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 | Solve the following system of equations:
$$
x y+x+y=34, \quad x^{2}+y^{2}-x-y=42 .
$$ | \begin{pmatrix}x_1=4,\quady_1=6,\\x_2=6,\quady_2=4\end{pmatrix} | 36 | 38 |
math | ## 194. Math Puzzle $7 / 81$
From a textbook by Adam Ries, who lived from 1492 to 1559, this problem was taken:
A son asks his father how old he is. The father answers: "If you were as old as I am, and half as old, and a quarter as old, and one year more, you would be 134 years old."
How old is the father? | 76 | 99 | 2 |
math | Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis.
1956 Tokyo Institute of Technology entrance exam | \frac{35 - 15\sqrt{5}}{12} | 46 | 18 |
math | 267. What relationship exists between the roots of the equations:
$$
\begin{aligned}
& a x^{2}+b x+c=0 \\
& c x^{2}+b x+a=0
\end{aligned}
$$ | x_{3},x_{4}=\frac{1}{x_{1}},\frac{1}{x_{2}} | 53 | 26 |
math | 13.308. What two-digit number is less than the sum of the squares of its digits by 11 and greater than their doubled product by 5? | 95or15 | 36 | 5 |
math | 6. $118 \quad x_{1}, x_{2}, \cdots, x_{1993}$ satisfy
$$\begin{array}{l}
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993, \\
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k},(k=1,2, \cdots, 1993)
\end{array}$$
Then, what is the maximum possible value of $\left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|+\cdots+\left|y_{1992}-y_{1993}\right|$? | 1992 | 195 | 4 |
math | 15. Let the quadratic function $f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)$ satisfy the following conditions:
(1) For $x \in \mathbf{R}$, $f(x-4)=f(2-x)$, and $f(x) \geqslant x$;
(2) For $x \in(0,2)$, $f(x) \leqslant\left(\frac{x+1}{2}\right)^{2}$;
(3) The minimum value of $f(x)$ on $\mathbf{R}$ is 0.
Find the largest $m(m>1)$ such that there exists $t \in \mathbf{R}$, for which, if $x \in[1, m]$, then $f(x+t) \leqslant x$. | 9 | 197 | 1 |
math | (a) What is the smallest (positive) multiple of 9 that is written only with the digits 0 and 1?
(b) What is the smallest (positive) multiple of 9 that is written only with the digits 1 and 2? | 111111111 | 52 | 9 |
math | # Task No. 5.1
## Condition:
A Dog, a Cat, and a Mouse are running around a circular lake. They all started simultaneously in the same direction from the same point and finished at the same time, each running at a constant speed. The Dog ran 12 laps, the Cat ran 6 laps, and the Mouse ran 4 laps. How many total overtakes were made from the start to the finish?
If two or more overtakes occur simultaneously, each overtake is counted separately. The start and finish moments are not counted as overtakes. | 13 | 119 | 2 |
math |
4. In a triangle $A B C$, points $D$ and $E$ are on segments $B C$ and $A C$ such that $B D=3 D C$ and $A E=4 E C$. Point $P$ is on line $E D$ such that $D$ is the midpoint of segment $E P$. Lines $A P$ and $B C$ intersect at point $S$. Find the ratio $B S / S D$.
| \frac{7}{2} | 102 | 7 |
math | 9. Two differentiable real functions $f(x)$ and $g(x)$ satisfy
$$
\frac{f^{\prime}(x)}{g^{\prime}(x)}=e^{f(x)-g(x)}
$$
for all $x$, and $f(0)=g(2003)=1$. Find the largest constant $c$ such that $f(2003)>c$ for all such functions $f, g$. | 1-\ln2 | 97 | 4 |
math | The volume of a parallelepiped is $V$. Find the volume of the polyhedron whose vertices are the centers of the faces of the given parallelepiped.
# | \frac{1}{6}V | 35 | 8 |
math | Example 3 Solve the equation $5^{x+1}=3^{x^{2}-1}$. | -1or\log_{3}5+1 | 22 | 11 |
math | $14 \cdot 72$ Find all real solutions of the equation $4 x^{2}-40[x]+51=0$.
(Canadian Mathematical Olympiad, 1999) | {\frac{\sqrt{29}}{2},\frac{\sqrt{189}}{2},\frac{\sqrt{229}}{2},\frac{\sqrt{269}}{2}} | 45 | 47 |
math | Example 7 Given that $n$ is a positive integer, and $n^{2}-71$ is divisible by $7 n+55$. Try to find the value of $n$.
(1994-1995, Chongqing and Four Other Cities Mathematics Competition) | 57 | 63 | 2 |
math | Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win. | n | 125 | 1 |
math | 14. From the 20 numbers 11, 12, 13, 14, ... 30, at least ( ) numbers must be taken to ensure that there are two numbers among the taken numbers whose sum is a multiple of ten. | 11 | 57 | 2 |
math | 10. When $x, y, z$ are positive numbers, the maximum value of $\frac{4 x z+y z}{x^{2}+y^{2}+z^{2}}$ is $\qquad$ - | \frac{\sqrt{17}}{2} | 49 | 11 |
math | \section*{Problem 7 - V01107 = V601215}
For which values of \(a\) does the curve
\[
y=\frac{1}{4}\left(a x-x^{3}\right)
\]
intersect the \(x\)-axis at an angle of \(45^{\circ}\) ? | \in{-4,2,4} | 75 | 9 |
math | 5. Misha painted all integers in several colors such that numbers whose difference is a prime number are painted in different colors. What is the smallest number of colors that Misha could have used? Justify your answer. | 4 | 44 | 1 |
math | Given a parallelepiped $A B C D A 1 B 1 C 1 D 1$. Points $M, L$, and $K$ are taken on the edges $A D, A 1 D 1$, and $B 1 C 1$ respectively, such that $B 1 K=\frac{1}{3} A 1 L, A M=\frac{1}{2} A 1 L$. It is known that $K L=2$. Find the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$. | \frac{3}{2} | 129 | 7 |
math | 1.59 (1) $f(x)$ is a monovariate quartic polynomial with integer coefficients, i.e.,
$$
f(x)=c_{4} x^{4}+c_{3} x^{3}+c_{2} x^{2}+c_{1} x+c_{0},
$$
where $c_{0}, c_{1}, c_{2}, c_{3}, c_{4}$ are all integers. Prove that if $a, b$ are integers and $a > b$, then $a - b$ divides $f(a) - f(b)$.
(2) Two people, A and B, are doing homework in the classroom.
A asks B: “How many problems have you finished?”
B replies: “The number of problems I have finished is a positive integer and is a root of a monovariate quartic polynomial with integer coefficients. Can you guess it?”
Then, A tries substituting 7 into the polynomial and gets a value of 77. At this point, B says: “The number of problems I have finished is more than 7.”
A says: “Alright, I will try a larger integer.” So A substitutes $B$ into the polynomial and gets a value of 85.
B looks and says: “The number of problems I have finished is more than $B$.” After some thought and calculation, A finally determines the number of problems B has finished.
Try to find the number of problems B has finished based on the conversation.
(China Hebei Province Mathematics Competition, 1979) | 14 | 333 | 2 |
math | 11. (6 points) A car rally has two equal-distance courses. The first course starts on a flat road, 26 kilometers from the midpoint it begins to climb; 4 kilometers after passing the midpoint, it is all downhill; The second course also starts on a flat road, 4 kilometers from the midpoint it begins to descend, 26 kilometers after passing the midpoint, it is all uphill. It is known that a certain race car used the same amount of time on these two courses; the speed at the start of the second course is $\frac{5}{6}$ of the speed at the start of the first course; and when encountering an uphill, the speed decreases by $25 \%$, and when encountering a downhill, the speed increases by $25 \%$. Therefore, the distance of each course is $\qquad$ kilometers. | 92 | 176 | 2 |
math | ## Task 1 - 040731
How many pages of a book are consecutively numbered from page 1 if a total of 1260 digits are printed? | 456 | 41 | 3 |
math | Let $a_1=1$ and $a_{n+1}=2/(2+a_n)$ for all $n\geqslant 1$. Similarly, $b_1=1$ and $b_{n+1}=3/(3+b_n)$ for all $n\geqslant 1$. Which is greater between $a_{2022}$ and $b_{2022}$?
[i]Proposed by P. Kozhevnikov[/i] | a_{2022} < b_{2022} | 106 | 16 |
math | 1. A professor from the Department of Mathematical Modeling at FEFU last academic year gave 6480 twos, thereby overachieving the commitments taken at the beginning of the year. In the next academic year, the number of professors increased by 3, and each of them started giving more twos. As a result, a new record was set for enclosed spaces: 11200 twos in a year. How many professors were there initially, if each professor gives the same number of twos as the others during the session? | 5 | 114 | 1 |
math | Three distinct numbers are given. The average of the average of the two smaller numbers and the average of the two larger numbers is equal to the average of all three numbers. The average of the smallest and the largest number is 2022.
Determine the sum of the three given numbers.
(K. Pazourek)
Hint. Express one of the numbers in terms of the remaining two. | 6066 | 80 | 4 |
math | 6. [6] Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6 -sided die. If she gets a 1 or 2 , she goes to DC. If she rolls a 3,4 , or 5 , she goes to Russia. If she rolls a 6 , she rolls again. What is the probability that she goes to DC? | \frac{2}{5} | 89 | 7 |
math | \section*{Problem 6B - 131236B}
Let \(M\) be the set of all points \(P(x, y)\) in a plane rectangular Cartesian coordinate system, where \(x, y\) are rational integers, and \(0 \leq x \leq 4\) and \(0 \leq y \leq 4\).
Determine the probability that the distance between two different points chosen arbitrarily from \(M\) is a rational integer (the unit of measurement being the unit of the coordinate system).
Hint: If \(n\) is the number of different ways to choose two points and \(m\) is the number of ways in which the distance is a rational integer, then the quotient \(\frac{m}{n}\) is the probability to be determined. Two selection possibilities are considered different if and only if the sets of points (each consisting of two points) selected in them are different. | \frac{9}{25} | 194 | 8 |
math | 11. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.
保留源文本的换行和格式如下：
11. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample. | (\frac{1}{4},0) | 224 | 9 |
math | 5. $\frac{\sin 7^{\circ}+\sin 8^{\circ} \cos 15^{\circ}}{\cos 7^{\circ}-\sin 8^{\circ} \sin 15^{\circ}}=$ | 2-\sqrt{3} | 55 | 6 |
math | ## Task B-4.4.
The following functions are given:
$$
f(x)=10^{10 x}, g(x)=\log \left(\frac{x}{10}\right), h_{1}(x)=g(f(x)), h_{n}(x)=h_{1}\left(h_{n-1}(x)\right)
$$
for all $n \geqslant 2$. Determine the sum of the digits of the number $h_{2015}(1)$. | 16121 | 107 | 5 |
math | $6 \cdot 109$ Given the family of curves
$$2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0$$
where $\theta$ is a parameter. Find the maximum value of the length of the chord intercepted by the line $y=2 x$ on this family of curves. | 8 \sqrt{5} | 86 | 6 |
math | Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$. | 4 | 85 | 1 |
math | 31. Find all positive integers $n$, such that
$$\frac{2^{n}-1}{3} \in \mathbf{N}^{*}$$
and there exists $m \in \mathbf{N}^{*}$, satisfying ${ }^{*}$ :
$$\left.\frac{2^{n}-1}{3} \right\rvert\,\left(4 m^{2}+1\right)$$ | n = 2^k, k \in \mathbf{N}^{*} | 95 | 19 |
math | 5. In the game "set," all possible four-digit numbers consisting of the digits $1,2,3$ (each digit appearing exactly once) are used. It is said that a triplet of numbers forms a set if, in each digit place, either all three numbers contain the same digit, or all three numbers contain different digits.
For example, the numbers 1232, 2213, 3221 form a set (in the first place, all three digits appear, in the second place, only the digit two appears, in the third place, all three digits appear, and in the fourth place, all three digits appear). The numbers $1123,2231,3311$ do not form a set (in the last place, two ones and a three appear).
How many sets exist in the game?
(Permuting the numbers does not create a new set: $1232,2213,3221$ and $2213,1232,3221$ are the same set.) | 1080 | 232 | 4 |
math | Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$? | 3 | 26 | 1 |
math | 5. Find the maximum value of the expression $(\sin 3 x+\sin 2 y+\sin z)(\cos 3 x+\cos 2 y+\cos z)$. (15 points) | 4.5 | 44 | 3 |
math | 4. (8 points) In another 12 days, it will be 2016, Hao Hao sighs: I have only experienced 2 leap years so far, and the year I was born is a multiple of 9. So how old will Hao Hao be in 2016? | 9 | 65 | 1 |
math | $[\quad$ Evaluation + example $\quad]$
The hostess baked a pie for her guests. She may have either 10 or 11 guests. Into what smallest number of slices should she cut the pie in advance so that it can be evenly divided among either 10 or 11 guests? | 20 | 65 | 2 |
math | 3. A non-empty finite set of numbers is called a trivial set if the sum of the squares of all its elements is odd.
Let the set $A=\{1,2, \cdots, 2017\}$. Then the number of trivial sets among all proper subsets of $A$ is $\qquad$ (powers of numbers are allowed in the answer). | 2^{2016}-1 | 79 | 8 |
math | Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$. | \{3, 7, 31, 127, 8191\} | 125 | 24 |
math | 6. 10 students (one is the team leader, 9 are team members) formed a team to participate in a math competition and won the first place. The organizing committee decided to award each team member 200 yuan, and the team leader received 90 yuan more than the average prize of the entire team of 10 members. Therefore, the team leader's prize is $\qquad$ yuan. | 300 | 87 | 3 |
math | A thousand points are the vertices of a convex thousand-sided polygon, inside which there are another five hundred points such that no three of the five hundred lie on the same line. This thousand-sided polygon is cut into triangles in such a way that all the specified 1500 points are vertices of the triangles and these triangles have no other vertices. How many triangles will result from such a cutting? | 1998 | 80 | 4 |
math | Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$. | \frac{1}{2} u | 88 | 8 |
math | How many integers between $2$ and $100$ inclusive [i]cannot[/i] be written as $m \cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$? Note that there are $25$ primes less than $100.$ | 35 | 75 | 2 |
math | Example 6 For any subset $A$ of $X_{n}=\{1,2, \cdots, n\}$, denote the sum of all elements in $A$ as $S(A), S(\varnothing)=0$, and the number of elements in $A$ as $|A|$. Find $\sum_{A \subseteq X_{n}} \frac{S(A)}{|A|}$. | \frac{1}{2}(n+1)\left(2^{n}-1\right) | 88 | 21 |
math | 31. [17] Given positive integers $a_{1}, a_{2}, \ldots, a_{2023}$ such that
$$
a_{k}=\sum_{i=1}^{2023}\left|a_{k}-a_{i}\right|
$$
for all $1 \leq k \leq 2023$, find the minimum possible value of $a_{1}+a_{2}+\cdots+a_{2023}$. | 2046264 | 110 | 7 |
math | 1. If the natural numbers $a, x, y$ satisfy $\sqrt{a-2 \sqrt{6}}=\sqrt{x} -\sqrt{y}$, then the maximum value of $a$ is $\qquad$. | 7 | 49 | 1 |
math | To celebrate her birthday, Ana is going to prepare pear and apple pies. At the market, an apple weighs $300 \mathrm{~g}$ and a pear, $200 \mathrm{~g}$. Ana's bag can hold a maximum weight of $7 \mathrm{~kg}$. What is the maximum number of fruits she can buy to be able to make pies of both fruits? | 34 | 85 | 2 |
math | Example 22. How many times do you need to roll two dice so that the probability of rolling at least one double six is greater than $1 / 2$? (This problem was first posed by the French mathematician and writer de Mere ( $1610-1684$ ), hence the problem is named after him). | 25 | 73 | 2 |
math | 3. There are ten small balls of the same size, five of which are red and five are white. Now, these ten balls are arranged in a row arbitrarily, and numbered from left to right as $1,2, \cdots, 10$. Then the number of arrangements where the sum of the numbers of the red balls is greater than the sum of the numbers of the white balls is. $\qquad$ | 126 | 87 | 3 |
math | ## Task B-2.4.
For real numbers $x, y$ if the equality $|x+y|+|x-y|=2$ holds, determine the maximum value of the expression $x^{2}-6 x+y^{2}$. | 8 | 52 | 1 |
math | 7. Let an "operation" be the act of randomly changing a known positive integer $n$ to a smaller non-negative integer (each number has the same probability). Then, the probability that the numbers $10$, $100$, and $1000$ all appear during the process of performing several operations to reduce 2019 to 0 is $\qquad$ | \frac{1}{1112111} | 81 | 13 |
math | For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$.
Find $I_1+I_2+I_3+I_4$.
[i]1992 University of Fukui entrance exam/Medicine[/i] | \frac{40}{9} | 77 | 8 |
math | 2. The farming proverb 'counting the nines in winter' refers to dividing every nine days into a segment, starting from the Winter Solstice, and sequentially naming them the first nine, second nine, ..., ninth nine, with the Winter Solstice being the first day of the first nine. December 21, 2012, was the Winter Solstice, so New Year's Day 2013 is the $\qquad$th nine's $\qquad$th day. | 3 | 103 | 1 |
math | ## Task 5 - V00505
How many matches (5 cm long, 2 mm wide, and 2 mm high) can fit into a cube with a side length of 1 m? | 5000000 | 45 | 7 |
math | 1. Calculate: $123456789 \times 8+9=$ | 987654321 | 21 | 9 |
math | 3. In the Magic and Wizardry club, all first and second-year students wear red robes, third-year students wear blue, and fourth-year students wear black.
Last year, at the general assembly of students, there were 15 red, 7 blue, and several black robes, while this year - blue and black robes are equal in number, and red robes are twice as many as blue ones.
(a) How many black robes will there be at the general assembly next year? (1 point)
(b) How many first-year students are there this year? (3 points)
(c) In what minimum number of years will the number of blue and black robes be equal again? (6 points)
(d) Suppose it is additionally known that each year the number of first-year students is 1 less than the number of fourth-year students. In how many years will the number of red robes first be three times the number of blue ones? (10 points) | 17 | 196 | 2 |
math | 18. Four different prime numbers $a, b, c, d$ satisfy the following properties:
(1) $a+b+c+d$ is also a prime number;
(2) The sum of two of $a, b, c, d$ is also a prime number:
(3) The sum of three of $a, b, c, d$ is also a prime number. The smallest value of $a+b+c+d$ that satisfies the conditions is $\qquad$ | 31 | 101 | 2 |
math | 10・11 If $a, b, c, d, e, f, p, q$ are Arabic numerals, and $b>c>d>a$. The difference between the four-digit numbers $\overline{c d a b}$ and $\overline{a b c d}$ is a four-digit number of the form $\overline{p q e f}$. If $\overline{e f}$ is a perfect square, and $\overline{p q}$ is not divisible by 5. Find the four-digit number $\overline{a b c d}$, and briefly explain the reasoning.
(China Beijing Junior High School Grade 3 Mathematics Competition, 1983) | 1983 | 146 | 4 |
math | 1. Simplify $\frac{a+1}{a+1-\sqrt{1-a^{2}}}+\frac{a-1}{\sqrt{1-a^{2}}+a-1}$ $(0<|a|<1)$ The result is $\qquad$ | 1 | 58 | 1 |
math | 110) Let $p$ be a given odd prime, and let the positive integer $k$ be such that $\sqrt{k^{2}-p k}$ is also a positive integer. Then $k=$ $\qquad$ | \frac{(p+1)^{2}}{4} | 48 | 13 |
math | ## Task A-2.2.
Determine all pairs $(p, q)$ of prime numbers for which the quadratic equation $x^{2}+p x+q=0$ has two distinct solutions in the set of integers. | (p,q)=(3,2) | 48 | 7 |
math | $\left[\begin{array}{l}\text { Combinations and Permutations } \\ {[\text { Systems of Points and Segments (other) }]}\end{array}\right]$
On two parallel lines $a$ and $b$, points $A_{1}, A_{2}, \ldots, A_{m}$ and $B_{1}, B_{2}, \ldots, B_{n}$ are chosen respectively, and all segments of the form $A_{i} B_{j}$
$(1 \leq i \leq m, 1 \leq j \leq n)$ are drawn. How many intersection points will there be, given that no three of these segments intersect at the same point? | C_{}^{2}C_{n}^{2} | 153 | 13 |
math | 7. Given that $P$ is a moving point on the ellipse $\frac{x^{2}}{12}+\frac{y^{2}}{4}=1$, and $F_{1} 、 F_{2}$ are the two foci of the ellipse. Then the range of $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$ is $\qquad$ | -4 \leqslant \overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}} \leqslant 4 | 87 | 34 |
math | 1. (17 points) When walking uphill, the tourist walks 2 km/h slower, and downhill 2 km/h faster, than when walking on flat ground. Climbing the mountain takes the tourist 10 hours, while descending the mountain takes 6 hours. What is the tourist's speed on flat ground? | 8 | 66 | 1 |
math | 4. In the Cartesian coordinate system $x O y$, the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has left and right foci $F_{1}$ and $F_{2}$, respectively. The chords $S T$ and $U V$ of ellipse $C$ are parallel to the $x$-axis and $y$-axis, respectively, and intersect at point $P$. It is known that the lengths of segments $P U, P S, P V, P T$ are $1, 2, 3, 6$, respectively. Then the area of $\triangle P F_{1} F_{2}$ is | \sqrt{15} | 160 | 6 |
math | 15. In how many ways can a million be factored into three factors? Factorizations that differ only in the order of the factors are considered the same. | 139 | 33 | 3 |
math | Anjanan A.
All possible non-empty subsets are taken from the set of numbers $1,2,3, \ldots, n$. For each subset, the reciprocal of the product of all its numbers is taken. Find the sum of all such reciprocal values. | n | 55 | 1 |
math | Example 1. Find the complex Fourier transform of the function
$$
f(x)=e^{-b^{2} x^{2}}
$$
using the formula
$$
e^{-x^{2}} \longmapsto \frac{1}{\sqrt{2}} e^{-p^{2} / 4}
$$ | F(p)=\frac{1}{b\sqrt{2}}e^{-p^{2}/(2b)^{2}} | 67 | 27 |
math | 12. Given vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ satisfy $|\boldsymbol{a}|=2,|\boldsymbol{b}|=1$, and the angle between them is $60^{\circ}$. Then the range of real number $\lambda$ that makes the angle between vectors $\boldsymbol{a}+\lambda \boldsymbol{b}$ and $\lambda \boldsymbol{a}-2 \boldsymbol{b}$ obtuse is $\qquad$ | (-1-\sqrt{3},-1+\sqrt{3}) | 104 | 14 |
math | Using the digits 1, 3, and 5, Mônica forms three-digit numbers that are greater than 150. How many numbers can Mônica form?
# | 21 | 37 | 2 |
math | 11. $[7]$ Define $\phi^{\prime}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when
$$
\sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n)
$$
is divided by 50 . | 12 | 94 | 2 |
math | \section*{Task 1 - 131021}
Determine all (in the decimal number system) three-digit prime numbers with the following properties!
(1) If each digit of the three-digit prime number is written separately, each one represents a prime number.
(2) The first two and the last two digits of the three-digit prime number represent (in this order) a two-digit prime number each. | 373 | 88 | 3 |
math | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | 476 | 27 | 3 |
math | 7.140. $\left\{\begin{array}{l}4^{x+y}=2^{y-x} \\ 4^{\log _{\sqrt{2}} x}=y^{4}-5\end{array}\right.$ | (\frac{1}{2};-\frac{3}{2}) | 52 | 14 |
math |
Problem 3. Find all functions $f(x)$ with integer values and defined in the set of the integers, such that
$$
3 f(x)-2 f(f(x))=x
$$
for all integers $x$.
| f(x)=x | 50 | 4 |
math | 1. In a regular quadrilateral pyramid $P-ABCD$, all four lateral faces are equilateral triangles. Let the dihedral angle between a lateral face and the base be $\theta$, then $\tan \theta=$ $\qquad$ | \sqrt{2} | 50 | 5 |
math | [ Identical Transformations ]
Represent the numerical expression $2 \cdot 2009^{2}+2 \cdot 2010^{2}$ as the sum of squares of two natural numbers.
# | 4019^{2}+1^{2} | 45 | 12 |
math | What is the probability that 3 angles chosen at random from the angles $1^{\circ}, 2^{\circ}, 3^{\circ}, \ldots 179^{\circ}$ are the 3 angles of a scalene triangle? | \frac{2611}{939929} | 54 | 15 |
math | 72. Find the product:
$$
\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \ldots\left(1-\frac{1}{225}\right)
$$ | \frac{8}{15} | 72 | 8 |
math | 10. (20 points) Find a point $A$ on the parabola $y^{2}=2 p x$ such that the normal line at point $A$ intersects the parabola to form the shortest chord $A B$. | A(p,\sqrt{2}p)orA(p,-\sqrt{2}p) | 52 | 20 |
math | Example 4. Factorize
\[
\begin{aligned}
1+2 x+3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+5 x^{6}+4 x^{7} \\
+3 x^{8}+2 x^{9}+x^{10} .
\end{aligned}
\] | (x+1)^{2}\left(1-x+x^{2}\right)^{2}\left(1+x+x^{2}\right)^{2} | 81 | 33 |
math | Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$. | 665 | 59 | 3 |
math | 4. On a $100 \times 100$ chessboard, 1975 rooks were placed (each rook occupies one cell, different rooks stand on different cells). What is the maximum number of pairs of rooks that could be attacking each other? Recall that a rook can attack any number of cells along a row or column, but does not attack a rook that is blocked by another rook. (I. Rubanov) | 3861 | 99 | 4 |
math | 3. Define the function on $\mathbf{R}$
$$
f(x)=\left\{\begin{array}{ll}
\log _{2}(1-x), & x \leqslant 0 ; \\
f(x-1)-f(x-2), & x>0 .
\end{array}\right.
$$
Then $f(2014)=$ $\qquad$ | 1 | 86 | 1 |
math | 5. Given the imaginary number $z$ satisfies $z^{3}+1=0$, then $\left(\frac{z}{z-1}\right)^{2018}+\left(\frac{1}{z-1}\right)^{2018}=$ | -1 | 60 | 2 |
math | 3. Natural numbers $m, n$ satisfy $8 m+10 n>9 m n$. Then
$$
m^{2}+n^{2}-m^{2} n^{2}+m^{4}+n^{4}-m^{4} n^{4}=
$$
$\qquad$ | 2 | 68 | 1 |
math | Problem 10.1. Solve the system
$$
\left\lvert\, \begin{aligned}
& 3 \cdot 4^{x}+2^{x+1} \cdot 3^{y}-9^{y}=0 \\
& 2 \cdot 4^{x}-5 \cdot 2^{x} \cdot 3^{y}+9^{y}=-8
\end{aligned}\right.
$$
Ivan Landjev | \frac{1}{2},1+\frac{1}{2}\log_{3}2 | 102 | 20 |
math | ## Task B-4.5.
Andro and Borna take turns rolling a fair die whose sides are marked with the numbers 1, 2, 3, 4, 5, and 6. The winner is the first one to roll a six. If Andro starts the game, what is the probability that Borna wins? | \frac{5}{11} | 72 | 8 |
math | Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$. | 936 | 64 | 3 |
math | 2+ [ Inscribed angle, based on the diameter ]
$A B$ is the diameter of the circle, $B C$ is the tangent. The secant $A C$ is divided by the circle at point $D$ in half. Find the angle $D A B$.
# | 45 | 61 | 2 |
math | 5. Find all real solutions of the system of equations
$$
\left\{\begin{array}{l}
(x+1)\left(x^{2}+1\right)=y^{3}+1 \\
(y+1)\left(y^{2}+1\right)=z^{3}+1 \\
(z+1)\left(z^{2}+1\right)=x^{3}+1
\end{array}\right.
$$ | (0,0,0)(-1,-1,-1) | 95 | 14 |
math | 68. $(5 a+2 b) x^{2}+a x+b=0$ is a linear equation in one variable $x$, and it has a unique solution, then $x=$ | \frac{5}{2} | 42 | 7 |
math | 6. [5] Let $p_{0}(x), p_{1}(x), p_{2}(x), \ldots$ be polynomials such that $p_{0}(x)=x$ and for all positive integers $n$, $\frac{d}{d x} p_{n}(x)=p_{n-1}(x)$. Define the function $p(x):[0, \infty) \rightarrow \mathbb{R} x$ by $p(x)=p_{n}(x)$ for all $x \in[n, n+1]$. Given that $p(x)$ is continuous on $[0, \infty)$, compute
$$
\sum_{n=0}^{\infty} p_{n}(2009)
$$ | e^{2010}-e^{2009}-1 | 166 | 15 |
math | Example 3 Given
$$
a^{2}(b+c)=b^{2}(a+c)=2010 \text {, and } a \neq b \text {. }
$$
Then $c^{2}(a+b)=$ $\qquad$ [2]
$(2010$, I Love Mathematics Junior High School Summer Camp Mathematics Competition) | 2010 | 76 | 4 |
math | 8.149. $\cos x - \cos 3x = \sin 2x$. | x_{1}=\frac{\pik}{2};x_{2}=(-1)^{n}\frac{\pi}{6}+\pin,k,n\inZ | 22 | 35 |
math |
1. Find all integers $n$ such that $n^{4}-3 n^{2}+9$ is prime.
(Aleš Kobza)
| n\in{-2,-1,1,2} | 33 | 12 |
math | 138. Solve the equation: $x^{4}-2 x^{2}-400 x=9999$ (solved elementarily). | x_1=11,\,x_2=-9,\,x^2+2x+101= | 34 | 26 |
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