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 | 5. Two right-angled triangles are similar. The larger triangle has short sides which are $1 \mathrm{~cm}$ and $5 \mathrm{~cm}$ longer than the short sides of the smaller triangle. The area of the larger triangle is $8 \mathrm{~cm}^{2}$ more than the area of the smaller triangle. Find all possible values for the side le... | (1.1,5.5)or(2,1) | 89 | 15 |
math | 3. Let the sequence $\left(x_{n}\right)_{n \in \mathbb{N}^{*}}$ be defined such that $x_{1}=1$ and $x_{n+1}=x_{n}+n$, for any $n \in \mathbb{N}^{*}$. Calculate $\lim _{n \rightarrow \infty} \sum_{k=2}^{n} \frac{1}{x_{k}-1}$. | 2 | 103 | 1 |
math | 13. Given that $n$ and $k$ are natural numbers, and satisfy the inequality $\frac{9}{17}<\frac{n}{n+k}<\frac{8}{15}$. If for a given natural number $n$, there is only one natural number $k$ that makes the inequality true, find the maximum and minimum values of the natural number $n$ that meet the condition. | 17, 144 | 86 | 7 |
math | $\begin{array}{l}\quad \text { 3. } \arcsin \frac{1}{\sqrt{10}}+\arccos \frac{7}{\sqrt{50}}+\operatorname{arctg} \frac{7}{31} \\ +\operatorname{arcctg} 10=\end{array}$ | \frac{\pi}{4} | 79 | 7 |
math | 10,11 | |
On the lateral edge of the pyramid, two points are taken, dividing the edge into three equal parts. Planes are drawn through these points, parallel to the base. Find the volume of the part of the pyramid enclosed between these planes, if the volume of the entire pyramid is 1. | \frac{7}{27} | 67 | 8 |
math | 25. Given that $x$ and $y$ are positive integers such that $56 \leq x+y \leq 59$ and $0.9<\frac{x}{y}<0.91$, find the value of $y^{2}-x^{2}$. | 177 | 63 | 3 |
math | 16. The lucky numbers are generated by the following sieving process. Start with the positive integers. Begin the process by crossing out every second integer in the list, starting your count with the integer 1. Other than 1 the smallest integer left is 3 , so we continue by crossing out every third integer left, start... | 1,3,7,9,13,15,21,25,31,33,37,43,49,51,63,67,69,73,75,79,87,93,99 | 156 | 64 |
math | 2. On some planets, there are $2^{N}(N \geqslant 4)$ countries, each with a flag composed of $N$ unit squares forming a rectangle of width $N$ and height 1. Each unit square is either yellow or blue. No two countries have the same flag. If $N$ flags can be arranged to form an $N \times N$ square such that all $N$ unit ... | 2^{N-2}+1 | 141 | 8 |
math | Task 2. Lazarot Petre has 151 liters of wine in a storm. Help Petre, as a friend, to give away 81 liters of wine if he has only 2 barrels that hold 51 and 91 liters. | 81 | 55 | 2 |
math | \section*{Problem 1 - 071241}
Determine all ordered quadruples of real numbers \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\) that satisfy the following system of equations:
\[
\begin{aligned}
& x_{1}+a x_{2}+x_{3}=b \\
& x_{2}+a x_{3}+x_{4}=b \\
& x_{3}+a x_{4}+x_{1}=b \\
& x_{4}+a x_{1}+x_{2}=b
\end{aligned}
\]
Here,... | \begin{aligned}&Case1:(,,b-,b-)\\&Case2.1.1:(\frac{b}{2+},\frac{b}{2+},\frac{b}{2+},\frac{b}{2+})\\&Case2.1.2:(,,, | 158 | 64 |
math | At what angle do the curves $x^{2}+y^{2}=16$ and $y^{2}=6 x$ intersect? What is the area of the region bounded by the two? | 19.06 | 42 | 5 |
math | 9. Let $y=f(x)$ be an odd function on $(-\infty,+\infty)$, $f(x+2)=-f(x)$, and when $-1 \leqslant x \leqslant 1$, $f(x)=x^{3}$.
(1) Find the analytical expression of $f(x)$ when $x \in[1,5]$;
(2) If $A=\{x \mid f(x)>a, x \in \mathbf{R}\}$, and $A \neq \varnothing$, find the range of real number $a$. | <1 | 132 | 2 |
math | A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtai... | 15 | 127 | 4 |
math | ## Task 4 - 270824
A cube $W$ is divided into volume-equal smaller cubes. The surface area of the cube $W$ is $A$, and the sum of the surface areas of the separated smaller cubes is $S$. Determine the ratio $A: S$
(a) if the cube $W$ has an edge length of $14 \mathrm{~cm}$ and the number of smaller cubes is 8,
(b) f... | 1:n | 167 | 2 |
math | 4. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}$ satisfying: $S_{n}+a_{n}=\frac{n-1}{n(n+1)}, n=1,2, \mathrm{~L}$, then the general term $a_{n}=$ $\qquad$ | a_{n}=\frac{1}{2^{n}}-\frac{1}{n(n+1)} | 81 | 23 |
math | Test B Find all solutions in the set of natural numbers for the equation $2^{m}-3^{n}=1$.
(Former Soviet Union Mathematical Competition) | =2,n=1 | 33 | 5 |
math | 7. Given that for any real numbers $x, y$, the function $f(x)$ satisfies $f(x)+f(y)=f(x+y)+xy$.
If $f(1)=m$, then the number of positive integer pairs $(m, n)$ that satisfy $f(n)=2019$ is $\qquad$. | 8 | 70 | 1 |
math | 1. If a two-digit natural number is decreased by 54, the result is a two-digit number with the same digits but in reverse order. In the answer, specify the median of the sequence of all such numbers.
# | 82 | 47 | 2 |
math | 4. Passing through point $A(1,-4)$, and parallel to the line $2 x+3 y+5=0$ the equation of the line is | 2x+3y+10=0 | 35 | 10 |
math | Let's calculate the volume of the tetrahedron that can be circumscribed around four mutually tangent, equal-radius spheres.
(Kövi Imre, teacher at the Fógymnasium, Iglón). | \frac{r^{3}}{3}(2\sqrt{3}+\sqrt{2})^{3} | 45 | 24 |
math | [ [Evenness and Oddness]
All the dominoes were laid out in a chain. At one end, there were 5 dots. How many dots are at the other end?
# | 5 | 40 | 1 |
math | 5. Calculate $f(2)$, if $25 f\left(\frac{x}{1580}\right)+(3-\sqrt{34}) f\left(\frac{1580}{x}\right)=2017 x$. Round the answer to the nearest integer. | 265572 | 63 | 6 |
math | Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$. | \frac{11}{4} | 72 | 8 |
math | 11.49 For what values of $\alpha$ and $\beta$ is the equality $\sin \alpha + \sin \beta = \sin (\alpha + \beta)$ possible? | \alpha+\beta=2\pin;\alpha=2\pin,\beta\in\mathbb{R};\beta=2\pin,\alpha\in\mathbb{R}(n\in\mathbb{Z}) | 39 | 49 |
math | Problem 10.6. A pair of natural numbers ( $a, p$ ) is called good if the number $a^{3}+p^{3}$ is divisible by $a^{2}-p^{2}$, and $a>p$.
(a) (1 point) Indicate any possible value of $a$ for which the pair $(a, 13)$ is good.
(b) (3 points) Find the number of good pairs for which $p$ is a prime number less than 20. | 24 | 110 | 2 |
math | Example 2 Find all solutions to the indeterminate equations
$$x^{2}-8 y^{2}=-1$$
and
$$x^{2}-8 y^{2}=1$$ | x+y \sqrt{8}= \pm(3 \pm \sqrt{8})^{j}, \quad j=0,1,2,3, \cdots | 40 | 36 |
math | 17. (GDR 3) A ball $K$ of radius $r$ is touched from the outside by mutually equal balls of radius $R$. Two of these balls are tangent to each other. Moreover, for two balls $K_{1}$ and $K_{2}$ tangent to $K$ and tangent to each other there exist two other balls tangent to $K_{1}, K_{2}$ and also to $K$. How many balls... | R=r(2+\sqrt{6}),R=r(1+\sqrt{2}),R=r[\sqrt{5-2\sqrt{5}}+(3-\sqrt{5})/2] | 111 | 41 |
math | 5. Which number is greater: $2023^{2023}$ or $2022^{2024} ?$
# | 2023^{2023}<2022^{2024} | 33 | 20 |
math | For every $A \subset S$, let
$$
S_{\mathrm{A}}=\left\{\begin{array}{ll}
(-)^{\mid \mathrm{A}} \mid \sum_{\mathbf{a} \in \mathrm{A}} a, & A \neq \varnothing, \\
0, & A=\varnothing .
\end{array}\right.
$$
Find $\sum_{\mathrm{A} \subset \mathrm{S}} S_{\mathrm{A}}$. | 0 | 109 | 1 |
math | Problem 4. (4 points) In how many different ways can the number 2004 be represented as the sum of natural numbers (one or several) that are approximately equal? Two numbers are called approximately equal if their difference is no more than 1. Sums that differ only in the order of the addends are considered the same. | 2004 | 72 | 4 |
math | ## Task 1 - 230731
Five girls, all older than 10 years and having their birthday on the same day, but no two of the same age, are asked about their age on their birthday. Each girl answers truthfully:
(1) Anja: "I am 5 years younger than Elke."
(2) Birgit: "I am younger than Carmen, but older than Dorit."
(3) Carme... | Anja:11,Birgit:13,Carmen:14,Dorit:12,Elke:16 | 181 | 29 |
math | . Find all sequences $\left(a_{n}\right)_{n \geqslant 1}$ of strictly positive real numbers such that for every integer $n$, we have:
$$
\sum_{i=1}^{n} a_{i}^{3}=\left(\sum_{i=1}^{n} a_{i}\right)^{2}
$$ | a_{n}=n | 79 | 5 |
math | 13. Find all integers $a$ such that the quadratic equation $a x^{2}+2 a x+a-9=0$ has at least one integer root. | a=1,9 | 37 | 5 |
math | 7. Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=2,\left|z_{2}\right|=3$. If the angle between the vectors they correspond to is $60^{\circ}$, then $\left|\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\right|=$ $\qquad$ . | \frac{\sqrt{133}}{7} | 87 | 12 |
math | 4. In the Cartesian coordinate plane, the number of integer points (i.e., points with both coordinates as integers) that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \leqslant 3 x, \\
y \geqslant \frac{1}{3} x, \\
x+y \leqslant 100
\end{array}\right.
$$
is | 2551 | 92 | 4 |
math | 11.026. The base of a right parallelepiped is a rhombus. A plane, passing through one side of the lower base and the opposite side of the upper base, forms an angle of $45^{\circ}$ with the base plane. The area of the resulting section is equal to $Q$. Determine the lateral surface area of the parallelepiped. | 2Q\sqrt{2} | 80 | 7 |
math | 4. If the system of inequalities about $x$ $\left\{\begin{array}{l}x^{3}+3 x^{2}-x-3>0, \\ x^{2}-2 a x-1 \leqslant 0\end{array},(a>0)\right.$ has exactly one integer solution, then the range of values for $a$ is $\qquad$ | [\frac{3}{4},\frac{4}{3}) | 86 | 14 |
math | Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$? | 34 | 40 | 2 |
math | 1085*. Several natural numbers form an arithmetic progression, starting with an even number. The sum of the odd terms of the progression is 33, and the sum of the even terms is 44. Find the progression and the number of its terms. List all solutions. | 2;5;8;11;14;17;20 | 59 | 17 |
math | $9 \cdot 55$ Let $n \geqslant 2$, find the maximum and minimum value of the product $x_{1} x_{2} \cdots x_{n}$ under the conditions $x_{i} \geqslant \frac{1}{n}, i=1,2, \cdots$, $n$, and $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$. | \frac{\sqrt{n^{2}-n+1}}{n^{n}} | 106 | 17 |
math | 9.4. Find all triples of prime numbers $p, q, r$ such that the numbers $|q-p|,|r-q|,|r-p|$ are also prime. | 2,5,7 | 40 | 5 |
math | 5. Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2}=36$ lines. Assume that all 36 lines drawn are distinct. What is the maximum possible number of points which lie on at ... | 462 | 84 | 3 |
math | Six. (25 points) Given
$$
f(x)=\lg (x+1)-\frac{1}{2} \log _{3} x .
$$
(1) Solve the equation: $f(x)=0$;
(2) Find the number of subsets of the set
$$
M=\left\{n \mid f\left(n^{2}-214 n-1998\right) \geqslant 0, n \in \mathbf{Z}\right\}
$$
(Li Tiehan, problem contributor) | 4 | 123 | 1 |
math | Problem 4. Mira used the digits $3,7,1,9,0$ and 4 to form the largest and smallest six-digit numbers using each digit exactly once, and then reduced their difference by 9 times. What number did Mira get? | 96759 | 55 | 5 |
math | Children at the camp threw a die and, depending on the result, performed the following tasks:
| 1 | go $1 \mathrm{~km}$ west |
| :--- | :--- |
| 2 | go $1 \mathrm{~km}$ east |
| 3 | go $1 \mathrm{~km}$ north |
| 4 | go $1 \mathrm{~km}$ south |
| 5 | stand still |
| 6 | go $3 \mathrm{~km}$ north |
After five throws, M... | 8,12,15,16,19,20,22 | 194 | 19 |
math | 9,10 |
| :---: | :---: | :---: |
| | [ Formulas for abbreviated multiplication (other).] | |
| | [ Properties of the modulus. Triangle inequality ] | |
| | equations with moduli | |
Solve the equation: $\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$. | 5\leqx\leq10 | 93 | 9 |
math | We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers. | 117, 156, 195 | 38 | 13 |
math | 1.1. A group of tourists set out on a route from a campsite. After 15 minutes, tourist Ivan remembered that he had forgotten a flashlight at the campsite and went back for it at a higher speed than the main group. After picking up the flashlight, he started to catch up with the group at the same increased speed and did... | 1.2 | 126 | 3 |
math | 8,9,10,11 |
Author: S $\underline{\text { Saghafian M. }}$.
In the plane, five points are marked. Find the maximum possible number of similar triangles with vertices at these points. | 8 | 50 | 1 |
math | Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes,
which form $1$ degree angles with one another at the origin. Determine the sum of the x-coordinates
of the points of intersection of these lines with the line $y = 100-x$ | 8950 | 65 | 4 |
math | Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide $\frac{L}{2016}$.
[i]Proposed by Yannick Yao[/i] | 44 | 76 | 2 |
math | 3. Given are the sides $a = BC$ and $b = CA$ of triangle $ABC$. Determine the length of the third side if it is equal to the length of the corresponding height. For which values of $a$ and $b$ does the problem have a solution? | ^{2}=\frac{1}{5}(^{2}+b^{2}\\sqrt{D}) | 59 | 22 |
math | Problem 4. In a singing competition, a Rooster, a Crow, and a Cuckoo participated. Each member of the jury voted for one of the three performers. The Woodpecker calculated that there were 59 judges in total, and that the Rooster and the Crow received a total of 15 votes, the Crow and the Cuckoo received 18 votes, and t... | 13 | 145 | 2 |
math | 7. Given that the parabola $C$ has the center of the ellipse $E$ as its focus, the parabola $C$ passes through the two foci of the ellipse $E$, and intersects the ellipse $E$ at exactly three points. Then the eccentricity of the ellipse $E$ is | \frac{2 \sqrt{5}}{5} | 66 | 12 |
math | 6. We use $S_{k}$ to denote an arithmetic sequence with the first term $k$ and a common difference of $k^{2}$, for example, $S_{3}$ is $3, 12, 21, \cdots$. If 306 is a term in $S_{k}$, the sum of all $k$ that satisfy this condition is $\qquad$. | 326 | 87 | 3 |
math | $4 \cdot 201$ (1) How much boiling water should be added to $a$ liters of water at $t_{1} \mathrm{C}$ to obtain water at $t_{2} \mathrm{C}$ $\left(t_{1}<100^{\circ} \mathrm{C}\right)$?
(2) If the approximate measured values are:
$a=3.641$ liters, $t_{1}=36.7^{\circ} \mathrm{C}, t_{2}=57.4^{\circ} \mathrm{C}$, how many ... | 1.769 | 138 | 5 |
math | 12. (16 points) Find all positive integers $n$ greater than 1, such that for any positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$, the inequality
$$
\left(x_{1}+x_{2}+\cdots+x_{n}\right)^{2} \geqslant n\left(x_{1} x_{2}+x_{2} x_{3}+\cdots+x_{n} x_{1}\right) \text {. }
$$
holds. | 2, 3, 4 | 119 | 7 |
math | $ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ . | 7 | 51 | 1 |
math | 1. The second term of a geometric progression is 5, and the third term is 1. Find the first term of this progression. | 25 | 29 | 2 |
math | A set of positive integers $X$ is called [i]connected[/i] if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$.
Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$. | 922 | 80 | 3 |
math | Example 1. Find the minimum value of $|x-1|+|x-3|+|x-5|$. | 4 | 28 | 1 |
math | Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit. | 1 | 90 | 1 |
math | 10.027. Each of the three equal circles of radius $r$ touches the other two. Find the area of the triangle formed by the common external tangents to these circles. | 2r^{2}(2\sqrt{3}+3) | 40 | 14 |
math | 8. The terms of a sequence are all 3 or 5, the first term is 3, and there are $2^{k-1}$ fives between the $k$-th 3 and the $(k+1)$-th 3, i.e., $3,5,3,5,5,3,5,5,5,5,3, \cdots$. Then the sum of the first 2004 terms of this sequence $S_{2004}=$ $\qquad$ . | 9998 | 114 | 4 |
math | Problem 11.6. Oleg wrote down several composite natural numbers less than 1500 on the board. It turned out that the greatest common divisor of any two of them is 1. What is the maximum number of numbers that Oleg could have written down? | 12 | 58 | 2 |
math | ## Task 2 - 090612
During the holidays, Klaus was in the countryside. From his observations, the following joke problem emerged:
$$
1 \frac{1}{2} \text { chickens lay in } 1 \frac{1}{2} \text { days } 1 \frac{1}{2} \text { eggs }
$$
Determine the total number of eggs that 7 chickens would lay in 6 days at the same l... | 28 | 103 | 2 |
math | Given a sector equal to a quarter of a circle with radius $R$. Find the length of the tangent line drawn at the midpoint of its arc to the intersection with the extensions of the extreme radii of the sector. | 2R | 44 | 2 |
math | ## Task A-2.4.
Determine all pairs of prime numbers $(p, q)$ for which $p^{q-1} + q^{p-1}$ is a square of a natural number. | (2,2) | 44 | 5 |
math | [b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that:
$$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds.
[i]Proposed by dangerousliri [/i] | n = 2 | 104 | 5 |
math | 12 Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with eccentricity $e=\sqrt{3}$, and the distance from its left focus $F_{1}$ to the asymptote is $\sqrt{2}$.
(1) Find the equation of the hyperbola $C$;
(2) If a line $l$ passing through the point $D(2,0)$ intersects the hyperbola $C$ at po... | x-2or-x+2 | 153 | 7 |
math | 1. Calculate the value of the numerical expression: $36+64 \cdot 17-(502-352: 8+511)$. | 155 | 38 | 3 |
math | 1. Let $A=\{1,2, \cdots, n\}$, then the sum of the elements in all non-empty proper subsets of $A$ is | \frac{n(n+1)(2^{n-1}-1)}{2} | 36 | 18 |
math | 3A. In the set of real numbers, solve the inequality
$$
\sqrt{x^{2}-x-12}>7 x
$$ | x\in(-\infty,-3] | 31 | 10 |
math | Find all odd integers $n \geqslant 1$ such that $n$ divides $3^{n}+1$.
untranslated text remains unchanged. | 1 | 34 | 1 |
math | ## Task 1 - 150621
A Soviet helicopter of the Mi-10 type can transport a payload of 15000 kp.
In a transport of bulky goods with three helicopters of this type, the first helicopter was loaded to $\frac{1}{3}$, the second to $\frac{7}{8}$, and the third to $\frac{3}{5}$ of its capacity.
Determine the total weight of... | 27125 | 107 | 5 |
math | 3. Consider the equation $x^{5}+y^{2}=z^{3}$, where $x, y$ and $z$ are non-zero natural numbers. A solution to this equation is a triplet of non-zero natural numbers $(a, b, c)$ with the property that $a^{5}+b^{2}=c^{3}$.
a) Show that the triplet $(3,10,7)$ is a solution to the given equation.
b) Determine a solution... | (3,10,7)(2^4,2^{10},2^7) | 154 | 21 |
math | 3. In $\triangle A B C$, it is known that $D$ is a point on side $B C$, $\frac{B D}{D C}=\frac{1}{3}$, $E$ is the midpoint of side $A C$, $A D$ intersects $B E$ at point $O$, and $C O$ intersects $A B$ at point $F$. Find the ratio of the area of quadrilateral $B D O F$ to the area of $\triangle A B C$ | \frac{1}{10} | 108 | 8 |
math | Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\] | f(n) = n | 84 | 6 |
math | The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan C... | 336 | 112 | 3 |
math | Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. | 11 | 57 | 4 |
math | 1st ASU 1961 problems Problem 10 A and B play the following game with N counters. A divides the counters into 2 piles, each with at least 2 counters. Then B divides each pile into 2 piles, each with at least one counter. B then takes 2 piles according to a rule which both of them know, and A takes the remaining 2 piles... | [N/2],[(N+1)/2],[N/2] | 207 | 15 |
math | In the class, there are 30 students: excellent students, C-grade students, and D-grade students. Excellent students always answer questions correctly, D-grade students always make mistakes, and C-grade students answer the questions given to them strictly in turn, alternating between correct and incorrect answers. All s... | 20 | 128 | 2 |
math | 35. The exchange rate of the cryptocurrency Chuhoin on March 1 was one dollar, and then it increased by one dollar every day. The exchange rate of the cryptocurrency Antonium on March 1 was also one dollar, and then each subsequent day it was equal to the sum of the previous day's rates of Chuhoin and Antonium, divided... | \frac{92}{91} | 101 | 9 |
math | Three, (20 points) Let $x, y, z \geqslant 0, x+y+z=1$. Find the maximum value of $A=\left(x-\frac{1}{6}\right)\left(y-\frac{1}{6}\right)\left(z-\frac{1}{6}\right)^{2}$.
| \frac{25}{1296} | 73 | 11 |
math | A7. Find the maximal value of
$$
S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}},
$$
where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. | \frac{8}{\sqrt[3]{7}} | 97 | 12 |
math | PROBLEM 4. - Determine all pairs of integers $(x, y)$ that satisfy the equation $x^{2}-y^{4}=2009$.
## Solution: | (x,y)=(\45,\2) | 38 | 9 |
math | 11. (20 points) Through a point $P$ on the directrix $l$ of the parabola $C: y^{2}=4 x$, draw two tangent lines to the parabola $C$, touching at points $A$ and $B$ respectively, and let $M$ be the intersection of the directrix $l$ with the $x$-axis.
(1) Let the focus of the parabola be $F$, prove:
$$
|P F|^{2}=|A F||B ... | P(-1, \pm 2), \text{ and the equation of line } A B \text{ is } y=x-1 \text{ or } y=-x+1 | 160 | 39 |
math | The mean number of days per month in $2020$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 63 | 46 | 2 |
math | 4. Given real numbers $x, y$ satisfy $x^{2}+y^{2}=20$. Then the maximum value of $x y+8 x+y$ is $\qquad$ . | 42 | 43 | 2 |
math | 5. [5] Rachelle picks a positive integer $a$ and writes it next to itself to obtain a new positive integer $b$. For instance, if $a=17$, then $b=1717$. To her surprise, she finds that $b$ is a multiple of $a^{2}$. Find the product of all the possible values of $\frac{b}{a^{2}}$. | 77 | 87 | 2 |
math | 10. (14 points) Let $n$ be a given integer greater than 2. There are $n$ bags that are indistinguishable in appearance, the $k$-th bag ($k=1,2, \cdots, n$) contains $k$ red balls and $n-k$ white balls. After mixing these bags, one bag is randomly selected, and three balls are drawn consecutively from it (without replac... | \frac{n-1}{2n} | 106 | 9 |
math | 2. In trapezoid $A B C D$, it is known that $A D / / B C, A D \perp$ $C D, B C=C D=2 A D, E$ is a point on side $C D$, $\angle A B E=45^{\circ}$. Then $\tan \angle A E B=$ $\qquad$
(2007, National Junior High School Mathematics Competition, Tianjin Preliminary Round) | 3 | 100 | 1 |
math | 7. Find the largest integer which cannot be expressed as sum of some of the numbers $135,136,137, \ldots, 144$ (each number may occur many times in the sum or not at all).
(1 mark)
某整數不能寫成 135、136、‥144 當中的某些數之和（在和式中每個數出現的次數不限, 亦可以不出現）。求該整數的最大可能值。 | 2024 | 112 | 4 |
math | 7th Swedish 1967 Problem 1 p parallel lines are drawn in the plane and q lines perpendicular to them are also drawn. How many rectangles are bounded by the lines? | \frac{pq(p-1)(q-1)}{4} | 38 | 15 |
math | 7.056. $7^{\lg x}-5^{\lg x+1}=3 \cdot 5^{\lg x-1}-13 \cdot 7^{\lg x-1}$. | 100 | 47 | 3 |
math | ## 181. Math Puzzle $6 / 80$
A gasoline car consumes 10.5 l of gasoline per $100 \mathrm{~km}$. It is known that the energy efficiency of a gasoline engine is 24 percent, while that of a diesel engine is 38 percent.
How much fuel would the car consume if it were powered by a diesel engine of the same performance and ... | 6.6 | 91 | 3 |
math | 14. (FRG 1) How many words with $n$ digits can be formed from the alphabet $\{0,1,2,3,4\}$, if neighboring digits must differ by exactly one? | x_{2n}=8\cdot3^{n-1},\quadx_{2n+1}=14\cdot3^{n-1} | 46 | 33 |
math | 10. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the value of $\sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right]$ is $\qquad$ . | 2015028 | 60 | 7 |
math | 5.39 It can be proven that for any given positive integer $n$, every complex number of the form $r + s i (r, s$ both integers) can be expressed as a polynomial in $(-n+i)$, and the coefficients of the polynomial all belong to $\{0,1,2, \cdots, n^{2}\}$.
That is, the equation
$$r+s i=a_{m}(-n+i)^{m}+a_{m-1}(-n+i)^{m-1}... | 490 | 333 | 3 |
math | 30. Several girls were picking mushrooms. They divided the collected mushrooms among themselves as follows: one of them was given 20 mushrooms and 0.04 of the remainder, another - 21 mushrooms and 0.04 of the new remainder, the third - 22 mushrooms and 0.04 of the remainder, and so on. It turned out that everyone recei... | 5 | 99 | 1 |
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