id
int64 20
101k
| problem
stringlengths 18
4.16k
| gt_ans
stringlengths 1
191
|
|---|---|---|
40,346 | A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3).
\end{align*}
The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$. | 40086 |
40,372 | 16. In $\triangle A B C$, the three interior angles are $\angle A$, $\angle B$, and $\angle C$, and $2 \angle C - \angle B = 180^{\circ}$. Additionally, the ratio of the perimeter of $\triangle A B C$ to its longest side is $m$. Therefore, the maximum value of $m$ is | \frac{9}{4} |
40,380 | Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \dots, k$. It is known that for $i = 1, 2, \dots, k$, Part $i$ contains $i$ multiple choice questions, each of which has $(i+1)$ answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly one question correct is equal to $2018$ times the probability that she gets no questions correct. Compute the number of questions that are on the exam.
[i]Proposed by Yannick Yao[/i] | 2037171 |
40,437 | As shown in Figure 1, given a square $A B C D$ with a side length of 1, the diagonals $A C$ and $B D$ intersect at point $O, E$ is a point on the extension of side $B C$, connect $A E$, which intersects $B D$ and $C D$ at points $P$ and $F$ respectively, connect $B F$ and extend it to intersect segment $D E$ at point $G$, connect $O E$ and intersect $C D$ at point $S$. If $P S / / A C$, find the length of $B G$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{2 \sqrt{5}}{3} |
40,452 | In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$. | 90^\circ |
40,453 | 16. Given the equations in $x$, $4 x^{2}-8 n x-3 n=2$ and $x^{2}-(n+3) x-2 n^{2}+2=0$. Does there exist a value of $n$ such that the square of the difference of the two real roots of the first equation equals an integer root of the second equation? If it exists, find such $n$ values; if not, explain the reason. | n=0 |
40,470 | Let $n>1$ be an integer. In a row, there are $n$ boxes, and we have $n+1$ identical stones. A distribution is a way to distribute the stones over the boxes, with each stone being in exactly one box. We say that two such distributions are a stone's throw away from each other if we can obtain one distribution from the other by moving exactly one stone to a different box. The sociability of a distribution $a$ is defined as the number of distributions that are a stone's throw away from $a$. Determine the average sociability over all possible distributions.
(Two distributions are the same if there are the same number of stones in the $i$-th box of both rows, for all $i \in\{1,2, \ldots, n\}$.)
Answer. $\frac{n^{2}-1}{2}$
In the solutions below, we call two distributions neighbors if they are a stone's throw away from each other. | \frac{n^{2}-1}{2} |
40,502 | 4. The expression $\frac{\left(2^{4}+\frac{1}{4}\right)\left(4^{4}+\frac{1}{4}\right)\left(6^{4}+\frac{1}{4}\right)}{\left(1^{4}+\frac{1}{4}\right)\left(3^{4}+\frac{1}{4}\right)\left(5^{4}+\frac{1}{4}\right)}$ $\times \frac{\left(8^{4}+\frac{1}{4}\right)\left(10^{4}+\frac{1}{4}\right)}{\left(7^{4}+\frac{1}{4}\right)\left(9^{4}+\frac{1}{4}\right)}$ represents a positive integer. This positive integer is | 221 |
40,524 | 5. Two natural numbers $x$ and $y$ sum to 111, such that the equation
$$
\sqrt{x} \cos \frac{\pi y}{2 x}+\sqrt{y} \sin \frac{\pi x}{2 y}=0
$$
holds. Then a pair of natural numbers $(x, y)$ that satisfies the condition is
$\qquad$ | (37,74) |
40,529 | The infinite sequence $a_0,a_1,\ldots$ is given by $a_1=\frac12$, $a_{n+1} = \sqrt{\frac{1+a_n}{2}}$. Determine the infinite product $a_1a_2a_3\cdots$. | \frac{3\sqrt{3}}{4\pi} |
40,533 | Example 1 Let $x, y, z \in \mathbf{R}^{+}, \sqrt{x^{2}+y^{2}}+z=1$, try to find the maximum value of $x y+2 x z$.
| \frac{\sqrt{3}}{3} |
40,561 | 4. As shown in Figure 2, the radius of hemisphere $O$ is $R$, and the rectangular prism $A B C D$ $A_{1} B_{1} C_{1} D_{1}$ is inscribed in the hemisphere $O$ with one of its faces $A B C D$ on the base of the hemisphere $O$. Then the maximum value of the sum of all the edges of the rectangular prism is | 12R |
40,596 | 10. Let $x \in\left(0, \frac{\pi}{2}\right)$. Then the function
$$
y=\frac{225}{4 \sin ^{2} x}+\frac{2}{\cos x}
$$
has a minimum value of | 68 |
40,600 | The [Binomial Expansion](https://artofproblemsolving.com/wiki/index.php?title=Binomial_Expansion&action=edit&redlink=1) is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$? | 428 |
40,602 | 12. Given real numbers $x, y$ satisfy
$$
\left\{\begin{array}{l}
x-y \leqslant 0, \\
x+y-5 \geqslant 0, \\
y-3 \leqslant 0 .
\end{array}\right.
$$
If the inequality $a\left(x^{2}+y^{2}\right) \leqslant(x+y)^{2}$ always holds, then the maximum value of the real number $a$ is $\qquad$ | \frac{25}{13} |
40,604 | [Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $BEF$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at [point](https://artofproblemsolving.com/wiki/index.php/Point) $Q.$ Find $PQ.$ | 259 |
40,608 | Example 2. A chord with slope $t$ is drawn through the endpoint $(-a, 0)$ of the real axis of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Find the equation of the line passing through the other endpoint of the chord and the endpoint $(a, 0)$ of the real axis. | y=\frac{b^{2}}{a^{2} t}(x-a) |
40,619 | Example 16 Solve the equation
$$
(2 x+1)^{7}+x^{7}+3 x+1=0 .
$$ | x=-\frac{1}{3} |
40,686 | In [tetrahedron](https://artofproblemsolving.com/wiki/index.php/Tetrahedron) $ABCD$, [edge](https://artofproblemsolving.com/wiki/index.php/Edge) $AB$ has length 3 cm. The area of [face](https://artofproblemsolving.com/wiki/index.php/Face) $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of the tetrahedron in $\mbox{cm}^3$. | 20 |
40,697 | Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 121 |
40,699 | Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$ | 12 |
40,709 | In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half. | 4 |
40,724 | 1. Let $[x]$ be the greatest integer not exceeding the real number $x$. For any $x \in \mathbf{R}$, we have $[x]+[x+a]=[2 x]$. Then the set of all possible values of the positive real number $a$ is | \frac{1}{2} |
40,727 | In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$. | 504 |
40,759 | Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$. | 4 - 2\sqrt{2} |
40,767 | 2. If the function $f(x)=\frac{2^{x+1}}{2^{x}+1}+\sin x$ has a range of $[n, M]$ on the interval $[-k, k](k>0)$, then $M+n$ $=$ $\qquad$ | 2 |
40,788 | 5. Given $f(x)=\left(x^{2}+3 x+2\right)^{\cos \pi x}$. Then the sum of all $n$ that satisfy the equation
$$
\left|\sum_{k=1}^{n} \log _{10} f(k)\right|=1
$$
is | 21 |
40,825 | $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$, with $AB=6$, $BC=7$, $CD=8$. Find $AD$. | \sqrt{51} |
40,841 | (25 points) (1) First, select $n$ numbers from $1,2, \cdots, 2020$, then choose any two numbers $a$ and $b$ from these $n$ numbers, such that $a \nmid b$. Find the maximum value of $n$.
| 1010 |
40,863 | Three. (25 points) Given the equation about $x$
$$
4 x^{2}-8 n x-3 n-2=0
$$
and $x^{2}-(n+3) x-2 n^{2}+2=0$.
Question: Is there such a value of $n$ that the square of the difference of the two real roots of equation (1) equals an integer root of equation (2)? If it exists, find such $n$ values; if not, explain the reason. | n=0 |
40,894 | $m$ boys and $n$ girls ($m>n$) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen?
[i]Hendrata Dharmawan, Boston, USA[/i] | m - n |
40,925 | Example 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 values for $\theta$ is $\qquad$ [3]
(2011, National High School Mathematics Joint Competition) | \left(\frac{\pi}{4}, \frac{5 \pi}{4}\right) |
40,958 | 6. Given in a Cartesian coordinate system there are two moving points $P\left(\sec ^{2} \alpha, \operatorname{tg} \alpha\right), Q(\sin \beta, \cos \beta+5)$, where $\alpha, \beta$ are any real numbers. Then the shortest distance between $P$ and $Q$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2 \sqrt{5}-1 |
40,981 | 3. As shown in the figure, $P$ is a point inside $\triangle ABC$. Lines are drawn through $P$ parallel to the sides of $\triangle ABC$.
The smaller triangles
$t_{1}, t_{2}$, and $t_{3}$
have areas of 4, 9,
and 49, respectively. Find the area of $\triangle ABC$. | 144 |
40,982 | 8. (40 points) In $\triangle A B C$, it is given that $B C=A C$, $\angle B C A=90^{\circ}$, points $D$ and $E$ are on sides $A C$ and $A B$ respectively, such that $A D=A E$, and $2 C D=B E$. Let $P$ be the intersection of segment $B D$ and the angle bisector of $\angle C A B$. Find $\angle P C B$.
---
The translation is provided as requested, maintaining the original formatting and structure. | 45^{\circ} |
40,986 | The cards in a stack of $2n$ cards are numbered [consecutively](https://artofproblemsolving.com/wiki/index.php/Consecutive) from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A,$ respectively. In this process, card number $(n+1)$ becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original positions. Find the number of cards in the magical stack in which card number 131 retains its original position. | 392 |
41,018 | $99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
[i]Proposed by N. Vlasova, S. Berlov[/i] | 98 |
41,045 | Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$. | n+1 |
41,056 | Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$.
[i]Proposed by Andrew Wen[/i] | 84 |
41,059 | The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and
$$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$ | p = 7 |
41,064 | For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime?
[i]Proposed by Alexander Wang[/i] | 22 |
41,071 | $25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants to give to each of the $24$ remaining donkeys a pot of one of six colours of the rainbow (except red), so that at least one pot of each colour is given to some donkey (but successive donkeys can receive pots of the same colour). Which of the two friends has more ways to get his plan implemented, and how many times more?
[i]Eeyore is a character in the Winnie-the-Pooh books by A. A. Milne. He is generally depicted as a pessimistic, gloomy, depressed, old grey stuffed donkey, who is a friend of the title character, Winnie-the-Pooh. His name is an onomatopoeic representation of the braying sound made by a normal donkey. Of course, Winnie-the-Pooh is a fictional anthropomorphic bear.[/i]
[i]Proposed by F. Petrov[/i] | 7 |
41,076 | Example 9 Five numbers $a, b, c, d, e$, their pairwise sums are $183, 186, 187, 190, 191, 192, 193, 194, 196, 200$. If $a<b<c<d<e$, then the value of $a$ is $\qquad$ | 91 |
41,099 | (F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$. | 2 |
41,123 | Three. (25 points) Let $a$ be a prime number, $b$ and $c$ be positive integers, and satisfy
$$
\left\{\begin{array}{l}
9(2 a+2 b-c)^{2}=509(4 a+1022 b-511 c), \\
b-c=2 .
\end{array}\right.
$$
Find the value of $a(b+c)$. | 2008 |
41,152 | Example 2 Let $x, y, z$ be real numbers, not all zero. Find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$.
| \frac{\sqrt{5}}{2} |
41,156 | 2. Given, $\odot \mathrm{O}_{1}$ and $\odot \mathrm{O}_{2}$ intersect at $\mathrm{A}$ and $\mathrm{B}$, a tangent line $\mathrm{AC}$ is drawn from point $\mathrm{A}$ to $\odot \mathrm{O}_{2}$, $\angle \mathrm{CAB}=45^{\circ}$, the radius of $\odot \mathrm{O}_{2}$ is $5 \sqrt{2} \mathrm{~cm}$, find the length of $\mathrm{AB}$ (Figure 2). | 10 |
41,168 | Calculate the following indefinite integrals.
[1] $\int \frac{x}{\sqrt{5-x}}dx$
[2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$
[3] $\int (\sin x+\cos x)^2dx$
[4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$
[5]$\int (\sin x+\sin 2x)^2 dx$ | x - \frac{1}{4} \sin 2x - \frac{1}{8} \sin 4x + \frac{4}{3} \sin^3 x + C |
41,185 | A company of $n$ soldiers is such that
(i) $n$ is a palindrome number (read equally in both directions);
(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.
Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$. | 515 |
41,193 | 6. In $\triangle A B C$, $\angle A \leqslant \angle B \leqslant \angle C$, if $\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\sqrt{3}$,
then the value of $\sin B+\sin 2 B$ is $\qquad$ | \sqrt{3} |
41,194 | 4. As shown in Figure 4, given that the radii of circles $\odot O_{1}$ and $\odot O_{2}$, which intersect at points $A$ and $B$, are $5$ and $7$ respectively, and $O_{1} O_{2}=6$. A line through point $A$ intersects the two circles at points $C$ and $D$. $P$ and $O$ are the midpoints of line segments $CD$ and $O_{1} O_{2}$, respectively. Then the length of $OP$ is | 2 \sqrt{7} |
41,200 | Example 2 Let $x, y \in \mathbf{R}, M=\max || x+y |$, $|x-y|,|1-x|,|1-y|\}$. Try to find the minimum value of $M$. | \frac{2}{3} |
41,223 | A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.
A rectangle with sides $m\ge3$ and $n\ge3$ consists of $mn$ unit squares. If in each of those unit squares exactly one number is written, such that any square with side $3$ is a magic square, then find the number of most different numbers that can be written in that rectangle. | 9 |
41,231 | 3. Let $a$, $b$, and $c$ be real numbers, $k$ be a positive constant, and
$$
\left\{\begin{array}{l}
a+b+c=0, \\
a b c=k .
\end{array}\right.
$$
(1) Find the minimum value of $\max \{a, b, c\}$;
(2) Find the minimum value of $|a|+|b|+|c|$. | 2 \sqrt[3]{4 k} |
41,274 | 6. Let $a, b, c, d$ be real numbers, satisfying
$$
a+2 b+3 c+4 d=\sqrt{10} \text {. }
$$
Then the minimum value of $a^{2}+b^{2}+c^{2}+d^{2}+(a+b+c+d)^{2}$ is $\qquad$ | 1 |
41,280 |
4. Let $l(n)$ denote the the greatest odd divisor of any natural number $n$. Find the sum
$$
l(1)+l(2)+l(3)+\cdots+l\left(2^{2013}\right) .
$$
(Michal Rolínek)
| \frac{4^{2013}+2}{3} |
41,284 | A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$. | 3995 |
41,286 | 1. Given the parabola
$$
y=x^{2}+(k+1) x+1
$$
intersects the $x$-axis at two points $A$ and $B$, not both on the left side of the origin. The vertex of the parabola is $C$. To make $\triangle A B C$ an equilateral triangle, the value of $k$ is $\qquad$ | -5 |
41,298 | 8. A. Given positive integers $a$, $b$, $c$ satisfy
$$
\begin{array}{l}
a+b^{2}-2 c-2=0, \\
3 a^{2}-8 b+c=0 .
\end{array}
$$
Then the maximum value of $a b c$ is $\qquad$ | 2013 |
41,301 | 8. There are 128 ones written on the blackboard. In each step, you can erase any two numbers $a$ and $b$ on the blackboard, and write the number $a \cdot b + 1$. After doing this 127 times, only one number remains. Denote the maximum possible value of this remaining number as $A$. Find the last digit of $A$.
| 2 |
41,302 | Initially 261 the perimeter of an integer-sided triangle is 75, and squares are constructed on each side. The sum of the areas of the three squares is 2009. Find the difference between the longest and shortest sides of this triangle. | 16 |
41,361 | For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$
a) Prove that $a_1+a_2+a_3$ is even.
b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes. | n = 7 |
41,387 | Let $n$ be a fixed positive integer and fix a point $O$ in the plane. There are $n$ lines drawn passing through the point $O$. Determine the largest $k$ (depending on $n$) such that we can always color $k$ of the $n$ lines red in such a way that no two red lines are perpendicular to each other.
[i]Proposed by Nikola Velov[/i] | \left\lceil \frac{n}{2} \right\rceil |
41,395 | If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$? | 2011 |
41,405 | Consider the polynomial $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$. Find a nonzero polynomial whose roots are the reciprocal values of the roots of $p(x)$. | 1 + 2x + 3x^2 + \ldots + 2000x^{1999} |
41,427 | 1. Given the general term of the sequence $\left\{a_{n}\right\}$
$$
a_{n}=\frac{(n+1)^{4}+n^{4}+1}{(n+1)^{2}+n^{2}+1} \text {. }
$$
Then the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$, $S_{n}=$ $\qquad$ | \frac{n\left(n^{2}+3 n+5\right)}{3} |
41,448 | 11. Given the parabola $y=x^{2}+m x+n$ passes through the point $(2,-1)$, and intersects the $x$-axis at points $A(a, 0)$ and $B(b, 0)$. If $P$ is the vertex of the parabola, find the equation of the parabola that minimizes the area of $\triangle P A B$. | y=x^{2}-4 x+3 |
41,453 | Four, (15 points) As shown in Figure 4, in $\triangle A B C$, $\angle B A C$ $=\angle B C A=44^{\circ}, M$
is a point inside $\triangle A B C$,
such that $\angle M C A=30^{\circ}$, $\angle M A C=16^{\circ}$. Find the measure of $\angle B M C$. | 150^{\circ} |
41,463 | Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$ | n = 11 |
41,470 | 6. In a convex quadrilateral $ABCD$, $\angle BAD + \angle ADC = 240^{\circ}$, $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $EF = \sqrt{7}$. If two squares $A_1$ and $A_2$ are drawn with sides $AB$ and $CD$, respectively, and a rectangle $A_3$ is drawn with length $AB$ and width $CD$, find the sum of the areas of the three figures $A_1$, $A_2$, and $A_3$. | 28 |
41,490 | $9 \cdot 51$ Let $m, n \in N$, find the minimum value of $\left|12^{m}-5^{n}\right|$. | 7 |
41,497 | 2. The sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}(n \geqslant 1)$ satisfy
$$
a_{n+1}=2 b_{n}-a_{n}, b_{n+1}=2 a_{n}-b_{n}
$$
$(n=1,2, \cdots)$. If $a_{1}=2007, a_{n}>0(n=2,3$, $\cdots$ ), then $b_{1}$ equals $\qquad$ . | 2007 |
41,504 | I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone? | 1 |
41,520 | Solve the system of equations for positive real numbers:
$$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$ | x = y = z = \frac{1}{\sqrt{2}} |
41,531 | 8. If the sum of the volumes of $n$ cubes with side lengths as positive integers is $2002^{2005}$. Find the minimum value of $n$.
| 4 |
41,554 | Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$. | 4\sqrt{14} |
41,562 | 17. (18 points) Among 200 small balls numbered $1, 2, \cdots, 200$, any $k$ balls are drawn such that there must be two balls with numbers $m$ and $n$ satisfying
$$
\frac{2}{5} \leqslant \frac{n}{m} \leqslant \frac{5}{2} \text {. }
$$
Determine the minimum value of $k$ and explain the reasoning. | 7 |
41,573 | In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$. | 60^\circ |
41,574 | One. (20 points) Let positive integers $a, b, c (a \geqslant b \geqslant c)$ be the lengths of the sides of a triangle, and satisfy
$$
a^{2}+b^{2}+c^{2}-a b-a c-b c=13 \text {. }
$$
Find the number of triangles that meet the conditions and have a perimeter not exceeding 30. | 11 |
41,579 | At a party there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of then women. For which $r$ and $s$ must there be a man and a woman who like each other? | r + s > n |
41,603 | Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$.
[i]2016 CCA Math Bonanza Individual #13[/i] | 133 |
41,620 | Find the largest positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$. | 12 |
41,679 | In [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$ | 463 |
41,721 | 2 Let $x_{i} \geqslant 0(1 \leqslant i \leqslant n, n \geqslant 4), \sum_{i=1}^{n} x_{i}=1$, find the maximum value of $F=\sum_{i=1}^{n} x_{i} x_{i+1}$. | \frac{1}{4} |
41,734 | Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$ | 1018081 |
41,759 | ## Zadatak A-2.1.
Dan je jednakokračni pravokutni trokut čije su katete duljine 10. Odredi najveću moguću površinu pravokutnika čija jedna stranica leži na hipotenuzi, a po jedan vrh na katetama danog trokuta.
| 25 |
41,790 | Three. (50 points) Given ten points in space, where no four points lie on the same plane. Connect some of the points with line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn. | 15 |
41,812 | Expanding $(1+0.2)^{1000}_{}$ by the binomial theorem and doing no further manipulation gives
${1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}$
$= A_0 + A_1 + A_2 + \cdots + A_{1000},$
where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k_{}^{}$ is $A_k^{}$ the largest? | 166 |
41,825 | One. (25 points) Let the sequence $\left\{x_{n}\right\}$ satisfy
$$
x_{1}=1, x_{n}=\sqrt{x_{n-1}^{2}+x_{n-1}}+x_{n-1}(n \geqslant 2) \text {. }
$$
Find the general term formula for the sequence $\left\{x_{n}\right\}$.
(Zhang Lei, provided) | x_{n}=\frac{1}{2^{2^{1-n}}-1} |
41,835 | Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 37 |
41,837 | The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ so that $\sin^2\theta = \tfrac{m}{n}$. Find $m + n$.
[asy]
import graph; size(3.2cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle);
draw((-1,3)--(-1,2));
draw((-1,2)--(-0.13,1.5));
draw((-0.13,1.5)--(0.73,2));
draw((0.73,2)--(0.73,3));
draw((0.73,3)--(-0.13,3.5));
draw((-0.13,3.5)--(-1,3));
draw((-1,3.5)--(0.73,3.5));
draw((0.73,3.5)--(0.73,1.5));
draw((-1,1.5)--(0.73,1.5));
draw((-1,3.5)--(-1,1.5));
label("$ A $",(-1.4,3.9),SE*labelscalefactor);
label("$ B $",(-1.4,3.28),SE*labelscalefactor);
label("$ C $",(-1.4,2.29),SE*labelscalefactor);
label("$ D $",(-1.4,1.45),SE*labelscalefactor);
label("$ E $",(-0.3,1.4),SE*labelscalefactor);
label("$ F $",(0.8,1.45),SE*labelscalefactor);
label("$ G $",(0.8,2.24),SE*labelscalefactor);
label("$ H $",(0.8,3.26),SE*labelscalefactor);
label("$ I $",(0.8,3.9),SE*labelscalefactor);
label("$ J $",(-0.25,3.9),SE*labelscalefactor); [/asy] | 55 |
41,839 | 8.82 Suppose there are 128 ones written on the blackboard. In each step, you can erase any two numbers $a$ and $b$ on the blackboard, and write $ab+1$. After 127 such steps, only one number remains. Let the maximum possible value of this remaining number be $A$. Find the last digit of $A$.
| 2 |
41,938 | 7. Arrange the three numbers $10 a^{2}+81 a+207, a+2$, and $26-2a$ in such a way that their common logarithms form an arithmetic sequence with a common difference of 1. Then, at this point, $a=$ $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{1}{2} |
41,949 | Determine all real solutions $x, y, z$ of the following system of equations: $\begin{cases}
x^3 - 3x = 4 - y \\
2y^3 - 6y = 6 - z \\
3z^3 - 9z = 8 - x\end{cases}$ | x = y = z = 2 |
41,968 | A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles. | 192 |
41,977 | 15. There is a 6-row $n$-column matrix composed of 0s and 1s, where each row contains exactly 5 ones, and the number of columns in which any two rows both have a 1 is at most 2. Find the minimum value of $n$. | 10 |
41,982 | Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right triangle), cutting it along the altitude to its hypotenuse and randomly discarding one of the two pieces once again, and continues doing this forever. As the number of iterations of this process approaches infinity, the total length of the cuts made in the paper approaches a real number $l$. Compute $[\mathbb{E}(l)]^2$, that is, the square of the expected value of $l$.
[i]Proposed by Matthew Kroesche[/i] | 64 |
42,007 | Find the number of permutations $(a_1, a_2, . . . , a_{2013})$ of $(1, 2, \dots , 2013)$ such that there are exactly two indices $i \in \{1, 2, \dots , 2012\}$ where $a_i < a_{i+1}$. | C_{2013} = 3^{2013} - (2014)2^{2013} + \frac{2013 \cdot 2014}{2} |
42,026 | Given a parallelogram $ABCD$, let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$, the distance from vertex $B$ to $\mathcal{P}$ is $25$, and the distance from vertex $C$ to $\mathcal{P}$ is $36$. Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$.
[i]Proposed by [b]HrishiP[/b][/i] | 220 |
42,035 | 1. In the complex number range, the equation
$$
x^{2}+p x+1=0(p \in \mathbf{R})
$$
has two roots $\alpha, \beta$. If $|\alpha-\beta|=1$, then $p=$ $\qquad$ | \pm \sqrt{3} \text{ or } \pm \sqrt{5} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.