task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | 35th IMO 1994 shortlist Problem A4 h and k are reals. Find all real-valued functions f defined on the positive reals such that f(x) f(y) = y h f(x/2) + x k f(y/2) for all x, y. Solution | f(x)=0orf(x)=2(\frac{x}{2})^ | 65 | 15 |
math | Pedrito's lucky number is $34117$. His friend Ramanujan points out that $34117 = 166^2 + 81^2 = 159^2 + 94^2$ and $166-159 = 7$, $94- 81 = 13$. Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of po... | 545 | 176 | 3 |
math | A and B are shooting at a shooting range, but they only have one six-chambered revolver with one bullet. Therefore, they agreed to take turns randomly spinning the cylinder and shooting. A starts. Find the probability that the shot will occur when the revolver is with A. | \frac{6}{11} | 56 | 8 |
math | For a positive integer $n$, let $\Omega(n)$ denote the number of prime factors of $n$, counting multiplicity. Let $f_1(n)$ and $f_3(n)$ denote the sum of positive divisors $d|n$ where $\Omega(d)\equiv 1\pmod 4$ and $\Omega(d)\equiv 3\pmod 4$, respectively. For example, $f_1(72) = 72 + 2 + 3 = 77$ and $f_3(72) = 8+12+18... | \frac{1}{10}(6^{2021} - 3^{2021} - 2^{2021} - 1) | 158 | 37 |
math | Example 3 Find all pairs of positive integers $(m, n)$ such that
$$n^{5}+n^{4}=7^{m}-1$$ | (m, n)=(2,2) | 33 | 8 |
math | Example 3. For the equation $t^{2} + z t + z i = 0$ with $z$ and $i$ as coefficients, it always has a real root $\alpha$. Find the locus equation of point $z$.
保持源文本的换行和格式如下：
Example 3. For the equation $t^{2} + z t + z i$ $=0$ with $z$ and $i$ as coefficients, it always has a real root $\alpha$. Find the locus equat... | x^{2}(1-y)=y^{3} | 114 | 11 |
math | 4. Find all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$, such that for all integers $m, n$, we have
$$
\begin{array}{r}
f(f(m)+n)+f(m) \\
=f(n)+f(3 m)+2014 .
\end{array}
$$ | f(n)=2n+1007 | 74 | 10 |
math | What is the largest integer that can be placed in the box so that $\frac{\square}{11}<\frac{2}{3}$ ?
## | 7 | 31 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{2+4+\ldots+2 n}{n+3}-n\right)$ | -2 | 45 | 2 |
math | 1. In the acute triangle $\triangle ABC$, the altitude $CE$ from $AB$ intersects with the altitude $BD$ from $AC$ at point $H$. The circle with diameter $DE$ intersects $AB$ and $AC$ at points $F$ and $G$, respectively. $FG$ intersects $AH$ at point $K$. Given $BC=25$, $BD=20$, $BE=7$, find the length of $AK$. | \frac{216}{25} | 98 | 10 |
math | Given are positive reals $x_1, x_2,..., x_n$ such that $\sum\frac {1}{1+x_i^2}=1$. Find the minimal value of the expression $\frac{\sum x_i}{\sum \frac{1}{x_i}}$ and find when it is achieved. | n-1 | 68 | 3 |
math | Example 4 Given an integer $n \geqslant 2$. Find the largest constant $\lambda(n)$ such that if the real sequence $a_{0}, a_{1}, \cdots, a_{n}$ satisfies:
$$
\begin{array}{l}
0=a_{0} \leqslant a_{1} \leqslant \cdots \leqslant a_{n}, \\
2 a_{i} \geqslant a_{i+1}+a_{i-1}, \\
\text { then }\left(\sum_{i=1}^{n} i a_{i}\rig... | \lambda(n)=\frac{n(n+1)^2}{4} | 173 | 15 |
math | 19. The function $f(x)$ is defined as $f(x)=\frac{x-1}{x+1}$.
The equation $f\left(x^{2}\right) \times f(x)=0.72$ has two solutions $a$ and $b$, where $a>b$. What is the value of $19 a+7 b$ ? | 134 | 78 | 3 |
math | 1. How many strikes do the clocks make in a day if they strike once every half hour, and at each hour $1,2,3 \ldots 12$ times? | 180 | 39 | 3 |
math | 3. Find all triples of pairwise distinct real numbers $x, y, z$ that are solutions to the system of equations:
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=-x+3 y+z \\
y^{2}+z^{2}=x+3 y-z \\
z^{2}+x^{2}=2 x+2 y-z
\end{array}\right.
$$ | 0,1,-2 | 93 | 5 |
math | 1. Person A and Person B are standing by the railway waiting for a train. It is known that the train is moving at a constant speed. At a certain moment, when the front of the train passes them, A starts walking in the same direction as the train at a constant speed, while B walks in the opposite direction at the same s... | 180 | 114 | 3 |
math | 421 ** Find the maximum value of the function $y=\frac{(x-1)^{5}}{(10 x-6)^{9}}$ for $x>1$.
| \frac{1}{2^{5}\cdot9^{9}} | 41 | 14 |
math | For how many positive integers $n$, $1\leq n\leq 2008$, can the set \[\{1,2,3,\ldots,4n\}\] be divided into $n$ disjoint $4$-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset? | 1004 | 82 | 4 |
math | 2. If real numbers $x, y$ satisfy $|x|+|y| \leqslant 1$, then the maximum value of $x^{2}-$ $xy+y^{2}$ is $\qquad$ | 1 | 49 | 1 |
math | 2. Let $M$ be a moving point on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$. Given points $F(1,0)$ and $P(3,1)$. Then the maximum value of $2|M F|-|M P|$ is $\qquad$. | 1 | 71 | 1 |
math | Problem 3. Several (more than one) consecutive natural numbers are written on the board, the sum of which is 2016. What can the smallest of these numbers be? | 1,86,220,285,671 | 39 | 16 |
math | 3. (3 points) Find all solutions of the inequality $\cos 5+2 x+x^{2}<0$, lying in the interval $\left[-2 ;-\frac{37}{125}\right]$. | x\in(-1-\sqrt{1-\cos5};-\frac{37}{125}] | 47 | 23 |
math | Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum po... | 5 \text{ consecutive primes} | 107 | 7 |
math | A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12\%$ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues? | 450 | 59 | 3 |
math | 3. For any pair of numbers, a certain operation «*» is defined, satisfying the following properties: $a *(b * c)=(a * b) \cdot c$ and $a * a=1$, where the operation «$\cdot$» is the multiplication operation. Find the root $x$ of the equation: $\quad x * 2=2018$. | 4036 | 80 | 4 |
math | Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$ | \frac{\sqrt{5}}{125} | 53 | 12 |
math | 2.1. Sasha solved the quadratic equation $3 x^{2}+b x+c=0$ (where $b$ and $c$ are some real numbers). In his answer, he got exactly one root: $x=-4$. Find $b$. | 24 | 55 | 2 |
math | \section*{Problem 2 - 081212}
a) On the sides \(AB, BC\), and \(CA\) of the triangle \(\triangle ABC\), there are points \(A_{1}, A_{2}, A_{3}\) respectively, \(B_{1}, B_{2}, B_{3}, B_{4}\) respectively, and \(C_{1}, C_{2}, C_{3}, C_{4}, C_{5}\) respectively, which are different from the vertices and pairwise differen... | 390 | 241 | 3 |
math | 25. Between 1 and 8000 inclusive, find the number of integers which are divisible by neither 14 nor 21 but divisible by either 4 or 6 . | 2287 | 41 | 4 |
math | 6.005. $\frac{1}{x(x+2)}-\frac{1}{(x+1)^{2}}=\frac{1}{12}$. | x_{1,2}\in\varnothing,x_{3}=-3,x_{4}=1 | 38 | 21 |
math | Example 15 (2000 National High School Competition Question) There are $n$ people, and it is known that any 2 of them make at most one phone call. The total number of calls made among any $n-2$ of them is equal, and is equal to $3^{k}$ ($k$ is a positive integer). Find all possible values of $n$.
---
The above text is... | 5 | 104 | 1 |
math | 2. Determine the values of $x \in Z$ for which $\sqrt[3]{x^{3}-6 x^{2}+12 x+29} \in Q$. | -2or5 | 40 | 4 |
math | 3. Find the last non-zero digit in 30 !.
(For example, $5!=120$; the last non-zero digit is 2 .) | 8 | 36 | 1 |
math | ## Aufgabe 2 - 260812
Uwe möchte mit einem Taschenrechner feststellen, ob 37 ein Teiler von 45679091 ist. Wenn er dabei den Rechner SR1 verwendet, könnte er folgendermaßen vorgehen: Er dividiert 45679091 durch 37. Der Rechner SR1 zeigt 1234570 an, also ein ganzzahliges Ergebnis. Zur Kontrolle multipliziert Uwe dieses ... | 45679091isnotdivisible37,butleavesremainderof1 | 197 | 20 |
math | Example 7 Given that $a, b, c, d$ take certain real values, the equation $x^{4}+a x^{3}+b x^{2}+c x+d=0$ has 4 non-real roots, where the product of 2 of the roots is $13+i$, and the sum of the other 2 roots is $3+4i$, where $i$ is the imaginary unit. Find $b$.
(13th American Invitational Mathematics Examination) | 51 | 107 | 2 |
math | It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$. | 1 | 104 | 1 |
math | 1. Each Kinder Surprise contains exactly 3 different gnomes, and there are 12 different types of gnomes in total. In the box, there are enough Kinder Surprises, and in any two of them, the triplets of gnomes are not the same. What is the minimum number of Kinder Surprises that need to be bought to ensure that after the... | 166 | 96 | 3 |
math | # Task 4.
## Maximum 10 points.
Calculate using trigonometric transformations
$$
\sin \frac{\pi}{22} \cdot \sin \frac{3 \pi}{22} \cdot \sin \frac{5 \pi}{22} \cdot \sin \frac{7 \pi}{22} \cdot \sin \frac{9 \pi}{22}
$$
# | \frac{1}{32} | 90 | 8 |
math | A list of integers consists of $(m+1)$ ones, $(m+2)$ twos, $(m+3)$ threes, $(m+4)$ fours, and $(m+5)$ fives. The average (mean) of the list of integers is $\frac{19}{6}$. What is $m$ ? | 9 | 71 | 1 |
math | Example 25 (28th IMO Candidate Problem) Given $x=-2272, y=10^{3}+10^{2} c+10 b+a, z=1$ satisfy the equation $a x+b y+c z=1$, where $a, b, c$ are positive integers, $a<b<c$, find $y$.
---
Here is the translation, maintaining the original text's line breaks and format. | 1987 | 96 | 4 |
math | [ Lines and Planes in Space (Miscellaneous).]
The base of the pyramid $S A B C D$ is an isosceles trapezoid $A B C D$, where $A B=B C=a, A D=2 a$. The planes of the faces $S A B$ and $S C D$ are perpendicular to the plane of the base of the pyramid. Find the height of the pyramid if the height of the face $S A D$, dra... | a | 111 | 1 |
math | ## Task B-1.3.
Marko spent 100 euros after receiving his salary. Five days later, he won $\frac{1}{4}$ of the remaining amount from his salary in a lottery and spent another 100 euros. After fifteen days, he received $\frac{1}{4}$ of the amount he had at that time and spent another 100 euros. In the end, he had 800 eu... | 2100 | 113 | 4 |
math | 340. Solve the systems of equations:
a) $\left\{\begin{array}{l}x+y+x y=5, \\ x y(x+y)=6 ;\end{array}\right.$
b) $\left\{\begin{array}{l}x^{3}+y^{3}+2 x y=4, \\ x^{2}-x y+y^{2}=1 .\end{array}\right.$ | (2;1),(1;2);(1;1) | 93 | 14 |
math | How many experiments do we need to perform at least so that the probability of rolling with 3 dice
a) once a 15,
b) once at least a 15,
is greater than $1 / 2$ | 15 | 48 | 2 |
math | 1. Let the set
$$
A=\left\{5, \log _{2}(a+3)\right\}, B=\{a, b\}(a, b \in \mathbf{R}) \text {. }
$$
If $A \cap B=\{1\}$, then $A \cup B=$ $\qquad$ | \{5,1,-1\} | 76 | 9 |
math | Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum? | 10 | 73 | 2 |
math | 3. (CZS) A tetrahedron $A B C D$ is given. The lengths of the edges $A B$ and $C D$ are $a$ and $b$, respectively, the distance between the lines $A B$ and $C D$ is $d$, and the angle between them is equal to $w$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $A B$ and $C D$. Calcul... | \frac{k^{3}+3k^{2}}{3k+1} | 140 | 18 |
math | 7. The range of the function $f(x)=x(\sqrt{1+x}+\sqrt{1-x})$ is $\qquad$ | \left[-\frac{8 \sqrt{3}}{9}, \frac{8 \sqrt{3}}{9}\right] | 30 | 29 |
math | 13. (25 points) Given that $a$, $b$, and $c$ are the sides opposite to the internal angles $\angle A$, $\angle B$, and $\angle C$ of $\triangle ABC$, respectively, and
\[ b \cos C + \sqrt{3} b \sin C - a - c = 0 \].
(1) Prove that $\angle A$, $\angle B$, and $\angle C$ form an arithmetic sequence;
(2) If $b = \sqrt{3}$... | 2 \sqrt{7} | 122 | 6 |
math | Agakhanovo $H . X$.
Different numbers $a, b$ and $c$ are such that the equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root. In addition, the equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$ have a common real root. Find the sum $a+b+c$. | -3 | 101 | 2 |
math | ## Task A-3.7. (10 points)
Find all two-digit natural numbers $a$ for which the equation
$$
2^{x+y}=2^{x}+2^{y}+a
$$
has a solution $(x, y)$ in natural numbers. | 14,20,30,44,48,62,92 | 60 | 20 |
math | Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let
$$F(n)=\sum_{d|n} f(d).$$
Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$ | F(n) = 1 | 87 | 7 |
math | 1. For numbering the pages of a dictionary, 6869 digits are used. How many pages does the dictionary have? | 1994 | 27 | 4 |
math | 2. If $f\left(\frac{1}{1-x}\right)=\frac{1}{x} f(x)+2$, then $f(3)=$ | -\frac{1}{2} | 36 | 7 |
math | For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i] | 511 | 48 | 3 |
math | Example 4 Find all odd prime numbers $p$ such that
$$
p \mid \sum_{k=1}^{2011} k^{p-1} .
$$ | 3 | 40 | 1 |
math | Question 2. Given non-negative real numbers $a_{1} \geq a_{2} \geq \ldots \geq a_{n}, b_{1} \leq b_{2} \leq \ldots \leq b_{n}$, satisfying
$$
a_{1} a_{n}+a_{2} a_{n-1}+\ldots+a_{n} a_{1}=b_{1} b_{n}+b_{2} b_{n-1}+\ldots+b_{n} b_{1}=1 \text {, }
$$
Find
$$
S=\sum_{1 \leq i<j \leq n} a_{i} b_{j}
$$
the minimum value. | \frac{n-1}{2} | 164 | 8 |
math | 28. Given a package containing 200 red marbles, 300 blue marbles and 400 green marbles. At each occasion, you are allowed to withdraw at most one red marble, at most two blue marbles and a total of at most five marbles out of the package. Find the minimal number of withdrawals required to withdraw all the marbles from ... | 200 | 83 | 3 |
math | [ [motion problem ]
In a cycling competition on a circular track, Vasya, Petya, and Kolya participated, starting simultaneously. Vasya completed each lap two seconds faster than Petya, and Petya completed each lap three seconds faster than Kolya. When Vasya finished the distance, Petya had one lap left to complete, an... | 6 | 97 | 1 |
math | Problem 5. Solve the system of equations
$$
\left\{\begin{array}{rl}
x^{2}+x y+y^{2} & =23 \\
x^{4}+x^{2} y^{2}+y^{4} & =253
\end{array} .\right.
$$ | (-\frac{\sqrt{29}+\sqrt{5}}{2};-\frac{\sqrt{29}-\sqrt{5}}{2}),(-\frac{\sqrt{29}-\sqrt{5}}{2};-\frac{\sqrt{29}+\sqrt{5}}{2}),(\frac{\sqrt{29}-\sqrt{5}}{2};\frac{} | 72 | 85 |
math | Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.
| 1008 | 41 | 4 |
math | 10. Given the circle $C:(x-1)^{2}+(y-2)^{2}=25$ and the line $l:(2 m+1) x+(m+1) y-7 m-4=0$ intersect, then the equation of $l$ when the chord length intercepted by the circle $C$ is the smallest is $\qquad$ . | 2x-y-5=0 | 82 | 7 |
math | 5. If $n$ is a positive integer greater than 1, then
$$
\begin{array}{l}
\cos \frac{2 \pi}{n}+\cos \frac{4 \pi}{n}+\cos \frac{6 \pi}{n}+\cdots+\cos \frac{2 n \pi}{n} \\
=
\end{array}
$$ | 0 | 82 | 1 |
math | Example 7 Let $f(n)=\frac{1}{\sqrt[3]{n^{2}+2 n+1}+\sqrt[3]{n^{2}-1}+\sqrt[3]{n^{2}-2 n+1}}$, find the value of $f(1)+f(3)+f(5)+\cdots$ $+f(999997)+f(999999)$. | 50 | 96 | 2 |
math | 14. Suppose there are 3 distinct green balls, 4 distinct red balls, and 5 distinct blue balls in an urn. The balls are to be grouped into pairs such that the balls in any pair have different colors. How many sets of six pairs can be formed? | 1440 | 57 | 4 |
math | 6.75 Given that $x, y, z$ are positive numbers, and satisfy the equation
$$
x y z(x+y+z)=1 \text{, }
$$
find the minimum value of the expression $(x+y)(y+z)$. | 2 | 54 | 1 |
math | For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$. | 2^{k-1}k + 1 | 54 | 12 |
math | 7. A terrace in the shape of a rectangle, 6 meters long and 225 centimeters wide, needs to be tiled with square stone tiles of the same size. What is the minimum number of tiles needed to tile the terrace?
MINISTRY OF SCIENCE, EDUCATION AND SPORT OF THE REPUBLIC OF CROATIA AGENCY FOR EDUCATION AND UPBRINGING
CROATIAN... | 24 | 104 | 2 |
math | Annie takes a $6$ question test, with each question having two parts each worth $1$ point. On each [b]part[/b], she receives one of nine letter grades $\{\text{A,B,C,D,E,F,G,H,I}\}$ that correspond to a unique numerical score. For each [b]question[/b], she receives the sum of her numerical scores on both parts. She kno... | 11 | 223 | 2 |
math | Example 2. Find $\int x \ln x d x$. | \frac{x^{2}}{2}\lnx-\frac{1}{4}x^{2}+C | 14 | 24 |
math | 11. For the function $f(x)=\sqrt{a x^{2}+b x}$, there exists a positive number $b$, such that the domain and range of $f(x)$ are the same. Then the value of the non-zero real number $a$ is $\qquad$. | -4 | 63 | 2 |
math | 11.16. (England, 66). Find the number of sides of a regular polygon if for four of its consecutive vertices \( A, B, C, D \) the equality
\[
\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A D}
\]
is satisfied. | 7 | 73 | 1 |
math | A1. If $r$ is a number for which $r^{2}-6 r+5=0$, what is the value of $(r-3)^{2}$ ? | 4 | 38 | 1 |
math | Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if ... | a = 8^{1/8} | 180 | 10 |
math | \section*{Problem 4 - 111244}
a) Determine all ordered triples \((x, y, z)\) of real numbers that satisfy the equation \(x^{3} z + x^{2} y + x z + y = x^{5} + x^{3}\).
b) Among the triples found in a), identify those in which exactly one of the three numbers \(x, y, z\) is positive, exactly one is negative, and exact... | (x,0,x^2)wherex<0 | 106 | 12 |
math | \section*{Problem 5 - 181045}
Determine all pairs of natural numbers \((n ; z)\) for which \(2^{n}+12^{2}=z^{2}-3^{2}\) holds! | (4,13) | 54 | 6 |
math | ## Task B-3.2.
Solve the inequality $|\cos x| \leqslant \cos x + 2 \sin x$ on the interval $[0, 2\pi\rangle$. | x\in[0,\frac{3\pi}{4}] | 46 | 14 |
math | Determine all possible values of $\frac{1}{x}+\frac{1}{y}$ if $x$ and $y$ are real numbers (not equal to $0$) that satisfy $x^{3}+y^{3}+3 x^{2} y^{2}=x^{3} y^{3}$. | -2 \text{ and } 1 | 70 | 9 |
math | Solve the following system of equations:
$$
\begin{aligned}
x+y & =2 a \ldots \\
x y\left(x^{2}+y^{2}\right) & =2 b^{4} \ldots
\end{aligned}
$$
What is the condition for obtaining real solutions?
Num. values: $\quad a=10, \quad b^{4}=9375$. | 15,5or5,15 | 89 | 9 |
math | 1. The sequence satisfies $a_{0}=\frac{1}{4}$, and for natural number $n$, $a_{n+1}=a_{n}^{2}+a_{n}$.
Then the integer part of $\sum_{n=0}^{201} \frac{1}{a_{n}+1}$ is $\qquad$.
(2011, National High School Mathematics League Gansu Province Preliminary) | 3 | 98 | 1 |
math | 10. Given that $p(x)$ is a 5th degree polynomial, if $x=0$ is a triple root of $p(x)+1=0$, and $x=1$ is a triple root of $p(x)-1=0$, then the expression for $p(x)$ is | 12x^{5}-30x^{4}+20x^{3}-1 | 64 | 20 |
math | A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$. A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more than $\ma... | 40 | 159 | 2 |
math | If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left \lfloor n \plus{} \sqrt {3n} \plus{} \frac {1}{2} \right \rfloor$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} \left \lfloor \frac {n^2 \plus{} 2n}{3} \right \rfloor$. | f(n) = \left\lfloor n + \sqrt{3n} + \frac{1}{2} \right\rfloor | 93 | 31 |
math | [ Area of a quadrilateral ]
The area of a quadrilateral is 3 cm², and the lengths of its diagonals are 6 cm and 2 cm. Find the angle between the diagonals.
# | 30 | 44 | 2 |
math | 7.3. It is known that all krakozyabrs have horns or wings (possibly both). According to the results of the world census of krakozyabrs, it turned out that $20 \%$ of the krakozyabrs with horns also have wings, and $25 \%$ of the krakozyabrs with wings also have horns. How many krakozyabrs are left in the world, if it i... | 32 | 112 | 2 |
math | Example 3 Given real numbers $x_{1}, x_{2}, \cdots, x_{10}$ satisfy $\sum_{i=1}^{10}\left|x_{i}-1\right| \leqslant 4, \sum_{i=1}^{10}\left|x_{i}-2\right| \leqslant 6$. Find the average value $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$. (2012, Zhejiang Province High School Mathematics Competition) | 1.4 | 124 | 3 |
math | 15. To prevent Xiaoqiang from being addicted to mobile games, Dad set a password on his phone. The phone password is 4 digits long, with each digit being a number between $0 \sim 9$. If the sum of the 4 digits used in the password is 20, Xiaoqiang would need to try at most $\qquad$ times to unlock the phone. | 633 | 82 | 3 |
math | In a city with $n$ inhabitants, clubs are organized in such a way that any two clubs have, and any three clubs do not have, a common member. What is the maximum number of clubs that can be organized this way? | [\frac{1}{2}+\sqrt{2n+\frac{1}{4}}] | 48 | 20 |
math | 4. Let $H$ be the orthocenter of $\triangle A B C$, and $O$ be the circumcenter of $\triangle A B C$. If $|\overrightarrow{H A}+\overrightarrow{H B}+\overrightarrow{H C}|=2$, then $|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}|=$ $\qquad$ . | 1 | 88 | 1 |
math | 8,9 On the hypotenuse $BC$ of the right triangle $ABC$, points $D$ and $E$ are marked such that $AD \perp BC$ and $AD = DE$. On the side $AC$, a point $F$ is marked such that $EF \perp BC$. Find the angle $ABF$.} | 45 | 74 | 2 |
math | 20. How many pairs of natural numbers ( $m, n$ ) exist such that $m, n \leqslant 1000$ and
$$
\frac{m}{n+1}<\sqrt{2}<\frac{m+1}{n} ?
$$
## 9th grade | 1706 | 69 | 4 |
math | 3. If the real number $x$ satisfies $\sin \left(x+20^{\circ}\right)=\cos \left(x+10^{\circ}\right)+\cos \left(x-10^{\circ}\right)$, then $\tan x=$ | \sqrt{3} | 58 | 5 |
math | 61.
A few days later, while wandering through the forest, Alice came across one of the brothers sitting under a tree. She asked him the same question and heard in response: “The true owner of the rattle speaks the truth today.”
Alice paused to think. She wanted to assess how likely it was that the brother who said t... | \frac{13}{14} | 125 | 9 |
math | 13. Let $a \in \mathbf{R}$, and the function $f(x)=a x^{2}+x-a(|x| \leqslant 1)$.
(1) If $|a| \leqslant 1$, prove: $|f(x)| \leqslant \frac{5}{4}$;
(2) Find the value of $a$ that makes the function $f(x)$ have a maximum value of $\frac{17}{8}$. | -2 | 110 | 2 |
math | 8. It is known that the equation $x^{3}+7 x^{2}+14 x-p=0$ has three distinct roots, and these roots form a geometric progression. Find p and solve this equation. | -8;rootsoftheequation:-1,-2,-4 | 47 | 15 |
math | 12.432 The side of the lower base of a regular truncated quadrilateral pyramid is 5 times the side of the upper base. The lateral surface area of the pyramid is equal to the square of its height. Find the angle between the lateral edge of the pyramid and the plane of the base. | \operatorname{arctg}\sqrt{9+3\sqrt{10}} | 63 | 19 |
math | 8. For a finite set
$$
A=\left\{a_{i} \mid 1 \leqslant i \leqslant n, i \in \mathbf{Z}_{+}\right\}\left(n \in \mathbf{Z}_{+}\right) \text {, }
$$
let $S=\sum_{i=1}^{n} a_{i}$, then $S$ is called the "sum" of set $A$, denoted as $|A|$. Given the set $P=\{2 n-1 \mid n=1,2, \cdots, 10\}$, all subsets of $P$ containing th... | 3600 | 193 | 4 |
math | 1. If on the interval $[2,3]$, the function $f(x)=x^{2}+b x+c$ and $g(x)=x+\frac{6}{x}$ take the same minimum value at the same point, then the maximum value of the function $f(x)$ on $[2,3]$ is | 15-4 \sqrt{6} | 70 | 9 |
math | 3. An Indian engineer thought of a natural number, listed all its proper natural divisors, and then increased each of the listed numbers by 1. It turned out that the new numbers are all the proper divisors of some other natural number. What number could the engineer have thought of? Provide all options and prove that t... | n=4n=8 | 95 | 6 |
math | $2 \cdot 84$ For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by
$$q_{i}=\left(q_{i-1}-1\right)^{3}+3 \quad(i=1,2, \cdots, n)$$
If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime, find the largest possible value of $n$. | 2 | 114 | 1 |
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