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 | 9. (16 points) Find the minimum value of $M$ such that the function
$$
f(x)=x^{2}-2 x+1
$$
for any partition of the interval $[0,4]$
$$
0=x_{0}<x_{1}<\cdots<x_{n-1}<x_{n}=4,
$$
satisfies
$$
\sum_{i=1}^{n}\left|f\left(x_{i}\right)-f\left(x_{i-1}\right)\right| \leqslant M .
$$ | 10 | 125 | 2 |
math | 4. (6 points) The older brother and the younger brother each bought several apples. The older brother said to the younger brother: “If I give you one apple, we will have the same number of apples.” The younger brother thought for a moment and said to the older brother: “If I give you one apple, your number of apples wi... | 12 | 91 | 2 |
math | B3. Find all real solutions of the equation $\left(x^{2}+x+3\right)^{2}+3\left(x^{2}+x-1\right)=28$. | x_{1}=-2,x_{2}=1 | 44 | 11 |
math | 116 The four vertices of a regular tetrahedron are $A, B, C, D$, with edge length $1, P \in AB, Q \in CD$, then the range of the distance between points $P, Q$ is $\qquad$ . | [\frac{\sqrt{2}}{2},1] | 58 | 12 |
math | 8. The product of $5.425 \times 0.63$ written in decimal form is | 3.4180 | 24 | 6 |
math | 3 . If $\mathrm{iog}_{2}\left(\log _{8} x\right)=\log _{8}\left(\log _{2} x\right)$, find $\left(\log _{2} x\right)^{2}$. | 27 | 56 | 2 |
math | Problem 1. A six-digit number ends with the digit 5. If this digit is moved to the beginning of the number, i.e., it is deleted from the end and appended to the start of the number, then the newly obtained number will be 4 times larger than the original number. Determine the original number. | 128205 | 66 | 6 |
math | G1.3 Given that $x$ and $y$ are non-zero real numbers satisfying the equations $\frac{\sqrt{x}}{\sqrt{y}}-\frac{\sqrt{y}}{\sqrt{x}}=\frac{7}{12}$ and $x-y=7$. If $w=x+y$, find the value of $w$. | 25 | 71 | 2 |
math | 2. Let $a$ and $b$ be real numbers such that $a>b>0$ and $a^{2}+b^{2}=6ab$. Determine the value of the expression $\frac{a+b}{a-b}$. | \sqrt{2} | 51 | 5 |
math | (4) $\sin 7.5^{\circ}+\cos 7.5^{\circ}=$ | \frac{\sqrt{4+\sqrt{6}-\sqrt{2}}}{2} | 24 | 19 |
math | ## Task 10/90
Which triples $(x ; y ; z)$ of prime numbers satisfy the equation $x^{3}-y^{3}-z^{3}=6 y(y+2)$? | (x;x-2;2) | 44 | 7 |
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 | Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Ga... | 19152 | 94 | 5 |
math | 7. Due to traffic congestion at the city exit, the intercity bus traveled the first third of the journey one and a half times slower than the calculated time. Will the bus be able to arrive at the destination on time if it increases its speed by a third for the remaining part of the journey? | Yes | 61 | 1 |
math | # 1. Task 1
What number should the asterisk be replaced with so that the equation $(2 x-7)^{2}+(5 x-*)^{2}=0$ has a root?
# | 17.5 | 45 | 4 |
math | Example 3 Calculate $\tan \frac{\pi}{7} \cdot \tan \frac{2 \pi}{7} \cdot \tan \frac{3 \pi}{7}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \sqrt{7} | 65 | 5 |
math | Given that the sum of 3 distinct natural numbers is 55, and the sum of any two of these numbers is a perfect square, then these three natural numbers are | 6,19,30 | 35 | 7 |
math | 4. The board has the number 5555 written in an even base $r$ ($r \geqslant 18$). Petya found out that the $r$-ary representation of $x^{2}$ is an eight-digit palindrome, where the difference between the fourth and third digits is 2. (A palindrome is a number that reads the same from left to right and from right to left... | 24 | 99 | 2 |
math | 22nd CanMO 1990 Problem 2 n(n + 1)/2 distinct numbers are arranged at random into n rows. The first row has 1 number, the second has 2 numbers, the third has 3 numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers. Solution | \frac{2^n}{(n+1)!} | 79 | 12 |
math | ## Task 1 - 290511
On April 30, Kerstin receives money as birthday gifts from several relatives. She now has exactly 35 marks in her piggy bank and resolves to diligently collect recyclables in the following months so that she can put exactly 5 marks into the piggy bank at the end of each month.
At the end of which m... | August | 99 | 1 |
math | ## Task 4 - 070914
Four teams $A, B, C$, and $D$ are participating in a football tournament. Each team plays exactly one match against each of the others, and teams are awarded 2, 1, or 0 "bonus points" for a win, draw, or loss, respectively.
The day after the tournament concludes, Peter hears the end of a radio repo... | 4 | 182 | 1 |
math | $n$ is composite. $1<a_1<a_2<...<a_k<n$ - all divisors of $n$. It is known, that $a_1+1,...,a_k+1$ are all divisors for some $m$ (except $1,m$). Find all such $n$. | n \in \{4, 8\} | 68 | 11 |
math | Question 224, Given a positive integer $n(n \geq 2)$, choose $m$ different numbers from $1, 2, \ldots, 3n$. Among these, there must be four pairwise distinct numbers $a, b, c, d$, satisfying $a=b+c+d$. Find the minimum value of $m$.
---
The translation maintains the original format and line breaks as requested. | 2n+2 | 89 | 4 |
math | Let's determine the sum of the first $n$ terms of the following sequence:
$$
1+7+11+21+\cdots+\left[n(n+1)+(-1)^{n}\right]
$$ | S_{n}=\frac{n(n+1)(n+2)}{3}+\frac{(-1)^{n}-1}{2} | 47 | 31 |
math | Example 1. $\mathrm{AC}$ and $\mathrm{CE}$ are two diagonals of a regular hexagon $\mathrm{ABCDEF}$, and points $\mathrm{M}$ and $\mathrm{N}$ internally divide $\mathrm{AC}$ and $\mathrm{CE}$, respectively, such that $\mathrm{M}: \mathrm{AC} = \mathrm{CN}: \mathrm{CE} = \mathrm{r}$. If points $\mathrm{B}$, $\mathrm{M}$... | \frac{\sqrt{3}}{3} | 132 | 10 |
math | 9.20 Find the largest binomial coefficient in the expansion of $\left(n+\frac{1}{n}\right)^{n}$, if the product of the fourth term from the beginning and the fourth term from the end is 14400. | 252 | 55 | 3 |
math | \section*{Exercise 3 - 071013}
A mathematician has lost the key to the compartment of a luggage machine. However, he still remembered the number of the compartment, which was a three-digit number divisible by 13, and that the middle digit was the arithmetic mean of the other two digits. The compartment could be quickl... | 234,468,741,975 | 89 | 15 |
math | $12 \cdot 34$ Find all square numbers $s_{1}, s_{2}$ that satisfy $s_{1}-s_{2}=1989$.
(30th International Mathematical Olympiad Shortlist, 1989) | (995^2,994^2),(333^2,330^2),(115^2,106^2),(83^2,70^2),(67^2,50^2),(45^2,6^2) | 56 | 66 |
math | 2. The solution set of the equation $\log _{5}\left(3^{x}+4^{x}\right)=\log _{4}\left(5^{x}-3^{x}\right)$ is | 2 | 46 | 1 |
math | 4.9 $n$ is the smallest positive integer that satisfies the following conditions:
(1) $n$ is a multiple of 75.
(2) $n$ has exactly 75 positive divisors (including 1 and itself).
Find $\frac{n}{75}$.
(8th American Invitational Mathematics Examination, 1990) | 432 | 76 | 3 |
math | 1. Given the sets
$$
A=\{x \mid 1 \leqslant x \leqslant 2\}, B=\left\{x \mid x^{2}-a x+4 \geqslant 0\right\} \text {. }
$$
If $A \subseteq B$, then the range of real number $a$ is
$\qquad$ | a \leqslant 4 | 86 | 8 |
math | The thief Rumcajs is teaching Cipísek to write numbers. They started writing from one and wrote consecutive natural numbers. Cipísek didn't find it very entertaining and pleaded for them to stop. Rumcajs eventually gave in and promised that he would stop writing as soon as Cipísek had a total of 35 zeros in the written... | 204 | 98 | 3 |
math | 5. (5 points) From the ten digits $0 \sim 9$, the pair of different numbers with the largest product is $\qquad$ multiplied, the pair with the smallest sum is $\qquad$ and $\qquad$ added, and the pairs that add up to 10 are $\qquad$ pairs. | 9,8,0,1,4 | 68 | 9 |
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 | Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$.
Find $f(1990, 31).$ | \frac{30! \cdot 1989!}{2020!} | 135 | 21 |
math | Problem 3. Six numbers are written in a row. It is known that among them there is a one and any three adjacent numbers have the same arithmetic mean. Find the maximum value of the geometric mean of any three adjacent numbers in this row, if the arithmetic mean of all 6 numbers is A. | \sqrt[3]{(3A-1)^{2}/4} | 62 | 16 |
math | ## Task 6B - 221246B
In the investigation of frequency distributions in mathematical statistics, functions appear that are defined for finitely many natural numbers and are required to satisfy so-called functional equations (equations between different function values).
An example of this is the following:
Given a n... | f(x)=\binom{n}{x}p^{x}(1-p)^{n-x} | 267 | 21 |
math | The lateral edge of a regular quadrilateral pyramid is $b$, and the plane angle at the vertex is $\alpha$. Find the radius of the sphere circumscribed about the pyramid.
# | \frac{b}{2\sqrt{\cos\alpha}} | 38 | 13 |
math | 8. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in[a, a+2]$, we have $f(x+a) \geqslant 2 f(x)$, then the range of the real number $a$ is . $\qquad$ | [\sqrt{2},+\infty) | 89 | 9 |
math | 3. If the real numbers $a, b, c, d, e$ satisfy the conditions
$$
\begin{array}{l}
a+b+c+d+e=8, \\
a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16 .
\end{array}
$$
Determine the maximum value of $e$. | \frac{16}{5} | 83 | 8 |
math | 1. Solve the equation $x^{\log _{2}\left(0.25 x^{3}\right)}=512 x^{4}$. | \frac{1}{2},8 | 35 | 8 |
math | 1. Given the inequality $a x+3 \geqslant 0$ has positive integer solutions of $1,2,3$. Then the range of values for $a$ is $\qquad$ . | -1 \leqslant a<-\frac{3}{4} | 45 | 16 |
math | Example 4 Let $n$ be a given positive integer. Try to find non-negative integers $k, l$, satisfying $k+l \neq 0$, and $k+l \neq n$, such that
$$
s=\frac{k}{k+l}+\frac{n-k}{n-(k+l)}
$$
takes the maximum value. | 2 | 73 | 1 |
math | Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$. | 99 | 87 | 2 |
math | 28. From a box containing $n$ white and $m$ black balls $(n \geqslant m)$, all balls are drawn one by one at random. Find the probability of the following events:
(a) there will be a moment when the number of white and black balls in the sequence of drawn balls will be equal
(b) at any moment, the number of white bal... | \frac{n-+1}{n+1} | 98 | 11 |
math | 63rd Putnam 2002 Problem B1 An event is a hit or a miss. The first event is a hit, the second is a miss. Thereafter the probability of a hit equals the proportion of hits in the previous trials (so, for example, the probability of a hit in the third trial is 1/2). What is the probability of exactly 50 hits in the first... | \frac{1}{99} | 92 | 8 |
math | 5. At $17-00$ the speed of the racing car was 30 km/h. Every 5 minutes thereafter, the speed increased by 6 km/h. Determine the distance traveled by the car from $17-00$ to $20-00$ of the same day. | 405 | 66 | 3 |
math | $PS$ is a line segment of length $4$ and $O$ is the midpoint of $PS$. A semicircular arc is drawn with $PS$ as diameter. Let $X$ be the midpoint of this arc. $Q$ and $R$ are points on the arc $PXS$ such that $QR$ is parallel to $PS$ and the semicircular arc drawn with $QR$ as diameter is tangent to $PS$. What is the ar... | 2\pi - 2 | 116 | 6 |
math | 7. The maximum value of the projection area of a rectangular prism with edge lengths $6,6,8$ on plane $\alpha$ is $\qquad$ | 12\sqrt{41} | 33 | 8 |
math | 128. Digits and Squares. One of Professor Rackbrain's small Christmas puzzles reads as follows: what are the smallest and largest squares containing all ten digits from 0 to 9, with each digit appearing only once? | 10267538499814072356 | 48 | 20 |
math | Suppose that $f(x)=x^{4}-x^{3}-1$ and $g(x)=x^{8}-x^{6}-2 x^{4}+1$. If $g(x)=f(x) h(x)$, determine the polynomial function $h(x)$. | (x)=x^{4}+x^{3}-1 | 59 | 12 |
math | 755. "Forty mice were walking, carrying forty groats,
Two poorer mice carried two groats each, Many mice - with no groats at all. The bigger ones carried seven each. And the rest carried four each.
How many mice walked without groats?" | 32 | 57 | 2 |
math | 3-4. Find the integer $a$, for which
$$
(x-a)(x-10)+1
$$
can be factored into the product $(x+b)(x+c)$ of two factors with integer $b$ and $c$. | =8or=12 | 53 | 6 |
math | Given the quadratic equation $x^{2}+p x+q$ with roots $x_{1}$ and $x_{2}$. Construct the quadratic equation whose roots are $x_{1}+1$ and $x_{2}+1$. | x^{2}+(p-2)x+q-p+1=0 | 53 | 16 |
math | 3.339. a) $\cos 36^{\circ}=\frac{\sqrt{5}+1}{4} ;$ b) $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$. | \cos36=\frac{\sqrt{5}+1}{4};\sin18=\frac{\sqrt{5}-1}{4} | 54 | 31 |
math | Křemílek and Vochomůrka found a treasure chest. Each of them took silver coins into one pocket and gold coins into the other. Křemílek had a hole in his right pocket and lost half of his gold coins on the way. Vochomůrka had a hole in his left pocket and lost half of his silver coins on the way home. At home, Vochomůrk... | Křemílektook12gold24silver,whileVochomůrkatook18gold24silver | 165 | 27 |
math | Tickets for the football game are \$10 for students and \$15 for non-students. If 3000 fans attend and pay \$36250, how many students went? | 1750 | 42 | 4 |
math | 7. The line $x-2 y-1=0$ intersects the parabola $y^{2}=4 x$ at points $A$ and $B$, and $C$ is a point on the parabola such that $\angle A C B=90^{\circ}$. Then the coordinates of point $C$ are $\qquad$. | (1,-2)or(9,-6) | 76 | 11 |
math | Example 5 A $98 \times 98$ chessboard is displayed on a computer screen, with its squares colored like a chessboard. You are allowed to select any rectangle with the mouse (the sides of the rectangle must lie on the grid lines), and then click the mouse button, which will change the color of each square in the rectangl... | 98 | 115 | 2 |
math | Find all the three-digit numbers $\overline{abc}$ such that the $6003$-digit number $\overline{abcabc\ldots abc}$ is divisible by $91$. | 91, 182, 273, 364, 455, 546, 637, 728, 819, 910 | 42 | 47 |
math | Example 6 Given a complex number $z$ satisfying $|z|=1$, try to find the maximum value of $u=\left|z^{3}-3 z+2\right|$.
(2000 Jilin Province High School Competition) | 3\sqrt{3} | 54 | 6 |
math | 7.98 Two people are playing a math game. Player A selects a set of 1-digit integers $x_{1}, x_{2}, \cdots, x_{n}$ as the answer, which can be positive or negative. Player B can ask: what is the sum $a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}$? Here, $a_{1}, a_{2}, \cdots, a_{n}$ can be any set of numbers. Determine the... | 1 | 131 | 1 |
math | 10. Simplify $\left(\cos 42^{\circ}+\cos 102^{\circ}+\cos 114^{\circ}+\cos 174^{\circ}\right)^{2}$ into a rational number.
10. 把 $\left(\cos 42^{\circ}+\cos 102^{\circ}+\cos 114^{\circ}+\cos 174^{\circ}\right)^{2}$ 化簡成一個有理數。 | \frac{3}{4} | 116 | 7 |
math | Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers and $a^2+\sqrt{2017-b^2}$ is a perfect square. | 67 | 46 | 2 |
math | 7. Find non-constant polynomials with integer coefficients $P(x)$ and $Q(x)$, satisfying
$$
P\left(Q^{2}(x)\right)=P(x) Q^{2}(x) \text {. }
$$ | \begin{pmatrix}P(x)=^{2},Q(x)=x;\\P(x)=^{2},Q(x)=-x;\\P(x)=^{2}-4,Q(x)=x-2;\\P(x)=^{2}-4,Q(x)=-x+20\end{pmatrix} | 50 | 67 |
math | Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$
$ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$.
$ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit.
$ (c)$ Determine $ \displaystyle\lim_{n\t... | 1 | 151 | 3 |
math | 1. Consider all the subsets of $\{1,2,3, \ldots, 2018,2019\}$ having exactly 100 elements. For each subset, take the greatest element. Find the average of all these greatest elements. | 2000 | 57 | 4 |
math | 4. Let $\left\{a_{n}\right\}$ be an arithmetic sequence with common difference $d(d \neq 0)$, and the sum of the first $n$ terms be $S_{n}$. If the sequence $\left\{\sqrt{8 S_{n}+2 n}\right\}$ is also an arithmetic sequence with common difference $d$, then the general term of the sequence $\left\{a_{n}\right\}$ is $a_{... | 4n-\frac{9}{4} | 109 | 9 |
math | The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country. | 7 | 56 | 1 |
math | Example 4 Find all real numbers $k$ such that $a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)$, for all $a, b, c, d \in[-1,+\infty)$: (2004 China Western Mathematical Olympiad) | \frac{3}{4} | 77 | 7 |
math | Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$. | \frac{\sqrt{21}}{2} | 65 | 11 |
math | 10. (20 points) Given $\angle A, \angle B, \angle C$ are the three interior angles of $\triangle ABC$, and the vector
$$
\boldsymbol{\alpha}=\left(\cos \frac{A-B}{2}, \sqrt{3} \sin \frac{A+B}{2}\right),|\boldsymbol{\alpha}|=\sqrt{2}.
$$
If when $\angle C$ is maximized, there exists a moving point $M$ such that $|\over... | \frac{2 \sqrt{3}+\sqrt{2}}{4} | 163 | 17 |
math | 7.248. $\log _{2} x \cdot \log _{3} x=\log _{3}\left(x^{3}\right)+\log _{2}\left(x^{2}\right)-6$. | 8;9 | 50 | 3 |
math | II. (50 points) For all $a, b, c \in \mathbf{R}^{+}$, find the minimum value of $f(a, b, c)=\frac{a}{\sqrt{a^{2}+8 b c}}+\frac{b}{\sqrt{b^{2}+8 a c}}+\frac{c}{\sqrt{c^{2}+8 a b}}$. | 1 | 92 | 1 |
math | 376. Given a triangle $A B C$, the angles of which are $\alpha, \beta$ and $\gamma$. Triangle $D E F$ is circumscribed around triangle $A B C$ such that vertices $A, B$ and $C$ are located on sides $E F$, $F D$ and $D E$ respectively, and $\angle E C A = \angle D B C = \angle F A B = \varphi$. Determine the value of th... | \operatorname{tg}\varphi_{0}=\operatorname{ctg}\alpha+\operatorname{ctg}\beta+\operatorname{ctg}\gamma | 124 | 35 |
math | Example 6. $f\left(\frac{2}{x}+1\right)=\lg x$, find $f(x)$. | f(x)=\lg 2-\lg (x-1) | 30 | 14 |
math | 7.4. How many six-digit natural numbers exist, each of which has adjacent digits with different parity | 28125 | 21 | 5 |
math | Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$. Determine the area of triangle $ AMC$. | \frac{1}{2} | 38 | 7 |
math | 7. (10 points) Households A, B, and C plan to subscribe to newspapers, with 5 different newspapers available for selection. It is known that each household subscribes to two different newspapers, and any two households have exactly one newspaper in common. How many different subscription methods are there for the three... | 180 | 66 | 3 |
math | Find all pairs $(p, q)$ of prime numbers with $p>q$ for which the number $$ \frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1} $$ is an integer. (Japan) Answer: The only such pair is $(3,2)$. | (3,2) | 85 | 5 |
math | Example 12. For $>0, \frac{1}{b}-\frac{1}{a}=1$. Compare the sizes of the following four numbers: $\sqrt{1+a}, \frac{1}{1-\frac{b}{2}}, 1+\frac{a}{2}$,
$$
\frac{1}{\sqrt{1-b}}
$$ | \frac{1}{1-\frac{b}{2}}<\frac{1}{\sqrt{1-b}}=\sqrt{1+a}<1+\frac{a}{2} | 78 | 39 |
math | 3. Find all possible values of the expression
$$
\frac{x+y}{x^{2}+y^{2}}
$$
where $x$ and $y$ are any real numbers satisfying the condition $x+y \geq 1$. | (0,2] | 52 | 5 |
math | 7.281. $\left\{\begin{array}{l}y \cdot x^{\log _{y} x}=x^{2.5} \\ \log _{3} y \cdot \log _{y}(y-2 x)=1 .\end{array}\right.$ | (3;9) | 65 | 5 |
math | What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$ | 35\% | 36 | 4 |
math | 4. (42nd IMO Shortlist) Let $\triangle ABC$ be an acute-angled triangle. Construct isosceles triangles $\triangle DAC$, $\triangle EAB$, and $\triangle FBC$ outside $\triangle ABC$ such that $DA = DC$, $EA = EB$, $FB = FC$, $\angle ADC = 2 \angle BAC$, $\angle BEA = 2 \angle ABC$, and $\angle CFB = 2 \angle ACB$. Let $... | 4 | 175 | 1 |
math | Example 1 The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., 1<\right.$ $x \in \mathbf{N}\}$ is $\qquad$
(1996, National High School Mathematics Competition) | 2^{90}-1 | 86 | 6 |
math | Example 5 Given $x, y, z \in \mathbf{R}_{+} \cup\{0\}$, and $x+y+z=\frac{1}{2}$. Find
$$\frac{\sqrt{x}}{4 x+1}+\frac{\sqrt{y}}{4 y+1}+\frac{\sqrt{z}}{4 z+1}$$
the maximum value. | \frac{3}{5} \sqrt{\frac{3}{2}} | 88 | 16 |
math | 4. Determine all values of the real parameter \( a \) for which the system of equations
$$
\begin{aligned}
(x+y)^{2} & =12 \\
x^{2}+y^{2} & =2(a+1)
\end{aligned}
$$
has exactly two solutions. | 2 | 66 | 1 |
math | Let $n$ be a positive integer. A frog starts on the number line at 0 . Suppose it makes a finite sequence of hops, subject to two conditions:
- The frog visits only points in $\left\{1,2, \ldots, 2^{n}-1\right\}$, each at most once.
- The length of each hop is in $\left\{2^{0}, 2^{1}, 2^{2}, \ldots\right\}$. (The hops ... | \frac{4^{n}-1}{3} | 146 | 11 |
math | 11. (3 points) Xiao Ming goes to school from home for class. If he walks 60 meters per minute, he can arrive 10 minutes early; if he walks 50 meters per minute, he will be 4 minutes late. The distance from Xiao Ming's home to the school is $\qquad$ meters. | 4200 | 71 | 4 |
math | ## Task $5 / 87$
Determine all three-digit (proper) natural numbers $z \in N$ in the decimal system that are represented by exactly $n$ digits 1 in the number system with base $n \in N$. | 781 | 52 | 3 |
math | 8.316. $\sin ^{4} x-\sin ^{2} x+4(\sin x+1)=0$.
8.316. $\sin ^{4} x-\sin ^{2} x+4(\sin x+1)=0$. | \frac{\pi}{2}(4k-1),k\inZ | 61 | 16 |
math | 1. Ana has three stamp albums. In the first one, there is one fifth of the stamps, in the second one, there are several sevenths, and in the third one, there are 303 stamps. How many stamps does Ana have? | 3535 | 53 | 4 |
math | 9. (20 points) Let the parabola $y^{2}=2 p x(p>0)$ intersect the line $x+y=1$ at points $A$ and $B$. If $O A \perp O B$, find the equation of the parabola and the area of $\triangle O A B$. | \frac{\sqrt{5}}{2} | 70 | 10 |
math | 3. Find the smallest positive number $x$, for which the following holds: If $a, b, c, d$ are any positive numbers whose product is 1, then
$$
a^{x}+b^{x}+c^{x}+d^{x} \geqq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}
$$
(Pavel Novotný) | 3 | 99 | 1 |
math | Example 9 The range of the function $y=\sqrt{1994-x}+\sqrt{x-1993}$ is __. (Example 11 in [1]) | [1, \sqrt{2}] | 40 | 8 |
math | A sequence $a_1, a_2,\dots, a_n$ of positive integers is [i]alagoana[/i], if for every $n$ positive integer, one have these two conditions
I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$
II- The number $a_n$ is the $n$-power of a positive integer.
Find all the sequence(s) [i]alagoana[/i]. | \{1\}_{n \in \mathbb{Z}_{>0}} | 107 | 17 |
math | 4. Given that $m$ and $n$ are rational numbers, and the equation
$$
x^{2}+m x+n=0
$$
has a root $\sqrt{5}-2$. Then $m+n=$ $\qquad$ .
(2001, National Junior High School Mathematics Competition, Tianjin Preliminary Round) | 3 | 73 | 1 |
math | Two workers, $A$ and $B$, completed a task assigned to them as follows. First, only $A$ worked for $\frac{2}{3}$ of the time it would take $B$ to complete the entire task alone. Then, $B$ took over from $A$ and finished the work. In this way, the work took 2 hours longer than if they had started working together and co... | A=6,B=3 | 143 | 6 |
math | Example 8 Find a natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$, it has a total of 10 divisors. | 5\cdot7^{4} | 42 | 7 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.