ae75f9295568040670 | 12,731 | 20260928_094616 |
1. Find all values of $x$, for each of which one of the three given numbers $\log _{x}\left(x-\frac{5}{2}\right)$, $\log _{x-\frac{5}{2}}(x-4)$, and $\log _{x-4} x$ is equal to the product of the other two. | math | 51 | 0.8 | openr1_math_220k | 694001a3b56d35b8bc06faa4a8ef6218aa645c1865a0cea63d5b3fdf40fcfef4 | 30,287 | 20260928_094616 |
6. If $2n+1, 20n+1 \left(n \in \mathbf{N}_{+}\right)$ are powers of the same positive integer, then all possible values of $n$ are | math | 32 | 0.8 | openr1_math_220k | a78c1b77ed97ebf1cc3f27b5237b5855c89eadd263a0b4c44fc0366c33be5c6f | 49,674 | 20260928_094616 |
2. There are two sequences of numbers arranged according to certain rules: (1) $1,4,7,10, \cdots, 997,1000$ (2) $2,6,10,14, \cdots, 994,998$ There are $\qquad$ numbers that appear in both sequences. | math | 49 | 0.8 | openr1_math_220k | 0405330634c1f9f79c1abdd819e86290d7edf2f3343fca5b9615aad42c1fcfa3 | 16,970 | 20260928_094616 |
Example 4.4.2 Color the 3 vertices of an equilateral triangle with red, blue, and green. How many different schemes are there? If (1) schemes that can be superimposed by rotation are considered the same, (2) schemes that can be superimposed by rotation and reflection are considered the same. | math | 73 | 0.8 | openr1_math_220k | e9254530ecd2e64fc320a19b367d001344e719da010aa06bdd0cbb8ceebb5944 | 41,121 | 20260928_094616 |
16. (6 points) A road is 400 meters long, and on both sides of the road, a trash can is placed every 20 meters, with the start and end points being bus stops where no trash cans are placed. How many trash cans are placed in total? | math | 57 | 0.8 | openr1_math_220k | 57879d9125a19a037daa60b65078be35ab9cd0735a30d034c60275821ade4e02 | 23,220 | 20260928_094616 |
10. (1 mark) How many positive integers less than 500 have exactly 15 positive integer factors? (1 分) 小於 500 的正整數中有多少個剛好有 15 個正因子? | math | 32 | 0.8 | openr1_math_220k | c6f7f39eece7424379650ec5c4ff641fb94e0d4174c05ddad213f65d0283388f | 64,807 | 20260928_094616 |
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $... | math | 127 | 0.8 | openr1_math_220k | 58d3c2a2eb818d8b6ea4e4d00b5a758820043e3ffec24de6d2802e468c1e0c0d | 35,759 | 20260928_094616 |
Example 1 Given real numbers $a, b (a \neq b)$, and they satisfy $$ \begin{array}{l} (a+1)^{2}=3-3(a+1), \\ 3(b+1)=3-(b+1)^{2} . \end{array} $$ Then the value of $b \sqrt{\frac{b}{a}}+a \sqrt{\frac{a}{b}}$ is ( ). (A) 23 (B) -23 (C) -2 (D) -13 | math | 60 | 0.65 | openr1_math_220k | 60cb9272ff4249681bb1dcc1c2a91ae0470992ca76bb7093062455412ff28dd7 | 57,628 | 20260928_094616 |
33. Let $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001, y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}, k=1,2, \cdots, 2001$. Find $\max \sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$ | math | 44 | 0.65 | openr1_math_220k | 7ba481a1b1e4598e3ec689ea46bfe1773e685554ef69fafe2fe7d1679b34a6d7 | 16,734 | 20260928_094616 |
Example 2. Find the area of the smallest circumscribed parallelogram around the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. | math | 33 | 0.8 | openr1_math_220k | f840a6fcad7df4e50e73d102a64a83dbcfdafd112dddd0a9f0ffed9eafd32dc2 | 55,895 | 20260928_094616 |
13.176. The total capacity of three tanks is 1620 liters. Two of them are filled with kerosene, while the third one is empty. To fill it, you need to use either all the contents of the first tank plus $1 / 5$ of the contents of the second, or all the contents of the second plus $1 / 3$ of the contents of the first. Fin... | math | 87 | 0.8 | openr1_math_220k | 819fd3d6ab966ff3cc28a02aca1c95e3f7f2b5b6f467a33f6d99dea6f592f669 | 9,691 | 20260928_094616 |
3. The set of positive odd numbers $\{1,3,5 \cdots\}$ is grouped in ascending order such that the $n$-th group contains $(2 n-1)$ odd numbers: $\{1\}, \quad\{3,5,7\}, \quad\{9,11,13,15,17\}, \cdots$ (First group) (Second group) (Third group) Then 1991 is in the group. | math | 67 | 0.8 | openr1_math_220k | 449d34583fa7c6fe34dff8992b0a5f29c33b1d2444f92b48f58c1daa3ecd5171 | 41,954 | 20260928_094616 |
Three, (25 points) Find the equation of the parabola obtained by rotating the parabola $y=x^{2}-2 x-3$ around point $A(3,0)$ by $90^{\circ}$ (Note: $x$ is a quadratic function of $y$). | math | 46 | 0.8 | openr1_math_220k | f84e6daea5607023cadd6a4e549e60d7f55fa1b00f4fec5844b654f5bba02e0b | 62,119 | 20260928_094616 |
Find all prime numbers $p, q, r$ such that $p$ divides $1+q^{r}$, $q$ divides $1+r^{p}$, and $r$ divides $1+p^{q}$. ## 2 Solutions | math | 32 | 0.75 | openr1_math_220k | 216700b387cbc8714cb712cebe689d14b6d577eff1824e12e24dc6a3965eba56 | 55,022 | 20260928_094616 |
[ Text problems (other).] Kolya and Katya study in the same class. There are twice as many boys as girls in this class. Kolya has 7 more classmates than classmates of the opposite gender. How many female classmates does Katya have? | math | 57 | 0.75 | openr1_math_220k | a922f25c49a802fec61d3658b533a5ac78a1dc99ba7a7b861aa0e346d74b670f | 9,656 | 20260928_094616 |
3. Given $0 \leq a_{k} \leq 1(k=1,2, \ldots, 2020)$, let $a_{2021}=a_{1}, a_{2022}=a_{2}$, then the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-\right.$ $\left.a_{k+1} a_{k+2}\right)$ is $\qquad$ | math | 49 | 0.65 | openr1_math_220k | 9b006bcade407e76c4f71b84c509d9c8db207ad63affd3a92036db1a8c27be56 | 5,466 | 20260928_094616 |
$18 \cdot 59$ Rectangle $A B C D$ has its diagonal $D B$ divided into three segments of length 1 each by two parallel lines $l$ and $l^{\prime}$, where $l$ and $l^{\prime}$ pass through points $A$ and $C$ respectively and are perpendicular to $B D$. The area of $A B C D$ is (保留一位小数) (A) 4.1 . (B) 4.2 . (C) 4.3 . (D) 4.... | math | 83 | 0.8 | openr1_math_220k | 9ce06fd916da9d75833a1f59727dfaebc160e51f17f0894df3247907a6b0e8da | 38,314 | 20260928_094616 |
1. Three non-zero real numbers form an arithmetic progression, and their squares taken in the same order form a geometric progression. Find all possible common ratios of the geometric progression. | math | 49 | 0.75 | openr1_math_220k | 124b1304bebe13480c9125937807bd8667c338526decff3458139cb99fe7e319 | 31,694 | 20260928_094616 |
$32 \cdot 6$ is the sum of all integers between 50 and 350 that end in 1 (A) 5880. (B) 5539. (C) 5208. (D) 4877. (E) 4566. (24th American High School Mathematics Examination, 1973) | math | 45 | 0.8 | openr1_math_220k | de048f03bf50ecc36f72cbf66c86ffb8175e13523f35e5934985dab53e6fc324 | 29,679 | 20260928_094616 |
11. Let the sequence $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $a_{1}=a_{2}=1, a_{3}=2$, and for any natural number $n$, $a_{n} a_{n+1} a_{n+2} \neq 1$, and $a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3}$, then the value of $a_{1}+a_{2}+\cdots+a_{100}$ is $\qquad$ . | math | 71 | 0.65 | openr1_math_220k | b5e2c301307f156984683b2cec3a4b552c44c7768d63fc27918e891550f3e052 | 19,494 | 20260928_094616 |
Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $\frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: $\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad... | math | 103 | 0.8 | openr1_math_220k | 8fcfa4e8d8a662ba95715400cfe7e5a3cf738e70be46eab7854aed7e50e38cf1 | 53,817 | 20260928_094616 |
14. Given that $\tan \alpha+\cot \alpha=4$, find $\sqrt{\sec ^{2} \alpha+\csc ^{2} \alpha-\frac{1}{2} \sec \alpha \csc \alpha}$. | math | 32 | 0.75 | openr1_math_220k | 39ffe8b126546e439300dbc7f37c8a6761131bb045ccc57fc6ab83693bf2dbb6 | 28,635 | 20260928_094616 |
20.3 .13 Let $F$ be the set of all ordered $n$-tuples $\left(A_{1}, A_{2}, \cdots, A_{n}\right)$, where $A_{i}(1 \leqslant i \leqslant n)$ are subsets of $\{1,2, \cdots, 1998\}$. Find $\sum_{\left(A_{1}, A_{2}, \cdots, A_{n}\right) \in F}\left|A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right|$. | math | 73 | 0.65 | openr1_math_220k | 6fad50ff296f307b2a80b0c86e0ce881abd7a19e03e75c09e342635caa7c48a7 | 22,383 | 20260928_094616 |
7. In a convex quadrilateral $ABCD$, the four interior angles satisfy $\angle A<\angle B<\angle C<\angle D$, and $\angle A, \angle B, \angle C, \angle D$ form an arithmetic sequence. Then the range of the common difference $d$ is $\qquad$ | math | 59 | 0.75 | openr1_math_220k | 007d3eca91fbe4c069f96176977492067546d852cddeca720e78ced1850d088a | 24,285 | 20260928_094616 |
5. What is the maximum number of rooks that can be placed on the cells of a $300 \times 300$ board so that each rook attacks no more than one other rook? (A rook attacks all cells it can reach according to chess rules, without passing through other pieces.) # | math | 64 | 0.8 | openr1_math_220k | 5c82c9311da5d0808915e5032013eb3df5edc724789563f52837156f6a15f4d7 | 11,713 | 20260928_094616 |
4. Let's write the natural number $a$ in its canonical form: $$ a=p_{1}^{s_{1}} \cdot p_{2}^{s_{2}} \cdot \ldots \cdot p_{n}^{s_{n}} $$ where $p_{1}, p_{2}, \ldots, p_{n}$ are distinct prime numbers, and $s_{1}, s_{2}, \ldots, s_{n}$ are natural numbers. It is known that the number of natural divisors of $a$, including... | math | 240 | 0.85 | openr1_math_220k | c5b7a959d10b0939ff9262cc2e6f71d6613d19d8ec0d5bce52581beecfe399b5 | 23,639 | 20260928_094616 |
7. Given that $z$ is a complex number, and $|z|=1$. When $\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\qquad$ or . $\qquad$ | math | 41 | 0.8 | openr1_math_220k | 59cfbbbb8a6eab2a95056fdc3effb632e867f07d2d90548cc4b3cf5b2ec724c4 | 23,094 | 20260928_094616 |
1. The number of intersection points between the graphs of the functions $y=\sin x$ and $y=\log _{2021}|x|$ is A. 1284 B. 1285 C. 1286 D. 1287 | math | 35 | 0.8 | openr1_math_220k | 00e24e498f2bf3e836b7123c5ff3eb14359da011dd328afe644eae84ca334ac6 | 40,924 | 20260928_094616 |
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$ | math | 70 | 0.65 | openr1_math_220k | 1d213fbf839ee5546d27053cfcd5384ae0410ac86ff350c7ad3f6365e7ed0eb8 | 35,054 | 20260928_094616 |
Three. (25 points) Find all positive integer triples $(x, y, z)$ such that $1+2^{x} \times 3^{y}=5^{z}$ holds. (Zhang Lei) | math | 30 | 0.8 | openr1_math_220k | 522c2139c7b103ace7d7a6c2b8c9085418167eb765cc7872f509ddc249785586 | 39,679 | 20260928_094616 |
[ Inscribed and Circumscribed Circles ] [ Inscribed Angle Subtended by a Diameter ] A circle with radius 1 is circumscribed around triangle $A P K$. The extension of side $A P$ beyond vertex $P$ intercepts a segment $B K$ from the tangent to the circle at vertex $K$, and $B K$ is equal to 7. Find the area of triangle $... | math | 98 | 0.8 | openr1_math_220k | 3e514e37e90549eb6952790d6fc7779943d5dca67345d95d72f82f92c81ec60b | 37,740 | 20260928_094616 |
5. Given the function $f(x)=x^{2}-x+\sqrt{2 x^{4}-6 x^{2}+8 x+16}$, then the minimum value of $f(x)$ is A. 4 B. $\frac{7}{2}$ C. 3 D. $\frac{5}{2}$ | math | 36 | 0.65 | openr1_math_220k | 00f22958dabfb2002f74b3b47f07a1c2aaf8340197609a67993bb27d2d617592 | 51,172 | 20260928_094616 |
11. (20 points) Given non-zero complex numbers $x, y$ satisfy $y^{2}\left(x^{2}-x y+y^{2}\right)+x^{3}(x-y)=0$. Find the value of $\sum_{m=0}^{29} \sum_{n=0}^{29} x^{18 m n} y^{-18 m n}$. | math | 46 | 0.65 | openr1_math_220k | 666164c15deb03034f11bedf72822c98c8909e66b6d193876d6f954288c1ed1b | 23,965 | 20260928_094616 |
## Problema I - 6 Se considera la inecuación $$ |x-1|<a x $$ donde $a$ es un parámetro real. a) Discutir la inecuación según los valores de $a$. b) Caracterizar los valores de $a$ para los cuales la inecuación tiene exactamente DOS soluciones enteras. | math | 62 | 0.75 | openr1_math_220k | a70120b9e150bd324e177cd109b82ef36ce0f0f235fc7c67c3077cfac15a07bf | 18,591 | 20260928_094616 |
106. Check that in 3-adic arithmetic 201 is a square root of the number ...112101 with an accuracy of three digits. Let us assume we have already found $B_{n}$. We will show how to find $B_{n+1}$ - the square root of $A$ with an accuracy of $n+1$ digits. We seek $B_{n+1}$ in the form $$ B_{n+1}=B_{n}+x \cdot 10^{n} $$ ... | math | 1,339 | 0.95 | openr1_math_220k | 041108483c3e924e0ecb3d72e7152eee329eae7a27acc31d152c413c5c37c0b2 | 15,660 | 20260928_094616 |
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S... | math | 86 | 0.8 | openr1_math_220k | 34cd1e64148dc6093d99c5707fd222217ad310bc25d73a953c19028e7f5b9c8b | 15,753 | 20260928_094616 |
Two numbers are randomly selected from interval $I = [0, 1]$. Given $\alpha \in I$, what is the probability that the smaller of the two numbers does not exceed $\alpha$? Is the answer $(100 \alpha)$%, it just seems too easy. :| | math | 56 | 0.8 | openr1_math_220k | 000799134e669e1ead098272212b07c7a6b2888558d02f4d8ffc36ad058ff217 | 10,565 | 20260928_094616 |
## Task 12/90 Determine all triples $(x, y, z)$ of nonnegative integers $x, y$, and $z$ that satisfy the Diophantine equation $3 x+4 y+5 z=30$ and whose sum $s=x+y+z$ is a prime number, through logical reasoning (thus, a solution approach based on systematic trial and error is excluded!). | math | 72 | 0.8 | openr1_math_220k | 86837cbbf139df18a6558da499babf4958406882ae963a864defdb34fe6079dd | 41,805 | 20260928_094616 |
LIX OM - I - Task 9 Determine the smallest real number a with the following property: For any real numbers $ x, y, z \geqslant a $ satisfying the condition $ x + y + z = 3 $ the inequality holds | math | 48 | 0.8 | openr1_math_220k | 87b16568997796a95f0293b31dee910de426d174a579098060d349edf21becf6 | 7,967 | 20260928_094616 |
B1. Find all pairs of natural numbers $(m, n)$ for which $$ m+3 n-5=2 v-11 d $$ where $v$ is the least common multiple of the numbers $m$ and $n$, and $d$ is the greatest common divisor of the numbers $m$ and $n$. (6 points) | math | 56 | 0.8 | openr1_math_220k | 05d2f6e560a08c02cb3a90b3879a921366649a551314c2c2f1d76ee73bc50881 | 12,256 | 20260928_094616 |
16. Given that the function $f(x)$ is defined on $\mathbf{R}^{+}$, and satisfies the following conditions: A. $f(x)$ is monotonically decreasing on $\mathbf{R}^{+}$, and $f(x)>\frac{1}{x^{2}}$; B. On $\mathbf{R}^{+}$, it always holds that $f^{2}(x) f\left(f(x)-\frac{1}{x^{2}}\right)=f^{3}(1)$. (1) Find the function val... | math | 99 | 0.8 | openr1_math_220k | 56ef34b7c626cd489b9c27f1dd186a9761019ad27825fc7d10a2bc94aa2b15ab | 62,107 | 20260928_094616 |
1. Given a non-constant sequence $\left\{a_{i}\right\}$ satisfies $a_{i+1}^{2}-a_{i} a_{i+1}+a_{i}^{2}=0$, and $a_{i+1} \neq a_{i-1}, i=1,2, \cdots, n$. For a given natural number $n, a_{1}=a_{n+1}=1$, then $\sum_{i=0}^{n-1} a_{i}$ equals A. 2 B. -1 C. 1 D. 0 | math | 64 | 0.65 | openr1_math_220k | 3b3cd80e3b2f6dfee5562df69b602d89bcf9f4e96e407a79b79a7158588fb1b9 | 21,741 | 20260928_094616 |
2. Find all values of $n, n \in N$, for which the sum of the first terms of the sequence $a_{k}=3 k^{2}-3 k+1, \quad k \in N, \quad$ is equal to the sum of the first $n$ terms of the sequence $b_{k}=2 k+89, k \in N$ (12 points) | math | 56 | 0.8 | openr1_math_220k | a4a2c0e0a5bd352afc032ced3bcbbc2bd80580bdd5629966ddbf88e376214639 | 3,539 | 20260928_094616 |
1. How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.) | math | 70 | 0.75 | openr1_math_220k | 98e462e927acb37a76f4a86cf1c70f650dc6492be9e701a287100e7563097480 | 26,211 | 20260928_094616 |
7. When rolling three dice simultaneously, the probability of at least one die showing a 6 is $\qquad$ (the result should be written as a reduced fraction). | math | 39 | 0.8 | openr1_math_220k | 64609dcbf9371f3de2781cae8d6874d17d0bd99dd086ae9901380a071d185ee9 | 58,393 | 20260928_094616 |
3. For a finite graph, the following operation can be performed: choose any cycle of length 4, select any edge in this cycle, and remove it from the graph. For a fixed integer $n(n \geqslant 4)$, if the complete graph with $n$ vertices is repeatedly subjected to the above operation, find the minimum number of edges in ... | math | 86 | 0.75 | openr1_math_220k | 1eef83914bad592f90bb67450bc7832142a0ce99e9e89de2c8be0324856a2707 | 33,159 | 20260928_094616 |
# Problem 4. The integer part $[x]$ of a real number $x$ is defined as the greatest integer $M$ such that $M \leq x$. For example, $[\sqrt{2}]=1,[2]=2,[\pi]=3$. Find all positive real numbers $x$ for which $$ x[x[x[x]]]<2018 $$ | math | 56 | 0.8 | openr1_math_220k | 4a118849fdba07faa977175295677b72aa3dfcf2e5d6f0a5c93cdd12c16a7a3e | 64,752 | 20260928_094616 |
3. If the length of a rectangle is increased by $p \%$, to keep the area of the rectangle unchanged, the width of the rectangle should be reduced by ( ). (A) $p \%$ (B) $(1-p) \%$ (C) $\frac{p}{1+p} \%$ (D) $\frac{100 p}{100+p} \%$ | math | 57 | 0.8 | openr1_math_220k | c317b6be6e7041eea251d43483d3f394c3d9358e68427e59b0479819f7a76eb9 | 34,389 | 20260928_094616 |
II. (25 points) Several boxes are unloaded from a cargo ship, with a total weight of 10 tons, and the weight of each box does not exceed 1 ton. To ensure that these boxes can be transported away in one go, how many trucks with a carrying capacity of 3 tons are needed at least? | math | 69 | 0.8 | openr1_math_220k | a4915e74f26f9c857cc9b4edc3e6ea9e8860b9ce8aba4f46d24107ae27044db9 | 63,755 | 20260928_094616 |
9. How many positive integers $n$ have the property that their representation in base 2 coincides with the base 3 representation of $2 n$? (A) None (B) 1 (C) 2 (D) more than 2, but finite (E) infinite. | math | 50 | 0.8 | openr1_math_220k | d2d46262bec87ec41939f2557b4ab51df4139ec592f360ca8f94ac42cd363027 | 12,261 | 20260928_094616 |
5.3 Given $a b c \neq 0$, and $\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=p$. Then the line $y=p x+p$ must pass through (A) the first and second quadrants. (B) the second and third quadrants. (C) the third and fourth quadrants. (D) the first and fourth quadrants. (China Junior High School Mathematics League, 1998) | math | 78 | 0.8 | openr1_math_220k | 65b4917a5e05cba27c10ded697b50055ad8d0854100fb0b350459043287f2f29 | 19,148 | 20260928_094616 |
Ana, Bento, and Lucas participate in a contest consisting of 20 questions, with the following rules. - Each correct answer is worth 5 points. - Each wrong answer results in a loss of 3 points. - Each unanswered question results in a loss of 2 points. | | correct | wrong | unanswered | | :--- | :---: | :---: | :---: | |... | math | 157 | 0.85 | openr1_math_220k | fe2ea1c5a6353019f032276b79120f5271ebea77c6704126efa0f3fbe919a6f3 | 24,603 | 20260928_094616 |
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