task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | 4. Find all integers $x$ and $y$ that satisfy the equation $x^{4}-2 y^{2}=1$. | x=\1,y=0 | 28 | 6 |
math | 68. From Pickleminster to Quickville. Trains $A$ and $B$ depart from Pickleminster to Quickville at the same time as trains $C$ and $D$ depart from Quickville to Pickleminster. Train $A$ meets train $C$ 120 miles from Pickleminster and train $D$ 140 miles from Pickleminster. Train $B$ meets train $C$ 126 miles from Qui... | 210 | 144 | 3 |
math | We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw? | 18 | 65 | 2 |
math | 1B. In the set of real numbers, solve the equation
$$
\sqrt{\frac{x^{2}-2 x+3}{x^{2}+2 x+4}}+\sqrt{\frac{x^{2}+2 x+4}{x^{2}-2 x+3}}=\frac{5}{2} \text {. }
$$ | x_{1}=2,x_{2}=\frac{4}{3} | 74 | 16 |
math | ## 12. Beautiful Lace
Two lacemakers need to weave a piece of lace. The first one would weave it alone in 8 days, and the second one in 13 days. How much time will they need for this work if they work together? | 4.95 | 56 | 4 |
math | 6. A point on the coordinate plane whose both horizontal and vertical coordinates are integers is called an integer point. The number of integer points in the region enclosed by the parabola $y=x^{2}+1$ and the line $2 x-y+81=0$ is $\qquad$ . | 988 | 64 | 3 |
math | # Problem 2. (2 points)
The sum of the sines of five angles from the interval $\left[0 ; \frac{\pi}{2}\right]$ is 3. What are the greatest and least integer values that the sum of their cosines can take?
# | 2;4 | 58 | 3 |
math | Example 1-17 5 girls and 7 boys are to form a group of 5 people, with the requirement that boy A and girl B cannot be in the group at the same time. How many schemes are there? | 672 | 48 | 3 |
math | G2.3 設 $a_{1} 、 a_{2} 、 a_{3} 、 a_{4} 、 a_{5} 、 a_{6}$ 為非負整數, 並霂足 $\left\{\begin{array}{c}a_{1}+2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}+6 a_{6}=26 \\ a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=5\end{array}\right.$ 。若 $c$ 為方程系統的解的數量, 求 $c$ 的值。
Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be non-negative integers and satisfy
$$
\left... | 5 | 315 | 1 |
math | 5. The force with which the airflow acts on the sail can be calculated using the formula
$F=\frac{A S \rho\left(v_{0}-v\right)^{2}}{2}$, where $A-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S$ $=4 \mathrm{M}^{2} ; \rho-$ is the density of air, $v_{0}$ - is the wind speed $v_{0}=4.8 \mathrm{~m... | 1.6\mathrm{M}/\mathrm{} | 229 | 11 |
math | Example 6. Solve the inequality
$$
3|x-1|+x^{2}>7
$$ | (-\infty;-1)\cup(2;+\infty) | 23 | 15 |
math | 4. The incident took place in 1968. A high school graduate returned from a written university entrance exam and told his family that he couldn't solve the following problem:
Several identical books and identical albums were bought. The books cost 10 rubles 56 kopecks, and the albums cost 56 kopecks. The number of book... | 8 | 231 | 1 |
math | 3. In a lake, a stream flows in, adding the same amount of water to the lake every day. 183 horses can drink all the water today, i.e., they would empty the lake in 24 hours, and 37 horses, starting from today, can drink all the water in 5 days. How many days, starting from today, would it take for one horse to drink a... | 365 | 89 | 3 |
math | ## 1. Jaja
Baka Mara has four hens. The first hen lays one egg every day. The second hen lays one egg every other day. The third hen lays one egg every third day. The fourth hen lays one egg every fourth day. If on January 1, 2023, all four hens laid one egg, how many eggs in total will Baka Mara's hens lay throughout... | 762 | 108 | 3 |
math | 1. For a right-angled triangle with a hypotenuse of 2009, if the two legs are also integers, then its area is $\qquad$ . | 432180 | 37 | 6 |
math | 1. A cyclist covered a certain distance in 1 hour and 24 minutes at a constant speed of $30 \mathrm{~km} / \mathrm{h}$. At what speed did he travel on the return trip if he traveled 12 minutes less? | 35\mathrm{~}/\mathrm{} | 57 | 10 |
math | 5. For what value of $z$ does the function $h(z)=$ $=\sqrt{1.44+0.8(z+0.3)^{2}}$ take its minimum value? | -0.3 | 44 | 4 |
math | Each of the thirty sixth-graders has one pen, one pencil, and one ruler. After their participation in the Olympiad, it turned out that 26 students lost a pen, 23 - a ruler, and 21 - a pencil. Find the smallest possible number of sixth-graders who lost all three items. | 10 | 68 | 2 |
math | 4. Let $a, b, c$ be the sides opposite to the interior angles $A, B, C$ of $\triangle A B C$, respectively, and the area $S=\frac{1}{2} c^{2}$. If $a b=\sqrt{2}$, then the maximum value of $a^{2}+b^{2}+c^{2}$ is . $\qquad$ | 4 | 87 | 1 |
math | Example 3. Find the integral $\int \frac{x}{x^{3}+1} d x$. | -\frac{1}{3}\ln|x+1|+\frac{1}{6}\ln(x^{2}-x+1)+\frac{1}{\sqrt{3}}\operatorname{arctg}\frac{2x-1}{\sqrt{3}}+C | 23 | 60 |
math | 4B. Solve the equation
$$
\log _{6}\left(3 \cdot 4^{-\frac{1}{x}}+2 \cdot 9^{-\frac{1}{x}}\right)+\frac{1}{x}=\log _{6} 5
$$ | -1 | 64 | 2 |
math | 15. The inequality $\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x$ holds for all $x \in \mathbf{R}$, find the range of real number $a$.
| \geqslant1or\leqslant-2 | 55 | 14 |
math | In the right triangle $ABC$ shown, $E$ and $D$ are the trisection points of the hypotenuse $AB$. If $CD=7$ and $CE=6$, what is the length of the hypotenuse $AB$? Express your answer in simplest radical form.
[asy]
pair A, B, C, D, E;
A=(0,2.9);
B=(2.1,0);
C=origin;
D=2/3*A+1/3*B;
E=1/3*A+2/3*B;
draw(A--B--C--cycle);
d... | 3\sqrt{17} | 182 | 7 |
math | 118. The random variable $X$ is given by the distribution function:
$$
F(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x \leq -c \\
\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c} & \text { if } & -c < x \leq c \\
1 & \text { if } & x > c
\end{array}\right.
$$
( arcsine law ).
Find the mathematical expectation of th... | 0 | 122 | 1 |
math | We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$.
(a) Determine the minimum possible number of bad chips.
(b) If we impose the ... | 1010 | 126 | 6 |
math | 2. If the polynomial $P=2 a^{2}-8 a b+17 b^{2}-16 a-4 b$ +2070, then the minimum value of $P$ is | 2002 | 45 | 4 |
math | 【4】The smallest natural number that leaves a remainder of 2 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 4 when divided by 7 is ( ). | 74 | 45 | 2 |
math | 3.2.7 * Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=3, a_{2}=8, a_{n+2}=2 a_{n+1}+2 a_{n}, n=1,2, \cdots$. Find the general term $a_{n}$ of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{2+\sqrt{3}}{2\sqrt{3}}(1+\sqrt{3})^{n}+\frac{\sqrt{3}-2}{2\sqrt{3}}(1-\sqrt{3})^{n} | 87 | 55 |
math | 1. Find the sum of the numbers $1-2+3-4+5-6+\ldots+2013-2014$ and $1+2-3+4-5+6-\ldots-2013+2014$. | 2 | 61 | 1 |
math | 6. The numbers $1,2, \ldots, 2016$ are written on a board. It is allowed to erase any two numbers and replace them with their arithmetic mean. How should one proceed to ensure that the number 1000 remains on the board? | 1000 | 60 | 4 |
math | 13.250. A brigade of lumberjacks was supposed to prepare $216 \mathrm{~m}^{3}$ of wood over several days according to the plan. For the first three days, the brigade met the daily planned quota, and then each day they prepared 8 m $^{3}$ more than planned, so by the day before the deadline, they had prepared 232 m $^{3... | 24\mathrm{~}^{3} | 113 | 10 |
math | 10.3. Given a trapezoid $A B C D$ and a point $M$ on the lateral side $A B$, such that $D M \perp A B$. It turns out that $M C=C D$. Find the length of the upper base $B C$, if $A D=d$. | \frac{}{2} | 69 | 6 |
math | Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation. | n = 1 \text{ and } n = 5 | 30 | 13 |
math | 12. Given that $x, y$ are real numbers, and $x+y=1$, find the maximum value of $\left(x^{3}+1\right)\left(y^{3}+1\right)$. | 4 | 48 | 1 |
math | 4B. Three cards are given. The number 19 is written on one, the number 97 on another, and a two-digit number on the third. If we add all the six-digit numbers obtained by arranging the cards in a row, we get the number 3232320. What number is written on the third card? | 44 | 74 | 2 |
math | 1. Vasya can get the number 100 using ten threes, parentheses, and arithmetic operation signs: $100=(33: 3-3: 3) \cdot(33: 3-3: 3)$. Improve his result: use fewer threes and get the number 100. (It is sufficient to provide one example). | 100=33\cdot3+3:3 | 82 | 13 |
math | 3. Given the real-coefficient equation $x^{3}+a x^{2}+b x+c=0$ whose three roots can serve as the eccentricities of an ellipse, a hyperbola, and a parabola, the range of $\frac{b}{a}$ is $\qquad$ | (-2,-\frac{1}{2}) | 65 | 10 |
math | 5.1. How many triangles with integer sides have a perimeter equal to 2017? (Triangles that differ only in the order of their sides, for example, 17, 1000, 1000 and 1000, 1000, 17, are counted as one triangle.) | 85008 | 74 | 5 |
math | 150. Find: $i^{28} ; i^{33} ; i^{135}$. | i^{28}=1;i^{33}=i;i^{135}=-i | 26 | 20 |
math | 3. If $x^{2}+y^{2}+2 x-4 y+5=0$, what is $x^{2000}+2000 y$? | 4001 | 42 | 4 |
math | Ana, Bia, Cátia, Diana, and Elaine work as street vendors selling sandwiches. Every day, they stop by Mr. Manoel's snack bar and take the same number of sandwiches to sell. One day, Mr. Manoel was sick and left a note explaining why he wasn't there, but asking each of them to take $\frac{1}{5}$ of the sandwiches. Ana a... | 75 | 225 | 2 |
math | We shuffle a 52-card French deck, then draw cards one by one from the deck until we find a black ace. On which draw is it most likely for the first black ace to appear? | 1 | 41 | 1 |
math | 3. A basketball championship has been played in a double round-robin system (each pair of teams play each other twice) and without ties (if the game ends in a tie, there are overtimes until one team wins). The winner of the game gets 2 points and the loser gets 1 point. At the end of the championship, the sum of the po... | 39 | 99 | 2 |
math | 2. Given $(a+b i)^{2}=3+4 i$, where $a, b \in \mathbf{R}, i$ is the imaginary unit, then the value of $a^{2}+b^{2}$ is $\qquad$ | 5 | 55 | 1 |
math | We randomly place points $A, B, C$, and $D$ on the circumference of a circle, independently of each other. What is the probability that the chords $AB$ and $CD$ intersect? | \frac{1}{3} | 43 | 7 |
math | Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already process... | 67 | 98 | 2 |
math | 7. (10 points) The last 8 digits of $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$ are | 87654321 | 55 | 8 |
math | 68. Find two numbers, the ratio of which is 3, and the ratio of the sum of their squares to their sum is 5. | 6,2 | 31 | 3 |
math | 23. Find the largest real number $m$ such that the inequality
$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+m \leqslant \frac{1+x}{1+y}+\frac{1+y}{1+z}+\frac{1+z}{1+x}$$
holds for any positive real numbers $x, y, z$ satisfying $x y z=x+y+z+2$. | \frac{3}{2} | 98 | 7 |
math | 1. Given the equation $\left|x^{2}-2 a x+b\right|=8$ has exactly three real roots, and they are the side lengths of a right triangle. Find the value of $a+b$.
(Bulgaria) | 264 | 50 | 3 |
math | 8.3. On the side $AC$ of triangle $ABC$, a point $M$ is taken. It turns out that $AM = BM + MC$ and $\angle BMA = \angle MBC + \angle BAC$. Find $\angle BMA$. | 60 | 56 | 2 |
math | 1A. Solve the equation in $\mathbb{R}$
$$
\left(x^{2}-3\right)^{3}-(4 x+6)^{3}+6^{3}=18(4 x+6)\left(3-x^{2}\right)
$$ | x_{1}=2+\sqrt{7},x_{2}=2-\sqrt{7},x_{3}=-3 | 61 | 26 |
math | 12.10*. On a circle with diameter $A B$, points $C$ and $D$ are taken. The line $C D$ and the tangent to the circle at point $B$ intersect at point $X$. Express $B X$ in terms of the radius of the circle $R$ and the angles $\varphi=\angle B A C$ and $\psi=\angle B A D$.
## §2. The Law of Cosines | BX=2R\sin\varphi\sin\psi/\sin|\varphi\\psi| | 96 | 21 |
math | ## Task 2 - 090612
During the holidays, Klaus was in the countryside. From his observations, the following joke problem emerged:
$$
1 \frac{1}{2} \text { chickens lay in } 1 \frac{1}{2} \text { days } 1 \frac{1}{2} \text { eggs }
$$
Determine the total number of eggs that 7 chickens would lay in 6 days at the same l... | 28 | 103 | 2 |
math | [ $\left.\begin{array}{c}\text { Equations in integers } \\ \text { [Examples and counterexamples. Constructions }\end{array}\right]$
Do there exist natural numbers $a, b, c, d$ such that $a / b + c / d = 1, \quad a / d + c / b = 2008$? | =2009\cdot(2008\cdot2009-1),b=2008\cdot2010\cdot(2008\cdot2009-1),=2008\cdot2009-1,=2008\cdot2010 | 82 | 74 |
math | 360. The sums of the terms of each of the arithmetic progressions, having $n$ terms, are equal to $n^{2}+p n$ and $3 n^{2}-2 n$. Find the condition under which the $n$-th terms of these progressions will be equal. | 4(n-1) | 65 | 5 |
math | 13.174. In four boxes, there is tea. When 9 kg were taken out of each box, the total that remained in all of them together was as much as there was in each one. How much tea was in each box? | 12\mathrm{} | 53 | 5 |
math | 1. Find all integers $n$ for which $n^{4}-3 n^{2}+9$ is a prime number. | n\in{-2,-1,1,2} | 28 | 12 |
math | ## Condition of the problem
Compose the equation of the tangent to the given curve at the point with abscissa $x_{0}$.
$$
y=\frac{x^{29}+6}{x^{4}+1}, x_{0}=1
$$ | 7.5x-4 | 56 | 6 |
math | 4.1. Mom baked four raisin buns for breakfast for her two sons. $V$ In the first three buns, she put 7, 7, 23 raisins, and some more in the fourth. It turned out that the boys ate an equal number of raisins and did not divide any bun into parts. How many raisins could Mom have put in the fourth bun? List all the option... | 9,23,37 | 89 | 7 |
math | G2.3 If $n \neq 0$ and $s=\left(\frac{20}{2^{2 n+4}+2^{2 n+2}}\right)^{\frac{1}{n}}$, find the value of $s$. | \frac{1}{4} | 57 | 7 |
math | 【Question 23】
How many consecutive "0"s are at the end of the product $5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015$? | 398 | 58 | 3 |
math | 3. If $n$ is a positive integer, and $n^{2}+9 n+98$ is exactly equal to the product of two consecutive positive integers, then all values of $n$ are $\qquad$ | 34,14,7 | 48 | 7 |
math | Task 3. A landscaping team worked on a large and a small football field, with the area of the large field being twice the area of the small field. In the part of the team that worked on the large field, there were 6 more workers than in the part that worked on the small field. When the landscaping of the large field wa... | 16 | 102 | 2 |
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 | Problem 10.7. At one meal, Karlson can eat no more than 5 kg of jam. If he opens a new jar of jam, he must eat it completely during this meal. (Karlson will not open a new jar if he has to eat more than 5 kg of jam together with what he has just eaten.)
Little Boy has several jars of raspberry jam weighing a total of ... | 12 | 120 | 2 |
math | ## 255. Math Puzzle $8 / 86$
Felix rides his bicycle on a straight route between two towns that are ten kilometers apart, and the round trip takes him one hour.
The next day, a strong wind blows, so he rides 5 km/h faster on the way there and 5 km/h slower on the way back compared to his average speed from the previo... | 4 | 95 | 1 |
math | 10. Let $S$ be the area of a triangle inscribed in a circle of radius 1. Then the minimum value of $4 S+\frac{9}{S}$ is $\qquad$ . | 7 \sqrt{3} | 44 | 6 |
math | In triangle $A B C$, angle $C$ is twice angle $A$ and $b=2 a$. Find the angles of this triangle.
# | 30,90,60 | 32 | 8 |
math | 5. Positive real numbers $a, b, c$ form a geometric sequence $(q \neq 1), \log _{a} b, \log _{b} c, \log _{c} a$ form an arithmetic sequence. Then the common difference $d=$ | -\frac{3}{2} | 60 | 7 |
math | 2. Let
$$
\begin{array}{l}
f(x)=x^{2}-53 x+196+\left|x^{2}-53 x+196\right| \\
\text { then } f(1)+f(2)+\cdots+f(50)=
\end{array}
$$ | 660 | 72 | 3 |
math | 4. The numbers $a_{1}, a_{2}, \ldots, a_{20}$ satisfy the conditions:
$$
\begin{aligned}
& a_{1} \geq a_{2} \geq \ldots \geq a_{20} \geq 0 \\
& a_{1}+a_{2}=20 \\
& a_{3}+a_{4}+\ldots+a_{20} \leq 20
\end{aligned}
$$
What is the maximum value of the expression:
$$
a_{1}^{2}+a_{2}^{2}+\ldots+a_{20}^{2}
$$
For which va... | 400 | 177 | 3 |
math | 13. Given $f(x)=\frac{x}{1+x}$. Find the value of the following expression:
$$
\begin{array}{l}
f\left(\frac{1}{2004}\right)+f\left(\frac{1}{2003}\right)+\cdots+f\left(\frac{1}{2}\right)+f(1)+ \\
f(0)+f(1)+f(2)+\cdots+f(2003)+f(2004) .
\end{array}
$$ | 2004 | 119 | 4 |
math | In a triangle, the base is $4 \mathrm{~cm}$ larger than the corresponding height. If we increase both the base and the height by $12 \mathrm{~cm}$, we get a triangle whose area is five times the area of the original triangle. What are the base and height of the original triangle? | 12 | 68 | 2 |
math | 15. [9] A cat is going up a stairwell with ten stairs. However, instead of walking up the stairs one at a time, the cat jumps, going either two or three stairs up at each step (though if necessary, it will just walk the last step). How many different ways can the cat go from the bottom to the top? | 12 | 73 | 2 |
math | Example 4. Find $\int \sqrt{a^{2}+x^{2}} d x$. | \frac{1}{2}x\sqrt{^{2}+x^{2}}+\frac{^{2}}{2}\ln|x+\sqrt{^{2}+x^{2}}|+C | 22 | 44 |
math | Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the ... | 486 | 142 | 3 |
math | Let's calculate the value of the expression under a) and simplify the expression under b) as much as possible;
a) $\left(1-\cos 15^{\circ}\right)\left(1+\sin 75^{\circ}\right)+\cos 75^{\circ} \cos 15^{\circ} \operatorname{cotg} 15^{\circ}$
b) $\sin \left(45^{\circ}-\alpha\right)-\cos \left(30^{\circ}+\alpha\right)+\sin... | 1 | 182 | 1 |
math | 4. [4 points] Solve the system of equations
$$
\left\{\begin{array}{l}
3 x^{2}+3 y^{2}-x^{2} y^{2}=3 \\
x^{4}+y^{4}-x^{2} y^{2}=31
\end{array}\right.
$$ | (\sqrt{5};\\sqrt{6}),(-\sqrt{5};\\sqrt{6}),(\sqrt{6};\\sqrt{5}),(-\sqrt{6};\\sqrt{5}) | 73 | 42 |
math | 1. (5 points) Find the degree measure of the angle
$$
\delta=\arccos \left(\left(\sin 3271^{\circ}+\sin 3272^{\circ}+\cdots+\sin 6871^{\circ}\right)^{\cos } 3240^{\circ}+\cos 3241^{\circ}+\cdots+\cos 6840^{\circ}\right)
$$ | 59 | 104 | 2 |
math | Given the following system of equations:
$$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
| \frac{341}{40} | 101 | 10 |
math | ## Task Condition
Find the derivative.
$y=\arccos \frac{x^{2}-4}{\sqrt{x^{4}+16}}$ | -\frac{2\sqrt{2}(4+x^{2})}{x^{4}+16} | 33 | 23 |
math | 7.073. $9^{x^{2}-1}-36 \cdot 3^{x^{2}-3}+3=0$. | -\sqrt{2};-1;1;\sqrt{2} | 33 | 14 |
math | Example 6 Solve the equation
\[
\begin{array}{l}
a^{4} \cdot \frac{(x-b)(x-c)}{(a-b)(a-c)}+b^{4} \cdot \frac{(x-c)(x-a)}{(b-c)} \\
+\epsilon^{4} \cdot \frac{(x-a)(x-b)}{(c-a)(c-b)}=x^{4} .
\end{array}
\] | x_{1}=a, x_{2}=b, x_{3}=c, x_{4}=-(a+b+c) | 95 | 27 |
math | 4. Given $P_{1}, P_{2}, \cdots, P_{100}$ as 100 points on a plane, satisfying that no three points are collinear. For any three of these points, if their indices are in increasing order and they form a clockwise orientation, then the triangle with these three points as vertices is called "clockwise". Question: Is it po... | 2017 | 100 | 4 |
math | 68. When Vasya Verkhoglyadkin was given this problem: "Two tourists set out from $A$ to $B$ simultaneously. The first tourist walked half of the total time at a speed of $5 \mathrm{km} /$ h, and the remaining half of the time at a speed of 4 km/h. The second tourist walked the first half of the distance at a speed of 5... | \frac{2a}{9}<\frac{9a}{40} | 137 | 17 |
math | 13. Given the function $f(x)$ $=\sin 2 x \cdot \tan x+\cos \left(2 x-\frac{\pi}{3}\right)-1$, find the intervals where the function $f(x)$ is monotonically decreasing. | [k\pi+\frac{\pi}{3},k\pi+\frac{\pi}{2}),(k\pi+\frac{\pi}{2},k\pi+\frac{5\pi}{6}](k\in{Z}) | 55 | 50 |
math | Example 13 (1993 Shanghai Competition Question) Given real numbers $x_{1}, x_{2}, x_{3}$ satisfy the equations $x_{1}+\frac{1}{2} x_{2}+\frac{1}{3} x_{3}=$ 1 and $x_{1}^{2}+\frac{1}{2} x_{2}^{2}+\frac{1}{3} x_{3}^{2}=3$, then what is the minimum value of $x_{3}$? | -\frac{21}{11} | 114 | 9 |
math | [ Generating functions Special polynomials (p) Find the generating functions of the Fibonacci polynomial sequence $F(x, z)=F_{0}(x)+F_{1}(x) z+F_{2}(x) z^{2}$ $+\ldots+F_{n}(x) z^{n}+\ldots$
and the Lucas polynomial sequence $L(x, z)=L_{0}(x)+L_{1}(x) z+L_{2}(x) z^{2}+\ldots+L_{n}(x) z^{n}+\ldots$
Definitions of Fibo... | F(x,z)=z(1-xz-z^{2})^{-1},L(x,z)=(2-xz)(1-xz-z^{2})^{-1} | 134 | 34 |
math | 6. We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ fields. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board such that the cut goes along the edge $k$. Calculate the last digit of the sum we get when we add all $N(k)$,... | 0 | 105 | 1 |
math | Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as:
$x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$.
$a)$ Prove that the sequence is converging.
$b)$ Find $\lim_{n\rightarrow \infty}{x_n}$. | a^{\frac{m+1}{m+2}} | 103 | 13 |
math | Let's find all right-angled triangles whose sides are integers, and when 6 is added to the hypotenuse, we get the sum of the legs. | (7,24,25),(8,15,17),(9,12,15) | 33 | 25 |
math |
4. Find all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ such that for all non-zero numbers $x, y$,
$$
x \cdot f(x y)+f(-y)=x \cdot f(x) .
$$
(Pavel Calábek)
| f(x)=(1+\frac{1}{x}) | 72 | 11 |
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 |
Problem 10.1. Find the least natural number $a$ such that the equation $\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0$ has a root.
| 6 | 84 | 1 |
math | 5. Is the number
$$
\frac{1}{2+\sqrt{2}}+\frac{1}{3 \sqrt{2}+2 \sqrt{3}}+\frac{1}{4 \sqrt{3}+3 \sqrt{4}}+\ldots+\frac{1}{100 \sqrt{99}+99 \sqrt{100}}
$$
rational? Justify! | \frac{9}{10} | 90 | 8 |
math | Find $x+y+z$ when $$a_1x+a_2y+a_3z= a$$$$b_1x+b_2y+b_3z=b$$$$c_1x+c_2y+c_3z=c$$ Given that $$a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9$$$$a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17$$$$a_1\left(bc_3-b_3c\right)-a\left(b_1c_3-b_3c_1... | \frac{16}{9} | 292 | 8 |
math | If $x$ is a positive integer, then the
$$
10 x^{2}+5 x, \quad 8 x^{2}+4 x+1, \quad 6 x^{2}+7 x+1
$$
are the lengths of the sides of a triangle, all of whose angles are less than $90^{\circ}$, and whose area is an integer. | 2x(x+1)(2x+1)(6x+1) | 86 | 16 |
math | Four fathers wanted to sponsor a skiing trip for their children.
The first promised: "I will give 11500 CZK,"
the second promised: "I will give a third of what you others give in total,"
the third promised: "I will give a quarter of what you others give in total,"
the fourth promised: "And I will give a fifth of wh... | 7500,6000,5000 | 102 | 14 |
math | Example 5 Given $a>0, b>0, a+2b=1$, find the minimum value of $t=\frac{1}{a}+\frac{1}{b}$. | 3+2\sqrt{2} | 42 | 8 |
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