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 | 10. Let $0 . \overline{a_{1} a_{2} a_{3} 1}$ be a four-digit decimal pure fraction, where $a_{i}(i=1,2,3)$ can only take 0 or 2. Let $T$ be the number of all such four-digit fractions, and $S$ be the sum of all such four-digit fractions. | 0.8888 | 85 | 6 |
math | 52. Find the matrix inverse of the matrix
$$
A=\left(\begin{array}{rr}
2 & -1 \\
4 & 3
\end{array}\right)
$$ | (\begin{pmatrix}3/10&1/10\\-2/5&1/5\end{pmatrix}) | 41 | 30 |
math | 612 * Find a polynomial $f(x)$ such that
$$
x^{2}+1 \mid f(x) \text { and } x^{3}+x^{2}+1 \mid f(x)+1 \text {. }
$$ | x^{4}+x^{3}+x-1 | 54 | 13 |
math | ## Zadatak B-4.2.
Pravac $p$ koji sadrži desni fokus hiperbole $4 x^{2}-5 y^{2}=20$ i okomit je na os $x$, siječe hiperbolu u točkama $A$ i $B$. Odredite opseg trokuta čiji su vrhovi $A, B$ i lijevi fokus hiperbole.
| \frac{36\sqrt{5}}{5} | 99 | 13 |
math | 4. In a family, there are six children. Five of them are respectively 2, 6, 8, 12, and 14 years older than the youngest, and the age of each child is a prime number. How old is the youngest? | 5 | 56 | 1 |
math | In each square of an $m \times n$ rectangular board there is a nought or a cross. Let $f(m,n)$ be the number of such arrangements that contain a row or a column consisting of noughts only. Let $g(m,n)$ be the number of arrangements that contain a row consisting of noughts only, or a column consisting of crosses only. W... | f(m,n) < g(m,n) | 96 | 10 |
math | ## Task Condition
Find the derivative of the specified order.
$y=\frac{\ln x}{x^{3}}, y^{IV}=?$ | \frac{-342+360\lnx}{x^{7}} | 30 | 18 |
math | ## Task 5
For the laying of a $6 \mathrm{~km}$ long gas pipeline, 40 days are scheduled. Each day, an equal length of the pipeline is completed.
How many $\mathrm{m}$ are still to be laid after the 30th day. | 1500\mathrm{~} | 61 | 9 |
math | Example 7 The range of the function $y=\frac{x^{2}-1}{x^{2}+2}$ is $\qquad$ . | [-\frac{1}{2},1) | 31 | 10 |
math | 4. Given twelve red points on a circle. Find the minimum value of $n$, such that there exist $n$ triangles with red points as vertices, so that every chord with red points as endpoints is a side of one of these triangles.
（Supplied by Tao Pingsheng） | 24 | 60 | 2 |
math | 10.074. One of the two parallel lines is tangent to a circle of radius $R$ at point $A$, while the other intersects this circle at points $B$ and $C$. Express the area of triangle $A B C$ as a function of the distance $x$ between the lines. | x\sqrt{2Rx-x^2} | 66 | 10 |
math | ## Task 1 - 150921
Klaus has to calculate $4^{2}-3^{2}$ for a homework assignment. He notices that the result, 7, is equal to the sum of the two numbers used, 4 and 3. When he checks his discovery with the numbers 10 and 11, he finds that $11^{2}-10^{2}=21=11+10$ as well.
Determine all pairs $(a, b)$ of natural numbe... | (,b)where1 | 141 | 6 |
math | Problem. For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair $\left(a_{i}, a_{j}\right)$ with $1 \leq i<j \leq n$ is called interesting if there exists a pair $\left(a_{k}, a_{l}\right)$ of integers with $1 \leq k<l \leq n$ such that
$$
\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2
$$
For each $n \geq 3$, find the larg... | \frac{1}{2}(n-1)(n-2)+1 | 162 | 16 |
math | 15. In the complex plane, the three vertices of $\triangle A B C$ correspond to the complex numbers $z_{1}, z_{2}, \dot{z}_{3}$. It is known that $\left|z_{1}\right|=3$, $z_{2}=\bar{z}_{1}, z_{3}=\frac{1}{z_{1}}$, find the maximum value of the area of $\triangle A B C$. | 4 | 95 | 1 |
math | 10th Brazil 1988 Problem 3 Let N be the natural numbers and N' = N ∪ {0}. Find all functions f:N→N' such that f(xy) = f(x) + f(y), f(30) = 0 and f(x) = 0 for all x = 7 mod 10. | f(n)=0foralln | 75 | 6 |
math | 7. The volume of a cube in cubic metres and its surface area in square metres is numerically equal to four-thirds of the sum of the lengths of its edges in metres.
What is the total volume in cubic metres of twenty-seven such cubes? | 216 | 50 | 3 |
math | 9. Find all solutions in positive integers of the diophantine equation $x^{2}+3 y^{2}=z^{2}$. | x=\frac{1}{2}\left(m^{2}-3 n^{2}\right), y=m n, z=\frac{1}{2}\left(m^{2}+3 n^{2}\right) | 30 | 45 |
math | 50 (1178). Simplify the expression
$$
\sin 2 \alpha+\sin 4 \alpha+\ldots+\sin 2 n \alpha
$$ | \frac{\sin((n+1)\alpha)\sin(n\alpha)}{\sin\alpha} | 39 | 21 |
math | ## Task B-3.2.
For the angles $\alpha, \beta$ and $\gamma$ of the acute-angled triangle $ABC$, it holds that $\operatorname{tg} \alpha: \operatorname{tg} \beta: \operatorname{tg} \gamma=1: 3: 12$. Calculate
$$
\sin \alpha \cdot \sin \beta \cdot \sin \gamma
$$ | \frac{32}{65} | 91 | 9 |
math | 2. In a planar quadrilateral $ABCD$, $AB=\sqrt{3}, AD=DC=CB=1$, the areas of $\triangle ABD$ and $\triangle BCD$ are $S$ and $T$ respectively, then the maximum value of $S^{2}+T^{2}$ is $\qquad$ . | \frac{7}{8} | 72 | 7 |
math | [ higher degree equations (other).]
Solve the equation $x^{3}-x-\frac{2}{3 \sqrt{3}}=0$. How many real roots does it have?
# | x_{1}=\frac{2}{\sqrt{3}},x_{2,3}=-\frac{1}{\sqrt{3}} | 41 | 31 |
math | [ Percentage and ratio problems ]
One time, a fisherman cast a net into a pond and caught 30 fish. Marking each fish, he released them back into the pond. The next day, the fisherman cast the net again and caught 40 fish, two of which were marked. How can we use this data to approximately calculate the number of fish ... | 600 | 80 | 3 |
math | 7 . In $\triangle \mathbf{A B C}$, the sides opposite to angles $\mathbf{A} 、 \mathbf{B} 、 \mathbf{C}$ are $a 、 \mathbf{b} 、 \mathbf{c}$ respectively. If $a^{2}+b^{2}=2019 c^{2}$, then $\frac{\cot C}{\cot A+\cot B}=$ | 1009 | 97 | 4 |
math | 3. The altitudes of an acute-angled triangle $A B C$ intersect at point $O$. Find the angle $A C B$, if $A B: O C=5$. | \operatorname{arctg}5 | 40 | 9 |
math | How many ways are there to color the five vertices of a regular 17-gon either red or blue, such that no two adjacent vertices of the polygon have the same color? | 0 | 37 | 1 |
math | 7. If the ellipse $x^{2}+k y^{2}=1$ and the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$ have the same directrix, then $k$ equals $\qquad$ | \frac{16}{7} | 59 | 8 |
math | $8 \cdot 21$ Find the integer $k$ such that $36+k, 300+k, 596+k$ are the squares of three consecutive terms in an arithmetic sequence. | 925 | 45 | 3 |
math | Example 1 Find all continuous functions $f(x)$ that satisfy the condition for any $x, y$ in $\mathbf{R}$
$$
f(x+y)=f(x)+f(y)
$$ | f(x)= | 43 | 3 |
math | 8.202. $2 \cos 13 x+3 \cos 3 x+3 \cos 5 x-8 \cos x \cos ^{3} 4 x=0$. | \frac{\pik}{12},k\in\mathbb{Z} | 45 | 18 |
math | 【Question 4】
The upper base, height, and lower base of a trapezoid form an arithmetic sequence, where the height is 12. What is the area of the trapezoid? $\qquad$.
| 144 | 50 | 3 |
math | Three. (Full marks 25 points) If the system of inequalities
$$
\left\{\begin{array}{l}
x^{2}-x-2>0, \\
2 x^{2}+(5+2 k) x+5 k<0
\end{array}\right.
$$
has only the integer solution $x=-2$, find the range of the real number $k$. | -3 \leqslant k < 2 | 86 | 11 |
math | Let $t$ be TNYWR.
Alida, Bono, and Cate each have some jelly beans.
The number of jelly beans that Alida and Bono have combined is $6 t+3$.
The number of jelly beans that Alida and Cate have combined is $4 t+5$.
The number of jelly beans that Bono and Cate have combined is $6 t$.
How many jelly beans does Bono hav... | 15 | 95 | 2 |
math | Eight children divided 32 peaches among themselves as follows. Anya received 1 peach, Katya - 2, Liza - 3, and Dasha - 4. Kolya Ivanov took as many peaches as his sister, Petya Grishin got twice as many peaches as his sister, Tolya Andreev - three times as many as his sister, and finally, Vasya Sergeev received four ti... | KolyaIvanov'sisterreceived3peaches,PetyaGrishin'sisterreceived4,TolyaAndreev'sisterreceived1peach,VasyaSergeev'sisterreceived2 | 113 | 44 |
math | Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$. | f(x) = cx | 72 | 6 |
math | 10. If for all positive real numbers $a, b, c, d$, the inequality $\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geqslant \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}$ always holds, find all real numbers $x$ that satisfy the condition. | \frac{15}{8} | 98 | 8 |
math | ## Task 11/66
The five numbers of a lottery draw (1 to 90) are sought, about which the following is stated:
a) All digits from 1 to 9 appear exactly once.
b) Only the three middle numbers are even.
c) The smallest number has a common divisor (different from itself) with the largest number.
d) The cross sum of one ... | 9,12,34,68,75 | 114 | 13 |
math | 477*. What is the greatest common divisor of all numbers $7^{n+2}+8^{2 n+1}(n \in N)$? | 57 | 34 | 2 |
math | ## Task B-2.4.
Upon arriving home, Professor Matko realized he had left his umbrella at one of the four places he visited that day: the bank, the post office, the pharmacy, and the store. The pharmacist noticed the umbrella in the pharmacy and, knowing the professor, realized that he would immediately set out to look ... | 16 | 108 | 2 |
math | 5. Let $f(x)=x^{2}-2 a x+2$, when $x \in[-1,+\infty)$, the inequality $f(x) \geqslant a$ always holds, the range of real number $a$ is $\qquad$ . | -3\leqslant\leqslant1 | 60 | 13 |
math | 8. There are 10 cards, each card has two different numbers from $1,2,3,4,5$, and no two cards have the same pair of numbers. These 10 cards are to be placed into five boxes labeled $1,2,3,4,5$, with the rule that a card with numbers $i, j$ can only be placed in box $i$ or box $j$. A placement is called "good" if the nu... | 120 | 129 | 3 |
math | 8. Task by L. Carroll. Couriers from places $A$ and $B$ move towards each other uniformly, but at different speeds. After meeting, one needed another 16 hours, and the other needed another 9 hours to reach their destination. How much time does it take for each to travel the entire distance from $A$ to $B$? | 28 | 76 | 2 |
math | 1. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal, and among all sets with this property, the sum of $S$ is the largest. Find the sum of the set $S$.
(4th American Invitational Mathem... | 61 | 87 | 2 |
math | 5.1. Find the smallest natural number $n$ such that the natural number $n^{2}+14 n+13$ is divisible by 68. | 21 | 37 | 2 |
math | 3. If $\left(x^{2}-x-2\right)^{3}=a_{0}+a_{1} x+\cdots+a_{6} x^{6}$, then $a_{1}+a_{3}+a_{5}=$ | -4 | 57 | 2 |
math | 22% ** Let $0<\theta<\pi$. Then the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$ is | \frac{4\sqrt{3}}{9} | 35 | 12 |
math | 7.1 Rays OA and OB form a right angle. Seventh-grader Petya drew rays OC and OD inside this angle, forming an angle of $10^{\circ}$, and then calculated all the acute angles between any pairs of the drawn rays (not only adjacent ones). It turned out that the sum of the largest and the smallest of the found angles is $8... | 65,15,10 | 104 | 8 |
math | $A B C$ triangle, perpendiculars are raised at $A$ and $B$ on side $AB$, which intersect $BC$ and $AC$ at $B_{1}$ and $A_{1}$, respectively. What is the area of the triangle if $AB=c$, $AB_{1}=m$, and $BA_{1}=n$? | \frac{mnc}{2(+n)} | 76 | 10 |
math | Example 4.19. Find the general solution of the equation
$$
y^{\prime \prime}+6 y^{\prime}+25 y=0 \text {. }
$$ | e^{-3x}(C_{1}\cos4x+C_{2}\sin4x) | 42 | 20 |
math | 8. Let $a_{1}=1, a_{2}=2$, for $n \geqslant 2$ we have
$$
a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1} .
$$
If for all positive integers $n \geqslant m$, we have $a_{n}>2+$ $\frac{2008}{2009}$, then the smallest positive integer $m$ is $\qquad$ . | 4019 | 122 | 4 |
math | Point $D$ lies on side $\overline{BC}$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC.$ The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ and $CA=6,$ the area of $\triangle AEF$ can be writ... | 36 | 148 | 2 |
math | For the New Year's celebration in 2014, Hongqi Mall is hosting a shopping and voucher return event. For every 200 yuan spent, a customer can participate in one round of a lottery. The lottery rules are as follows: take out the 13 hearts cards $(A, 2, 3, \cdots, Q, K)$ from a deck of cards, shuffle them, and randomly dr... | \frac{365}{2014} | 220 | 12 |
math | Three. (50 points) Find all positive integers $n$, for which there exists a permutation of 1, 2, $\cdots, n$ such that $\left|a_{i}-i\right|$ are all distinct.
Find all positive integers $n$, for which there exists a permutation of 1, 2, $\cdots, n$ such that $\left|a_{i}-i\right|$ are all distinct. | 4kor4k+1 | 93 | 6 |
math | Problem 4. Let $n \geq 2$ be a natural number. Determine the set of values that the sum
$$
S=\left[x_{2}-x_{1}\right]+\left[x_{3}-x_{2}\right]+\ldots+\left[x_{n}-x_{n-1}\right]
$$
can take, where $x_{1}, x_{2}, \ldots, x_{n}$ are real numbers with integer parts $1,2, \ldots, n$. | {0,1,2,\ldots,n-1} | 109 | 13 |
math | How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$)? | 16 | 41 | 2 |
math | Example 1 Convert the rational number $\frac{67}{29}$ to a continued fraction. | [2,3,4,2] | 21 | 9 |
math | 143. For which $n \in \boldsymbol{N}$ does the equality
$$
\sqrt[n]{17 \sqrt{5}+38}+\sqrt[n]{17 \sqrt{5}-38}=\sqrt{20} ?
$$ | 3 | 59 | 1 |
math | ## Task 4 - 231224
Determine all natural numbers $n$ for which the number $2^{n}+5$ is a perfect square. | 2 | 38 | 1 |
math | Subject III
A natural number of the form $\overline{a b c d}$ is called superb if $4 \cdot \overline{a b}=\overline{c d}$.
a) How many superb numbers exist?
b) Show that any superb number is divisible by 13. | 15 | 63 | 2 |
math | Example 7. Find the Wronskian determinant for the functions $y_{1}(x)=e^{k_{1} x}$, $y_{2}(x)=e^{k_{2} x}, y_{3}(x)=e^{k_{3} x}$.
1) The linear dependence of the functions $\sin x, \sin \left(x+\frac{\pi}{8}\right), \sin \left(x-\frac{\pi}{8}\right)$ can be established by noting that $\sin \left(x+\frac{\pi}{8}\right)... | e^{(k_{1}+k_{2}+k_{3})x}(k_{2}-k_{1})(k_{3}-k_{1})(k_{3}-k_{2}) | 196 | 43 |
math | Task 1 - 110831 A vessel (without a drain) with a capacity of 1000 liters was initially filled with exactly 30 liters of water per second at a constant flow rate, and from a later time $t$ onwards, exactly 15 liters of water per second. After exactly $40 \mathrm{~s}$, measured from the beginning, the vessel was full.
... | \frac{4}{5} | 106 | 7 |
math | 2. Find the area of a right triangle, one of whose legs is 6, and the hypotenuse is 10. | 24 | 28 | 2 |
math | Find all integer pairs $(x, y)$ for which
$$
(x+2)^{4}-x^{4}=y^{3}
$$ | (x,y)=(-1,0) | 30 | 8 |
math | 1. In $\triangle A B C$,
$$
\begin{array}{l}
z=\frac{\sqrt{65}}{5} \sin \frac{A+B}{2}+\mathrm{i} \cos \frac{A-B}{2}, \\
|z|=\frac{3 \sqrt{5}}{5} .
\end{array}
$$
Then the maximum value of $\angle C$ is | \pi-\arctan\frac{12}{5} | 89 | 14 |
math | 8.386. Given $(1+\operatorname{tg} x)(1+\operatorname{tg} y)=2$. Find $x+y$. | x+\frac{\pi}{4}(4k+1),k\inZ | 33 | 17 |
math | 106. Prime factorization. Find the prime divisors of the number 1000027. | 103\cdot7\cdot73\cdot19 | 25 | 14 |
math | 8. Given that $S$ is an infinite subset of the set of positive integers, satisfying that for any $a, b, c \in S, a b c \in S$, the elements of $S$ are arranged in ascending order to form the sequence $\left\{a_{n}\right\}$, and it is known that $a_{1}=2, a_{2031}=2^{4061}$, then $a_{2017}=$ $\qquad$ . | 2^{4033} | 108 | 7 |
math | 2. If three distinct real numbers $a$, $b$, and $c$ satisfy
$$
a^{3}+b^{3}+c^{3}=3 a b c \text {, }
$$
then $a+b+c=$ $\qquad$ | 0 | 56 | 1 |
math | 1. Let $n$ be a two-digit number such that the square of the sum of digits of $n$ is equal to the sum of digits of $n^{2}$. Find the sum of all possible values of $n$.
(1 mark)
設 $n$ 為兩位數, 而 $n$ 的數字之和的平方等於 $n^{2}$ 的數字之和 。求 $n$ 所有可能值之和。 | 170 | 102 | 3 |
math | 3. Find the equation of the parabola that is tangent to the $x$-axis and $y$-axis at points $(1,0)$ and $(0,2)$, respectively, and find the axis of symmetry and the coordinates of the vertex of the parabola. | (2 x-y)^{2}-4(2 x+y)+4=0, 10 x-5 y-6=0, \left(\frac{16}{25}, \frac{2}{25}\right) | 60 | 51 |
math | [Inclusion-Exclusion Principle] [Arithmetic. Mental calculation, etc.]
In a group of 50 children, some know all the letters except "r", which they simply skip when writing, while others know all the letters except "k", which they also skip. One day, the teacher asked 10 students to write the word "кот" (cat), 18 other... | 8 | 140 | 1 |
math | 4. Determine all natural numbers of the form $\overline{9 a 6 b 9}$ divisible by 3, where the digits in the tens and thousands places are prime numbers. | 92679,93639,95679,97629,97659 | 39 | 29 |
math | [ Numerical inequalities. Comparing numbers.]
It is known that the values of the expressions $b / a$ and $b / c$ are in the interval $(-0.9, -0.8)$. In which interval do the values of the expression $c / a$ lie? | (\frac{8}{9},\frac{9}{8}) | 61 | 14 |
math | [ Numerical inequalities. Comparing numbers.]
What is greater
a) $2^{300}$ or $3^{200}$ ?
b) $2^{40}$ or $3^{28} ?$
c) $5^{44}$ or $4^{53}$ ? | )3^{200};b)3^{28};)4^{53} | 64 | 20 |
math | 1. At the end of the school year $u$ in a sixth grade, the conduct grades were distributed as follows: $50 \%$ of the students had good conduct, $\frac{1}{5}$ of the students had poor conduct, and the remaining 9 students had exemplary conduct. How many students were in this sixth grade? | 30 | 70 | 2 |
math | 11.2. In the piggy bank, there are 1000 coins of 1 ruble, 2 rubles, and 5 rubles, with a total value of 2000 rubles. How many coins of each denomination are in the piggy bank, given that the number of 1-ruble coins is a prime number. | 1-ruble-3,2-ruble-996,5-ruble-1 | 77 | 19 |
math | Question 4 Given that $x, y, z$ are all positive numbers, and
$$
x y z=\frac{1}{2}, x^{2}+y^{2}+z^{2} \leqslant 2 \text{. }
$$
Then the maximum value of $x^{4}+y^{4}+z^{4}$ is | \frac{3}{2} | 80 | 7 |
math | 8 Given that $f(x)$ is a decreasing function defined on $(0,+\infty)$, if $f\left(2 a^{2}+a+\right.$ 1) $<f\left(3 a^{2}-4 a+1\right)$ holds, then the range of values for $a$ is $\qquad$. | 0<<\frac{1}{3}or1<<5 | 74 | 13 |
math | ## Problem B1
A 98 x 98 chess board has the squares colored alternately black and white in the usual way. A move consists of selecting a rectangular subset of the squares (with boundary parallel to the sides of the board) and changing their color. What is the smallest number of moves required to make all the squares b... | 98 | 71 | 2 |
math | Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer. | 3n^2 | 41 | 6 |
math | 16. Suppose $x$ and $y$ are two real numbers such that $x+y=10$ and $x^{2}+y^{2}=167$. Find the value of $x^{3}+y^{3}$. | 2005 | 54 | 4 |
math | 1. $(1 \mathbf{1}, 7-9)$ A and B are shooting at a shooting range, but they only have one six-chamber 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} | 70 | 8 |
math | ## Task Condition
Find the derivative.
$y=\ln \cos \frac{1}{3}+\frac{\sin ^{2} 23 x}{23 \cos 46 x}$ | \frac{\sin46x}{\cos^{2}46x} | 43 | 17 |
math | 8. $[\mathbf{5}]$ Compute the number of sequences of numbers $a_{1}, a_{2}, \ldots, a_{10}$ such that
I. $a_{i}=0$ or 1 for all $i$
II. $a_{i} \cdot a_{i+1}=0$ for $i=1,2, \ldots, 9$
III. $a_{i} \cdot a_{i+2}=0$ for $i=1,2, \ldots, 8$. | 60 | 119 | 2 |
math | Two sums, each consisting of $n$ addends , are shown below:
$S = 1 + 2 + 3 + 4 + ...$
$T = 100 + 98 + 96 + 94 +...$ .
For what value of $n$ is it true that $S = T$ ? | 67 | 73 | 2 |
math | 4. ( 1 mark) A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get -1 mark. He will get 0 marks for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)... | 495 | 153 | 3 |
math | 3. There are 49 children, each with a number on their chest, ranging from 1 to 49, all different. Please select some of the children and arrange them in a circle so that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? | 18 | 72 | 2 |
math | A $2.9 \, \text{cm}$ diameter measuring cylinder is filled with water to a height of $4 \, \text{cm}$, then dense copper cubes with an edge length of $2 \, \text{cm}$ are placed into the cylinder. What is the maximum number of cubes that the water in the cylinder can submerge? | 5 | 74 | 1 |
math | 7. Let $AB$ be a chord through the center of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and $F$ be a focus. Then the maximum value of the area of $\triangle ABF$ is $bc\left(c^{2}=a^{2}-b^{2}\right)$. | bc | 87 | 1 |
math | Problem 4.6.2 A rectangle with a perimeter of 100 cm is divided into 70 identical smaller rectangles by six vertical and nine horizontal cuts. What is the perimeter of each of them, if the total length of all the cuts is 405 cm $?$ | 13 | 60 | 2 |
math | Example 8. When $\mathrm{n}=1,2,3, \cdots, 1988$, find the sum $\mathrm{s}$ of the lengths of the segments cut off by the x-axis for all functions
$$
y=n(n+1) x^{2}-(2 n+1) x+1
$$ | \frac{1988}{1989} | 71 | 13 |
math | 25. Find the polynomial expression in $Z=x-\frac{1}{x}$ of $x^{5}-\frac{1}{x^{5}}$. | Z^{5}+5Z^{3}+5Z | 34 | 13 |
math | 2. Two bottles of equal volume are filled with a mixture of water and juice. In the first bottle, the ratio of the quantities of water and juice is $2: 1$, and in the second bottle, it is $4: 1$. If we pour the contents of both bottles into a third bottle, what will be the ratio of the quantities of water and juice in ... | 11:4 | 80 | 4 |
math | Example: Given $a_{1}=a_{2}=1, a_{n+1}=a_{n}+a_{n-1}(n \geqslant 2)$, find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n}-(\frac{1-\sqrt{5}}{2})^{n}] | 61 | 46 |
math | II. (40 points) Given a prime number $p$ and a positive integer $n (p \geqslant n \geqslant 3)$, the set $A$ consists of different sequences of length $n$ with elements from the set $\{1,2, \cdots, p\}$ (not all elements in the same sequence are the same). If any two sequences $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$... | p^{n-2} | 191 | 6 |
math | Example 8. Calculate: $\operatorname{tg} 5^{\circ}+\operatorname{ctg} 5^{\circ}-2 \sec 80^{\circ}$. (79 National College Entrance Examination Supplementary Question) | 0 | 52 | 1 |
math | 8. It is known that Team A and Team B each have several people. If 90 people are transferred from Team A to Team B, then the total number of people in Team B will be twice that of Team A; if some people are transferred from Team B to Team A, then the total number of people in Team A will be 6 times that of Team B. Then... | 153 | 91 | 3 |
math | Let $a, b$, and $c$ be integers that satisfy $2a + 3b = 52$, $3b + c = 41$, and $bc = 60$. Find $a + b + c$ | 25 | 52 | 2 |
math | Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer. | 600 | 69 | 3 |
math | Let $N$ be the smallest number that has 378 divisors and is of the form $2^{a} \times 3^{b} \times 5^{c} \times 7^{d}$. What is the value of each of these exponents? | =6,b=5,=2,=2 | 59 | 11 |
math | [ Pascal's Triangle and Newton's Binomial ]
How many rational terms are contained in the expansion of
a) $(\sqrt{2}+\sqrt[4]{3})^{100}$
b) $(\sqrt{2}+\sqrt[8]{3})^{300}$? | 26 | 63 | 2 |
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