task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying
\[m(m + k) = n(n + 1).\] | \mathbb{N} \setminus \{2, 3\} | 37 | 17 |
math | 2. An old problem. A landlord, having calculated that a cow is four times more expensive than a dog, and a horse is four times more expensive than a cow, took 200 rubles with him to the city and spent all the money to buy a dog, two cows, and a horse. How much does each of the purchased animals cost? | the\dog\costs\8\rubles,\the\cow\32\rubles,\\the\horse\128\rubles | 74 | 33 |
math | Fear, from the seven digits $0,1,2,3,4,5,6$, many seven-digit numbers without repeated digits can be formed, among which some are multiples of 55. Among these multiples of 55, find the largest and the smallest. (Write out the reasoning process) | 1042635, 6431205 | 64 | 16 |
math | Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angl... | \frac{\alpha}{2} | 105 | 7 |
math | Find all ordered triplets $(p,q,r)$ of positive integers such that $p$ and $q$ are two (not necessarily distinct) primes, $r$ is even, and
\[p^3+q^2=4r^2+45r+103.\] | (7, 2, 4) | 62 | 10 |
math | Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$. | 32 | 54 | 2 |
math | 7. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1} a_{n}-1=a_{n}^{2}$.
(1) Prove: $\sqrt{2 n-1} \leqslant a_{n} \leqslant \sqrt{3 n-2}$;
(2) Find the integer $m$, such that $\left|a_{2005}-m\right|$ is minimized. (2005 Hebei Province High School Mathematics Competition Problem) | 63 | 122 | 2 |
math | Example 15. Two equally strong chess players are playing chess. What is more likely: a) to win one game out of two or two games out of four? b) to win at least two games out of four or at least three games out of five? Draws are not considered. | \frac{11}{16}>\frac{8}{16} | 60 | 17 |
math | 2. If the planar vectors $\vec{a}=\left(2^{m},-1\right)$ and $\vec{b}=\left(2^{m}-1,2^{m+1}\right)$ are perpendicular, where $m$ is a real number, then the magnitude of $\vec{a}$ is . $\qquad$ | \sqrt{10} | 74 | 6 |
math | ## Task A-4.4.
A natural number is said to be quirky if in its decimal representation it has 100 digits and if by removing any of its digits, a 99-digit number divisible by 7 is formed. How many quirky natural numbers are there? | 2^{98} | 58 | 5 |
math | 10 . 18 A decimal natural number $a$ consists of $n$ identical digits $x$, and the number $b$ consists of $n$ identical digits $y$, while the number $c$ consists of $2n$ identical digits $z$. For any $n \geqslant 2$, find the digits $x, y, z$ such that $a^{2}+b=c$ holds.
（20th All-Soviet Union Mathematical Olympiad, 19... | 3,2,1or6,8,4or8,3,7 | 110 | 17 |
math | 4. Let $0<\theta<\pi$, then the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$ is $\qquad$ . | \frac{4 \sqrt{3}}{9} | 38 | 12 |
math | 2. The two positive integers $a, b$ with $a>b$ are such that $a \%$ of $b \%$ of $a$ and $b \%$ of $a \%$ of $b$ differ by 0.003 . Find all possible pairs $(a, b)$. | (5,2),(5,3),(6,1),(6,5) | 65 | 17 |
math | The volume of a certain rectangular solid is $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid. | 96 | 63 | 2 |
math | Example 2. Find the particular solution of the equation
$$
\left(1+e^{x}\right) y y^{\prime}=e^{x}
$$
satisfying the initial condition $\left.y\right|_{x=0}=1$. | \sqrt{1+\ln(\frac{1+e^{x}}{2})^{2}} | 56 | 21 |
math | Find the smallest possible value of the expression
$$
\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor,
$$
where \(a, b, c\) and \(d\) vary over the set of positive integers.
(Here \(\lfloor x\rfloor\) ... | 9 | 131 | 1 |
math | 6. Let $n$ be the smallest positive integer satisfying the following conditions: (1) $n$ is a multiple of 75; (2) $n$ has exactly 75 positive divisors (including 1 and itself). Find $\frac{n}{75}$. | 432 | 60 | 3 |
math | 4. It is known that the continued fraction of $\pi$ is
$$x=[3,7,15,1,292,1,1, \ldots]$$
Try to find its first seven convergents and their approximate values. | 3.1415926534<\pi<3.1415926540 | 55 | 28 |
math | 2. Divide the positive integers $1,2, \cdots, 2008$ into 1004 groups: $a_{1}, b_{1} ; a_{2}, b_{2} ; \cdots ; a_{1004}, b_{1004}$, and satisfy
$$
a_{1}+b_{1}=a_{2}+b_{2}=\cdots=a_{1004}+b_{1004} \text {. }
$$
For all $i(i=1,2, \cdots, 1004), a_{i} b_{i}$ the maximum value is | 1009020 | 146 | 7 |
math | The circle inscribed in triangle $ABC$ touches side $AB$ at point $M$, with $AM=1, BM=4$. Find $CM$, given that $\angle BAC=120^{\circ}$. | \sqrt{273} | 48 | 7 |
math | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. | k+1 | 109 | 3 |
math | Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ that satisfy $f(x+y)-f(x-y)=2y(3x^2+y^2)$ for all $x,y{\in}R$
______________________________________
Azerbaijan Land of the Fire :lol: | f(x) = x^3 + a | 67 | 10 |
math | The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $. | p > 2 | 111 | 5 |
math | Solve the following equation if $x$ and $y$ are integers:
$$
5 x^{2}+5 x y+5 y^{2}=7 x+14 y
$$ | \begin{pmatrix}x_{1}=-1,&y_{1}=3\\x_{2}=0,&y_{2}=0\\x_{3}=1,&y_{3}=2\end{pmatrix} | 41 | 48 |
math | For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation
$$8^k=x^3+y^3+z^3-3xyz$$ | k + 1 | 52 | 5 |
math | 3. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+a_{n}, b_{n}=\frac{1}{1+a_{n}}, S_{n}=b_{1}+b_{2}+\cdots+b_{n}, P_{n}=$ $b_{1} b_{2} \cdots b_{n}$, try to find the value of $2 P_{n}+S_{n}$. | 2 | 120 | 1 |
math | 1. Given are vectors $\vec{a}=(2,1, p)$ and $\vec{b}=(2, p+1,4)$.
a) Determine all possible values of the parameter $p$ for which there exists a vector $\vec{v}$ such that:
$$
\begin{aligned}
\vec{a} \cdot \vec{v} & =|\vec{b}| \\
\vec{a} \times \vec{v} & =\vec{b}
\end{aligned}
$$
b) For each such value of $p$, deter... | p=-1,\vec{v}=(\frac{2\sqrt{5}-2}{3},\frac{5+\sqrt{5}}{3},\frac{1-\sqrt{5}}{3}),\varphi=45 | 156 | 52 |
math | 2.14. Solve the equation
$$
\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1
$$ | 5\leqslantx\leqslant10 | 41 | 14 |
math | 1. If we subtract the unit digit from any natural number with at least two digits, we get a number that is one digit "shorter." Find all original numbers that are equal to the absolute value of the difference between the square of the "shorter" number and the square of the removed digit. | 48,100,147 | 62 | 10 |
math | $1 \cdot 49$ integers $1,2, \cdots, n$ are arranged in a permutation such that each number is either greater than all the numbers before it or less than all the numbers before it. How many such permutations are there? | 2^{n-1} | 54 | 6 |
math | 7、Xiaoming and Xiaohong are running on a 600-meter circular track. The two start from the same point at the same time and run in opposite directions. The time interval between the first and second meeting is $\mathbf{50}$ seconds. It is known that Xiaohong's speed is $\mathbf{2}$ meters/second slower than Xiaoming's. T... | 7 | 95 | 1 |
math | 10. Given that line segment $A B$ is the diameter of sphere $O$ with radius 2, points $C, D$ are on the surface of sphere $O$, $C D=2, A B \perp C D$, $45^{\circ} \leqslant \angle A O C \leqslant 135^{\circ}$, then the range of the volume of tetrahedron $A B C D$ is $\qquad$ . | [\frac{4}{3},\frac{4\sqrt{3}}{3}] | 107 | 19 |
math | Original: $f\left(\frac{2000}{2001}\right)+f\left(\frac{1999}{2001}\right)+\cdots+f\left(\frac{1}{2001}\right)$. | 1000 | 58 | 4 |
math | 7. Represent the number 2015 as the sum of some number of natural numbers so that their product is the largest.
Answer. $2015=3+3+\ldots+3+2$ (671 threes and one two). | 2015=671\cdot3+2 | 56 | 13 |
math | 8. (Question 2 of the 5th Test, *The American Mathematical Monthly*, pages 119 to 121, 1953)
Given a positive integer $n \geqslant 3$, for $n$ complex numbers $z_{1}, z_{2}, \cdots, z_{n}$ with modulus 1, find
$$
\min _{x_{1}, z_{2} \cdots, s_{n}}\left[\max _{\omega \in C,|\omega|=1} \prod_{j=1}^{n}\left|\omega-z_{j}\... | 2 | 184 | 1 |
math | In a football team (11 people), a captain and his deputy need to be selected. How many ways can this be done
# | 110 | 28 | 3 |
math | 13.357. A cyclist traveled 96 km 2 hours faster than he had planned. During this time, for each hour, he traveled 1 km more than he had planned to travel in 1 hour and 15 minutes. At what speed did he travel? | 16 | 60 | 2 |
math | 4. Determine all pairs $a, b$ of real numbers for which each of the quadratic equations
$$
a x^{2}+2 b x+1=0, \quad b x^{2}+2 a x+1=0
$$
has two distinct real roots, and exactly one of them is common to both equations. | (,--\frac{1}{4}), | 72 | 10 |
math | Question 24 Let $m \geqslant 14$ be an integer, and the function $f: \mathbf{N} \rightarrow \mathbf{N}$ is defined as follows:
$$
f(n)=\left\{\begin{array}{ll}
n-m+14 . & n>m^{2} ; \\
f(f(n+m-13)) & n \leqslant m^{2} .
\end{array}\right.
$$
Find all $m$ such that $f(1995)=1995$. | 14or45 | 123 | 5 |
math | 5. For each value of the parameter $a$, solve the equation $\log _{2} \frac{3 \sqrt{3}+\cos a(\sin x+4)}{3 \sin a \cos x}=|3 \sin a \cos x|-|\cos a(\sin x+4)+3 \sqrt{3}|$. | \frac{\pi}{6}+2\pik,k\inZ | 72 | 16 |
math | Example 6 Find the smallest positive integer $k$, such that for all $a$ satisfying $0 \leqslant a \leqslant 1$ and all positive integers $n$, the inequality
$$a^{k}(1-a)^{n} \leqslant \frac{1}{(n+1)^{3}}$$
holds. | 4 | 78 | 1 |
math | 7. Let $z=x+y \mathrm{i} (x, y \in \mathbf{R}, \mathrm{i}$ be the imaginary unit), the imaginary part of $z$ and the real part of $\frac{z-\mathrm{i}}{1-z}$ are both non-negative. Then the area of the region formed by the points $(x, y)$ on the complex plane that satisfy the conditions is $\qquad$ . | \frac{3 \pi+2}{8} | 90 | 11 |
math | 7. Remove the joker cards from a deck of playing cards, and randomly draw 5 cards from the remaining 52 cards. The probability that at least two of the cards have the same number (or letter $J, Q, K, A$) is $\qquad$ (calculate this probability value, accurate to 0.01). | 0.49 | 73 | 4 |
math | 4. There are 306 different cards with numbers $3,19,3^{2}, 19^{2}, \ldots, 3^{153}, 19^{153}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer? | 17328 | 90 | 5 |
math | B2. Jure wrote down the natural numbers from 1 to 2015 on a board. Urška then reviewed the written numbers from the smallest to the largest and erased every number that was not divisible by 3. Among the numbers that remained on the board, she then erased every number that was not divisible by $3^{2}$ from the smallest ... | 1458 | 120 | 4 |
math | 6. Find the smallest positive integer $n$, such that every $n$-element subset of $S=\{1,2, \cdots, 150\}$ contains 4 pairwise coprime numbers (it is known that $S$ contains 35 prime numbers). | 111 | 61 | 3 |
math | 6. A certain intelligence station has four different passwords $A, B, C, D$, and uses one of them each week. Each week, one of the three passwords not used in the previous week is chosen with equal probability. If the first week uses password $A$, then the probability that the seventh week also uses password $A$ is $\q... | \frac{61}{243} | 82 | 10 |
math | Problem 1. In a store, there are 800 notebooks and it operates from Monday to Friday. In one week, all the notebooks were sold such that each day 40 more notebooks were sold than the previous day. How many notebooks were sold each day of that week? | 80,120,160,200,240 | 59 | 18 |
math | 5.3. Find the limit of the sequence:
$$
\lim _{n \rightarrow \infty}(\sqrt{(n+2)(n+7)}-n)
$$ | \frac{9}{2} | 39 | 7 |
math | 3. In the known sequence $1,4,8,10,16,19,21,25,30,43$, the number of subarrays whose sum is divisible by 11 is $\qquad$ . | 7 | 53 | 1 |
math | 3. Let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a permutation of the numbers $1,2, \ldots, n$, where $n \geq 2$. Determine the maximum possible value of the expression
$$
\sum_{k=1}^{n-1}\left|a_{k+1}-a_{k}\right|
$$
(Romania) | [\frac{n^{2}}{2}]-1 | 92 | 11 |
math | Example 7 The equation in terms of $x$
$$
x^{3}-a x^{2}-2 a x+a^{2}-1=0
$$
has only one real root. Then the range of values for $a$ is | a<\frac{3}{4} | 51 | 9 |
math | 8. Equilateral triangle $A B C$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $A B$ and $A C$ of $\Omega$ respectively such that $B C=D E$. Given that triangle $A B E$ has area 3 and triangle $A C D$ has area 4 , find the area of triangle $A B C$.
Proposed by: Yuan Yao | \frac{37}{7} | 92 | 8 |
math | 1. Given the sum of 12 distinct positive integers is 2010. Then the maximum value of the greatest common divisor of these positive integers is . $\qquad$ | 15 | 38 | 2 |
math | 2.9. Solve the equation $\sqrt{2 x-6}+\sqrt{x+4}=5$.
### 2.10. Solve the equation
$$
\sqrt[m]{(1+x)^{2}}-\sqrt[m]{(1-x)^{2}}=\sqrt[m]{1-x^{2}}
$$ | 5 | 69 | 1 |
math | 6. Variant 1 Sasha wrote the number 765476547654 on the board. He wants to erase several digits so that the remaining digits form the largest possible number divisible by 9. What is this number? | 7654765464 | 52 | 10 |
math | (1) If the function $f(x)=\frac{x}{\sqrt{1+x^{2}}}$ and $f^{(n)}(x)=\underbrace{f[f[f \cdots f}_{n}(x)]]$, then $f^{(99)}(1)=$ $\qquad$ . | \frac{1}{10} | 68 | 8 |
math | Question 74, Given $a \geq b \geq c \geq d>0, a^{2}+b^{2}+c^{2}+d^{2}=\frac{(a+b+c+d)^{2}}{3}$, find the maximum value of $\frac{a+c}{b+d}$. | \frac{7+2\sqrt{6}}{5} | 72 | 14 |
math | Let $f(x)=x^2-kx+(k-1)^2$ for some constant $k$. What is the largest possible real value of $k$ such that $f$ has at least one real root?
[i]2020 CCA Math Bonanza Individual Round #5[/i] | 2 | 64 | 1 |
math |
4. Find all positive real numbers $x, y, z$ such that
$$
2 x-2 y+\frac{1}{z}=\frac{1}{2014}, \quad 2 y-2 z+\frac{1}{x}=\frac{1}{2014}, \quad 2 z-2 x+\frac{1}{y}=\frac{1}{2014}
$$
| 2014,\quad2014,\quad2014 | 94 | 16 |
math | In the 1448th exercise, let the circles centered at $A$ and $B$ be denoted by $k_{1}$ and $k_{2}$, their radii by $r_{1}$ and $r_{2}$, respectively, and the radius of the circle $k$ that touches $k$ from the inside and $k_{1}$ and $k_{2}$ from the outside by $r_{3}$. What does the ratio $r_{3} / r_{1}$ tend to, if - wi... | \frac{1}{2} | 143 | 7 |
math | 6. Given a cyclic quadrilateral $ABCD$ with side lengths $AB=1, BC=3, CD=DA=2$. Then the area of quadrilateral $ABCD$ is $\qquad$ . | 2\sqrt{3} | 45 | 6 |
math | 7. Given a hyperbola with the two coordinate axes as axes of symmetry, the foci on the $y$-axis, the real axis length is $2 \sin \theta, \theta \in\left[\frac{\pi}{4}, \frac{\pi}{3}\right]$, and the shortest distance from any point $P(x, y)$ on the hyperbola to the point $M(1,0)$ is $\frac{1}{\sin \theta}$, then the ra... | (1,\frac{2\sqrt{21}}{7}] | 121 | 15 |
math |
Problem 4. Find the number of all natural numbers $n, 4 \leq n \leq$ 1023, such that their binary representations do not contain three consecutive equal digits.
Emil Kolev
| 228 | 50 | 3 |
math | ## Task B-1.2.
Determine the periodic numbers $0 . \dot{x}, 0 . \dot{x} \dot{y}$ and $0 . \dot{x} y \dot{z}$ for which
$$
0 . \dot{x}+0 . \dot{x} \dot{y}+0 . \dot{x} y \dot{z}=\frac{445}{333}
$$
where $x, y, z$ (not necessarily distinct) are digits. | 00\dot{4},00\dot{4}\dot{4},00\dot{4}4\dot{7} | 111 | 31 |
math | ## Task Condition
Find the derivative.
$y=\frac{2 x-5}{4} \cdot \sqrt{5 x-4-x^{2}}+\frac{9}{4} \cdot \arcsin \sqrt{\frac{x-1}{3}}$ | \sqrt{5x-4-x^{2}} | 57 | 11 |
math |
4. Let $X=\{1,2,3, \ldots, 11\}$. Find the the number of pairs $\{A, B\}$ such that $A \subseteq X$, $B \subseteq X, A \neq B$ and $A \cap B=\{4,5,7,8,9,10\}$.
| 3^5-1 | 80 | 5 |
math | 342. Each natural number from 1 to 50000 is replaced by the number equal to the sum of its digits. The same operation is performed with the resulting numbers, and this is repeated until all numbers become single-digit. How many times will each of the integers from 0 to 8 appear among these single-digit numbers? | 1,2,3,4,5-5556 | 72 | 14 |
math | 42. $\int(2 x-1)^{3} d x$.
The integral of $(2x-1)^3$ with respect to $x$. | 2x^{4}-4x^{3}+3x^{2}-x+C | 35 | 18 |
math | ## Task 1 - 280821
A cargo ship needs exactly 12 days for a shipping route from port $A$ to port $B$. A tanker travels this route in the opposite direction and takes 15 days for it.
The cargo ship departs from port $A$ 6 days later than the tanker from port $B$.
a) How many days after the departure of the cargo ship... | 4 | 117 | 1 |
math | Example 11. Solve the inequality
$$
\sqrt{\frac{9}{10}} \cdot\left(\frac{9}{10}\right)^{x-1}>\frac{10^{(3 / 4) x-1}}{\sqrt{10}}
$$ | x<4\frac{\lg3-2}{8\lg3-7} | 63 | 18 |
math | 1. Calculate the value of the numerical expression:
$$
2025+720:(72-9 \cdot 7)-(4 \cdot 6-6) \cdot 5+1
$$ | 2016 | 46 | 4 |
math | (4) If $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right), \theta \in[0,2 \pi)$, then the range of $\theta$ is $\qquad$ . | (\frac{\pi}{4},\frac{5\pi}{4}) | 68 | 16 |
math | Example 1 Given a cyclic quadrilateral $ABCD$ with side lengths $AB=2, BC=6, CD=DA=4$, find the area of quadrilateral $ABCD$. | 8\sqrt{3} | 40 | 6 |
math | 9. Given non-negative real numbers $u, v, w$ satisfy $u+v+w=$ 2, then the range of $u^{2} v^{2}+v^{2} w^{2}+w^{2} u^{2}$ is $\qquad$ | [0,1] | 59 | 5 |
math | 22. Find the value of $\frac{\tan 40^{\circ} \tan 60^{\circ} \tan 80^{\circ}}{\tan 40^{\circ}+\tan 60^{\circ}+\tan 80^{\circ}}$. | 1 | 64 | 1 |
math | Five. (12 points) Try to find the largest and smallest seven-digit numbers that can be divided by 165, formed by the 7 digits $0,1,2,3,4,5,6$ without repetition (require the reasoning process to be written out).
Find the largest and smallest seven-digit numbers that can be formed using the digits $0,1,2,3,4,5,6$ witho... | 1042635, 6431205 | 109 | 16 |
math | 3. If real numbers $x, y$ satisfy $4 x^{2}-4 x y+2 y^{2}=1$, then the sum of the maximum and minimum values of $3 x^{2}+x y+y^{2}$ is $\qquad$ | 3 | 56 | 1 |
math | ## Task Condition
Find the derivative.
$y=x^{3^{x}} \cdot 2^{x}$ | x^{3^{x}}\cdot2^{x}\cdot(3^{x}\cdot\ln3\cdot\ln(x)+\frac{3^{x}}{x}+\ln2) | 23 | 42 |
math | 11. In a regular 2017-gon, all diagonals are drawn. Petya randomly selects some $\mathrm{N}$ diagonals. What is the smallest $N$ such that among the selected diagonals, there are guaranteed to be two of the same length? | 1008 | 60 | 4 |
math | Find all polynomials $P(x)$ of degree $1$ such that
$\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$ | P(x) = x + c \text{ and } P(x) = -x + c | 73 | 20 |
math | 1. Let the polynomial $p_{k}(x)=\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2}$, where $k$ is any given positive integer, find the coefficient of $x^{2}$ in $p_{k}(x)$. | \frac{1}{3}(4^{2k-1}-4^{k-1}) | 83 | 20 |
math | 1. $\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}=$ | \frac{3}{4} | 47 | 7 |
math | At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges?
[i]R... | 64 | 81 | 2 |
math | 5. Given the inequality $\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta<3 a+6$, for $\theta \in\left[0, \frac{\pi}{2}\right]$ to always hold. Find the range of $\theta$. (1st China Southeast Mathematical Olympiad) | a>3 | 96 | 3 |
math | Example 27 ([38.5]) Find all positive integer pairs $\{a, b\}$, satisfying the equation
$$a^{b^{2}}=b^{a}.$$ | \{1,1\}, \{16,2\}, \{27,3\} | 40 | 23 |
math | Example 1 Suppose the lengths of the three sides of a triangle are integers $l$, $m$, and $n$, and $l > m > n$. It is known that
$$
\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\},
$$
where $\{x\}=x-[x]$, and $[x]$ represents the greatest integer not exceedin... | 3003 | 151 | 4 |
math | Determine the remainder of the Euclidean division of the polynomial $x+x^{3}+x^{9}+x^{27}+x^{81}+$ $x^{243}$ by $x-1$ and by $x^{2}-1$. | R(x)=6R(x)=6x | 58 | 9 |
math | Example 1 (Fill in the blank Question 1) Given $\frac{1}{4}(b-c)^{2}=$ $(a-b)(c-a)$ and $a \neq 0$. Then $\frac{b+c}{a}=$ $\qquad$ . | 2 | 58 | 1 |
math | Exercise 2. An East-West maritime line sees 10 ships depart each morning at distinct times, 5 ships depart from the West side and 5 from the East side. We assume that they all sail at the same speed and that as soon as two ships meet, they turn around and head back in the direction they came from, always at the same sp... | 25 | 85 | 2 |
math | Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i]) | (0, 0, 1), (0, 1, 3), (2, 1, 5) | 56 | 28 |
math | The numbers $p$ and $q$ are prime and satisfy
\[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\]
for some positive integer $n$. Find all possible values of $q-p$.
[i]Luxembourg (Pierre Haas)[/i] | \{2, 3, 5\} | 82 | 12 |
math | Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$. | 135^\circ | 70 | 5 |
math | Divide the numbers 2, 3, 5, 7, 11, 13, and 17 into two groups in such a way that, by multiplying all the numbers in one group and all in the other, we find consecutive numbers. | 714=2\times3\times7\times17\quad\text{}\quad715=5\times11\times13 | 56 | 34 |
math | Example 5.23. Using differentiation and integration, find the Taylor series expansion for the given function $f(x)=\operatorname{arc} \operatorname{tg} x$ and specify the intervals in which these expansions are valid. | \operatorname{arctg}x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots+(-1)^{n}\frac{x^{2n+1}}{2n+1}+\ldots | 50 | 63 |
math | 13. Factor $(a+1)(a+2)(a+3)(a+4)-120$ completely into factors with integer coefficients. | (^{2}+5+16)(-1)(+6) | 32 | 16 |
math | Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R... | \frac{\sqrt{2R^2 - d^2}}{2} | 154 | 17 |
math | 75. Arrange 10 small balls labeled with numbers 1 to 10 in a row, such that the sum of the numbers on every 3 adjacent balls is a multiple of 3. The total number of arrangements is $\qquad$. | 1728 | 52 | 4 |
math | Let's determine the $x$ for which the
$$
|13-x|+|58-x|+|71-x|+|75-x|+|79-x|
$$
sum is minimal! | 71 | 48 | 2 |
math | G3.2 Let $x_{1}, x_{2}, \ldots, x_{10}$ be non-zero integers satisfying $-1 \leq x_{i} \leq 2$ for $i=1,2, \ldots, 10$.
If $x_{1}+x_{2}+\ldots+x_{10}=11$, find the maximum possible value for $x_{1}^{2}+x_{2}^{2}+\cdots x_{10}^{2}$. | 31 | 117 | 2 |
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