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39,150 | 2. Let $a, b, c \in \mathbf{R}$, and satisfy the system of equations
$$
\left\{\begin{array}{l}
a^{2}+b^{2}+c^{2}-10 a-11=0, \\
a^{2}-b c-4 a-5=0 .
\end{array}\right.
$$
Then the range of values for $a b+b c+c a$ is | [-40,72] |
39,181 | We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$. We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$, thus defining the sets $M_1,M_2,...,M_n$. Find the minimum value of $n$. | 8 |
39,184 | [b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $
[b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $ | 8 |
39,198 | 3. Among $m$ students, it is known that in any group of three, two of them know each other, and in any group of four, two of them do not know each other. Then the maximum value of $m$ is $\qquad$ | 8 |
39,200 | Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$. $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$, then the area of $ABC$ can be expressed as $m\sqrt n$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Determine $m+n$.
[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,dir(B--D));
label("$E$",E,dir(C--E));
[/asy] | 10 |
39,226 | 14. The domain of the function $y=\sqrt{9-x^{2}} \cdot \lg (1-\sqrt{2} \cos x)$ is $\qquad$ . | [-3, -\frac{\pi}{4}) \cup (\frac{\pi}{4}, 3] |
39,229 |
Problem 8'2. Let $M$ be the midpoint of the side $B C$ of $\triangle A B C$ and $\angle C A B=45^{\circ} ; \angle A B C=30^{\circ}$.
a) Find $\angle A M C$.
b) Prove that $A M=\frac{A B \cdot B C}{2 A C}$.
Chavdar Lozanov
| 45 |
39,230 | In [convex](https://artofproblemsolving.com/wiki/index.php/Convex_polygon) [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are [right angles](https://artofproblemsolving.com/wiki/index.php/Right_angle), and $\angle B, \angle C, \angle E,$ and $\angle F$ are [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent). The area of the hexagonal region is $2116(\sqrt{2}+1).$ Find $AB$. | 46 |
39,245 | 17. If $0<a, b, c<1$ satisfies $a b+b c+c a=1$, find the minimum value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$. (2004 Sichuan Province Mathematics Competition Problem) | \frac{3(3+\sqrt{3})}{2} |
39,250 | 4. Find the largest odd number that cannot be expressed as the sum of three distinct composite numbers.
untranslated text:
4.求一个不能用三个不相等的合数之和表示的最大奇数。 | 17 |
39,275 | 57. If $x, y, z$ are all positive real numbers, and $x^{2}+y^{2}+z^{2}=1$, find the minimum value of $S=\frac{(z+1)^{2}}{2 x y z}$. | 3+2\sqrt{2} |
39,284 | Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation
$$(n^2+1)a_{n+1} - a_n = n^3+n^2+1$$
for all positive integers $n$. | a_n = n |
39,289 | 15. As shown in Figure $2, A$ is a fixed point on the circle $\odot O$ with radius 1, and $l$ is the tangent line of $\odot O$ passing through point $A$. Let $P$ be a point on $\odot O$ different from $A$, and $P Q \perp$ $l$, with the foot of the perpendicular being $Q$. When point $P$ moves on $\odot O$, the maximum value of the area of $\triangle P A Q$ is $\qquad$ | \frac{3 \sqrt{3}}{8} |
39,291 | Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a$ be the three-letter string received when $aaa$ is transmitted and let $S_b$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a$ comes before $S_b$ in alphabetical order. When $p$ is written as a fraction in lowest terms, what is its numerator? | 532 |
39,314 | Example 2 If $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right)$, where $\theta \in[0,2 \pi)$, then the range of values for $\theta$ is $\qquad$ [2]
(2011, National High School Mathematics Joint Competition) | \left(\frac{\pi}{4}, \frac{5 \pi}{4}\right) |
39,319 | Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$. | a_n = 2n - 1 |
39,343 | Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a+b+c=d$. | 11 |
39,361 | 1. Let $x^{2}+y^{2} \leqslant 2$. Then the maximum value of $\left|x^{2}-2 x y-y^{2}\right|$ is $\qquad$ . | 2 \sqrt{2} |
39,375 | One, (20 points) In a certain competition, each player plays exactly one game against every other player. The winner of each game gets 1 point, the loser gets 0 points, and in the case of a draw, both get 0.5 points. After the competition, it is found that each player's score is exactly half from games played against the 10 lowest-scoring players (the 10 lowest-scoring players each have exactly half of their points from games played against each other). Find the number of participants in the competition. | 25 |
39,378 | 11. Given positive numbers $a, b, c, d, e, f$, which simultaneously satisfy
$$
\begin{array}{l}
\frac{b c d e f}{a}=\frac{1}{2}, \frac{a c d e f}{b}=\frac{1}{4}, \frac{a b d e f}{c}=\frac{1}{8}, \\
\frac{a b c e f}{d}=2, \frac{a b c d f}{e}=4, \frac{a b c d e}{f}=8 .
\end{array}
$$
Then the value of $a+b+c+d+e+f$ is $\qquad$ | \frac{5}{2}+\frac{15 \sqrt{2}}{4} |
39,394 | A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$, and the slope of line $AB$ is $-\frac 23$. Given that $AB=CD$ and $BC< AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac mn$, where $m,n$ are two relatively prime positive integers. Find $100m+n$.
[i] Proposed by Yannick Yao [/i] | 1706 |
39,422 | 5. In the Cartesian coordinate system $x o y$, the area of the figure formed by points whose coordinates satisfy the condition $\left(x^{2}+y^{2}+2 x+2 y\right)\left(4-x^{2}-y^{2}\right) \geqslant 0$ is
保留源文本的换行和格式，直接输出翻译结果如下：
5. In the Cartesian coordinate system $x o y$, the area of the figure formed by points whose coordinates satisfy the condition $\left(x^{2}+y^{2}+2 x+2 y\right)\left(4-x^{2}-y^{2}\right) \geqslant 0$ is | 2 \pi + 4 |
39,430 | 1. Given a positive integer $n$ such that the last two digits of $3^{n}$ form a two-digit prime number. Then the sum of all $n$ that satisfy this condition and do not exceed 2010 is $\qquad$ . | 909128 |
39,446 | Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\frac mn$, where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive integers, find $m+n^{}_{}$. | 164 |
39,449 | Example 2 Calculate
$$
\left(1+\cos \frac{\pi}{7}\right)\left(1+\cos \frac{3 \pi}{7}\right)\left(1+\cos \frac{5 \pi}{7}\right)
$$
the value.
$(2017$, Peking University Boya Talent Program Exam) | \frac{7}{8} |
39,451 | II. (15 points) As shown in Figure 4, in the right triangle $\triangle ABC$, $\angle ACB=90^{\circ}$, point $D$ is on side $CA$ such that
$$
\begin{array}{l}
CD=1, DA=3, \text { and } \\
\angle BDC=3 \angle BAC .
\end{array}
$$
Find the length of $BC$. | \frac{4\sqrt{11}}{11} |
39,454 | 2. In a regular tetrahedron $ABCD$, let
$$
\overrightarrow{AE}=\frac{1}{4} \overrightarrow{AB}, \overrightarrow{CF}=\frac{1}{4} \overrightarrow{CD} \text {, }
$$
$\vec{U} \overrightarrow{DE}$ and $\overrightarrow{BF}$ form an angle $\theta$. Then $\cos \theta=$ | -\frac{4}{13} |
39,460 | 2. In an isosceles right $\triangle ABC$, $\angle C=90^{\circ}$, points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, $FD=FE=\frac{AC}{2}$, $\angle DFE=90^{\circ}$. Then $FC: CE: EF=$ $\qquad$ . | 3: 4: 5 |
39,463 | 13. (15 points) In the sequence $\left\{a_{n}\right\}$,
$$
a_{n}=2^{n} a+b n-80\left(a 、 b \in \mathbf{Z}_{+}\right) \text {. }
$$
It is known that the minimum value of the sum of the first $n$ terms $S_{n}$ is obtained only when $n=6$, and $7 \mid a_{36}$. Find the value of $\sum_{i=1}^{12}\left|a_{i}\right|$. | 8010 |
39,532 | Five, color the numbers in $S=\{0,1,2, \cdots, n\}$ with two colors arbitrarily. Find the smallest positive integer $n$, such that there must exist $x, y, z \in S$ of the same color, satisfying $x+y=2 z$. | 8 |
39,541 | The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex [roots of unity](https://artofproblemsolving.com/wiki/index.php/Roots_of_unity). The set $C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C_{}^{}$? | 144 |
39,552 | In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event? | 43890 |
39,557 | ## C2.
There are 2016 costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all of them were in the shop at a specific time instance or no two of them were both in the shop at any time instance.
| 45 |
39,577 | Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$. | (2, 2) |
39,581 | 5. In $\triangle A B C$, $A B=A C=7, B C=4$, point $M$ is on $A B$, and $B M=\frac{1}{3} A B$. Draw $E F \perp B C$, intersecting $B C$ at $E$ and the extension of $C A$ at $F$. Then the length of $E F$ is $\qquad$ | 5 \sqrt{5} |
39,584 | 7. If the three sides of $\triangle A B C$ are all unequal, the area is $\frac{\sqrt{15}}{3}$, and the lengths of the medians $A D$ and $B E$ are 1 and 2, respectively, then the length of the median $C F$ is $\qquad$. | \sqrt{6} |
39,592 | Three. (25 points) Let the quadratic function $f(x)=x^{2}+a x+b$, $F=\max _{|x| \leq 1} |f(x)|$. When $a, b$ traverse all real numbers, find the minimum value of $F$.
| \frac{1}{2} |
39,597 | 2. Let $x, y, z$ be real numbers, not all zero. Then the maximum value of the function $f(x, y, z)=\frac{x y+y z}{x^{2}+y^{2}+z^{2}}$ is $\qquad$ | \frac{\sqrt{2}}{2} |
39,644 | Example 2 Given that the volume of a rectangular prism is 1, the sum of its length, width, and height is $k$, and its surface area is $2k$. Find the range of the real number $k$.
| [3,+\infty) |
39,652 | 1. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n+1} \leqslant \frac{a_{n+2}+a_{n}}{2}, a_{1}=1, a_{403}=2011 \text {. }
$$
Then the maximum value of $a_{5}$ is $\qquad$ | 21 |
39,656 | A positive integer $n\geq 4$ is called [i]interesting[/i] if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$ | 404 |
39,659 | Question 1 Find the minimum value of the function $y=2 \sqrt{(x-1)^{2}+4}+$ $\sqrt{(x-8)^{2}+9}$. | 5 \sqrt{5} |
39,667 | Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$. | 50 |
39,668 | Example 3 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} |
39,670 | A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library? | \frac{95}{159} |
39,672 | Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square. | 121 |
39,682 | Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by [i]Nikola Velov, Macedonia[/i] | 100 |
39,692 | 30. Let $f(x, y, z)=\sin ^{2}(x-y)+\sin ^{2}(y-z)+\sin ^{2}(z-x), x, y, z \in \mathbf{R}$, find the maximum value of $f(x, y, z)$. (2007 Zhejiang Province Mathematics Competition Problem) | \frac{9}{4} |
39,714 | 6. It is known that the selling price of a certain model of car is 230,000 yuan per unit. A factory's total cost for producing this model of car in a year consists of fixed costs and production costs. The fixed cost for one year is 70,000,000 yuan. When producing $x$ units of this car in a year, the production cost for each car is $\frac{70-\sqrt{x}}{3 \sqrt{x}}$ ten thousand yuan $(0<x<1000)$. To ensure that the factory's sales revenue from producing this model of car in a year is no less than the total cost, the factory needs to produce at least $\qquad$ units of this car. | 318 |
39,719 | 2. Let the volume of tetrahedron $ABCD$ be $V$, $E$ be the midpoint of edge $AD$, and point $F$ be on the extension of $AB$ such that $BF = AB$. The plane through points $C$, $E$, and $F$ intersects $BD$ at point $G$. Then the volume of tetrahedron $CDGE$ is | \frac{1}{3} V |
39,720 | Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$, rounded to the nearest integer? | 167 |
39,722 |
2. Let $a_{1}, a_{2}, \ldots, a_{2 n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let
(i) $a_{1}^{2}+a_{3}^{2}+\cdots+a_{2 n-1}^{2}=x$,
(ii) $a_{2}^{2}+a_{4}^{2}+\cdots+a_{2 n}^{2}=y$, and
(iii) $a_{n}+a_{n+1}=z$.
Express $d$ in terms of $x, y, z, n$.
| \frac{y-x}{nz} |
39,726 | 10. If $0<a, b, c<1$ satisfy the condition $ab+bc+ca=1$, then the minimum value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is $\qquad$ | \frac{3(3+\sqrt{3})}{2} |
39,727 | Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$? | 120 |
39,730 | 24. If the inequality $\cos ^{2} x+2 p \sin x-2 p-2<0$ holds for any real number $x$, find the range of $p$.
| (1-\sqrt{2},+\infty) |
39,732 | For the NEMO, Kevin needs to compute the product
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
[i]Proposed by Evan Chen[/i] | 870 |
39,733 | Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 71 |
39,749 | 3. The set of positive integer solutions $(x, y)$ for the indeterminate equation $3 \times 2^{x}+1=y^{2}$ is $\qquad$ | (3,5),(4,7) |
39,754 | To 9. On a plane, there is a fixed point $P$, consider all possible equilateral triangles $ABC$, where $AP=3, BP=2$. What is the maximum length of $CP$? (1961 Autumn Competition) | 5 |
39,775 | 1. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E$, $F$, and $G$ are the midpoints of edges $B C$, $C C_{1}$, and $C D$ respectively. Then the angle formed by line $A_{1} G$ and plane $D E F$ is $\qquad$ | 90^{\circ} |
39,817 | Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$ | 2^{1994} |
39,819 | 23 Let $k$ and $m$ be positive integers. Find the minimum possible value of $\left|36^{k}-5^{m}\right|$. | 11 |
39,832 | Question 2 In the tetrahedron $P-ABC$, the base is an equilateral triangle with side length 3, $PA=3, PB=4, PC=5$. Then the volume $V$ of the tetrahedron $P-ABC$ is $\qquad$ | \sqrt{11} |
39,836 | Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish. | 9802 |
39,839 | Three, (50 points) Find the smallest positive integer $n$ such that there exists an $(n+1)$-term sequence $a_{0}, a_{1}, \cdots, a_{n}$, satisfying $a_{0}=0, a_{n}=2008$, and
$$
\left|a_{i}-a_{i-1}\right|=i^{2}(i=1,2, \cdots, n) .
$$ | 19 |
39,843 | 21. (1) Given that $a, b, c, d, e, f$ are real numbers, and satisfy the following two equations: $a+b+c+d+e+f=10, (a-1)^{2}+(b-1)^{2}+(c-1)^{2}+(d-1)^{2}+(e-1)^{2}+(f-1)^{2}=6$, find the maximum value of $f$. (1993 Balkan Mathematical Olympiad)
(2) Given that $a, b, c, d \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, and satisfy the following two equations: $\sin a+\sin b+\sin c+\sin d=1, \cos 2a+\cos 2b+\cos 2c+\cos 2d \geqslant \frac{10}{3}$, prove that $0 \leqslant a, b, c, d \leqslant \frac{\pi}{6}$. (1985 Balkan Mathematical Olympiad) | \frac{10}{3} |
39,858 | Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells. Answer. $2 n$.
Translate the above text into English, please retain the line breaks and format of the source text, and output the translation result directly.
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells. Answer. $2 n$. | 2 n |
39,867 | In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament? | 25 |
39,895 | 3. Given $a, b, c \in \mathbf{R}_{+}$, and $abc=4$. Then the minimum value of the algebraic expression $a^{a+b} b^{3b} c^{c+b}$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 64 |
39,901 | 5. In the non-decreasing sequence of positive odd numbers $\{1,3,3,3,5,5,5, 5,5, \cdots\}$, each positive odd number $k$ appears $k$ times. It is known that there exist integers $b$, $c$, and $d$, such that for all integers $n$, $a_{n}=$ $b[\sqrt{n+c}]+d$, where $[x]$ denotes the greatest integer not exceeding $x$. Then $b+c+d$ equals | 2 |
39,911 | Example 10 In $\triangle A B C$, $\angle A B C=50^{\circ}$, $\angle A C B=30^{\circ}, R$ is a point inside the triangle, $\angle R A C=$ $\angle R C B=20^{\circ}$. Find the degree measure of $\angle R B C$. | 20^{\circ} |
39,917 | ## Aufgabe 4 - 110934
In einem Rechteck $A B C D$ mit $A B=C D=a$ und $B C=D A=b,(a>b)$ schneide die Halbierende des Winkels $\angle B A D$ die Seite $C D$ in $S_{1}$. Weiter sei $S_{2}$ der Mittelpunkt von $A B$.
Ermitteln Sie das Verhältnis $a: b$ der Seitenlängen eines solchen Rechtecks, bei dem die Halbierende des Winkels $\angle A S 2 C$ die Seite $C D$ in $S_{1}$ schneidet!
| :b=8:3 |
39,952 | Find all odd integers $n \geq 1$ such that $n$ divides $3^{n}+1$.
| 1 |
39,965 | Example 2 Let the side length of the equilateral $\triangle ABC$ be $2, M$ is the midpoint of $AB$, $P$ is any point on $BC$, and $PA+PM$ are denoted as $s$ and $t$ for their maximum and minimum values, respectively. Then $s^{2}-t^{2}=$ $\qquad$
(2000, National Junior High School Mathematics League) | 4 \sqrt{3} |
39,966 | II. (30 points) Given positive numbers $m, n$ are the roots of the quadratic equation $x^{2}+$ $p x+q=0$, and $m^{2}+n^{2}=3, m n=1$. Find the value of the polynomial $x^{3}-(\sqrt{5}-1) x^{2}-(\sqrt{5}-1) x+1994$. | 1993 |
39,968 | 1. Given that $a, b, x, y$ are positive real numbers, satisfying:
$$
a+b+\frac{1}{a}+\frac{9}{b}=8, a x^{2}+b y^{2}=18 \text {. }
$$
Then the range of values for $a x+b y$ is . $\qquad$ | (3 \sqrt{2}, 6 \sqrt{2}] |
39,975 | Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$. What is the maximum possible value for $n$?
[i]Proposed by James Lin[/i] | 5 |
39,988 | 4. Let $a$ be a positive number. Then the function
$$
f(x)=a \sin x+\sqrt{a} \cos x\left(-\frac{\pi}{2} \leqslant x \leqslant \frac{\pi}{2}\right)
$$
the sum of the maximum and minimum values $g(a)(a>0)$ is in the range $\qquad$ . | \left(0, \frac{1}{2}\right) |
39,993 | Example 9 On the square $A B C D$: there are 10 points, 8 of which are inside $\triangle A B C$, and 2 points are on the side $\mathrm{I}$ of the square (not at the vertices). [L These 10 points, together with points $A, B, C, D$, are not collinear. Find how many small triangles these 10 points, along with the 4 vertices of the square, can divide the square into. | 20 |
40,011 | 7. Let set $S$ contain $n$ elements, $A_{1}, A_{2}, \cdots, A_{k}$ are different subsets of $S$, they all have non-empty intersections with each other, and no other subset of $S$ can intersect with all of $A_{1}, A_{2}, \cdots, A_{k}$. Express this in terms of $n$:
| 2^{n-1} |
40,025 | 4. Given that the graph of a quadratic function is tangent to the lines containing the three sides of $\triangle A B C$. If the coordinates of the vertices of the triangle are $A\left(\frac{1}{4}, \frac{3}{2}\right), B\left(\frac{1}{2}, 1\right), C\left(\frac{3}{4}, 1\right)$, then the expression of the quadratic function is | y=x^{2}-2 x+2 |
40,041 | a) Prove that, whatever the real number x would be, the following inequality takes place
${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$
b) Solve the following system in the set of real numbers:
${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$.
The Mathematical Gazette | (1, 1, 1) |
40,045 | Example 2 In the acute triangle $\triangle ABC$, $\angle A=30^{\circ}$, a circle is constructed with $BC$ as its diameter, intersecting $AB$ and $AC$ at $D$ and $E$ respectively. Connecting $DE$ divides $\triangle ABC$ into $\triangle ADE$ and quadrilateral $DBCE$, with areas denoted as $S_{1}$ and $S_{2}$ respectively. Then $S_{1}: S_{2}=$ $\qquad$ | 3:1 |
40,048 | Calculate the following indefinite integrals.
[1] $\int \sin x\sin 2x dx$
[2] $\int \frac{e^{2x}}{e^x-1}dx$
[3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$
[4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$
[5] $\int \frac{e^x}{e^x+1}dx$ | \ln |e^x + 1| + C |
40,057 | Example 4. Find the length of the chord obtained by the intersection of the line $\left\{\begin{array}{l}x=-3+2 t, \\ y=3 t\end{array}\right.$ and the ellipse $2 x^{2}+4 x y+5 y^{2}-4 x-22 y+7=0$. | \frac{4}{77} \sqrt{5330} |
40,079 | Let the four-digit number $\overline{a b c d}$ be a perfect square, and its arithmetic square root can be expressed as $\sqrt{\overline{a b c d}}=\overline{a b}+\sqrt{\overline{c d}}$. How many such four-digit numbers are there? | 9 |
40,082 | Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$. | 5 |
40,131 | 5. Given the equation in $x$
$$
x^{4}-6 x^{3}+a x^{2}+6 x+b=0
$$
the left side can be divided by $x-1$, and the remainder when divided by $x+2$ is 72. Then all the solutions of this equation (in ascending order) are $\qquad$ | -1,1,2,4 |
40,132 | 3. Consider a complete graph with $n$ vertices. The vertices and edges of this complete graph are colored according to the following rules:
(1) Two edges emanating from the same vertex have different colors;
(2) The color of each vertex is different from the colors of the edges emanating from it.
For each fixed $n$, find the minimum number of colors required. | n |
40,136 | 8. Let $p, q$ be prime numbers, and satisfy $p^{3}+q^{3}+1=p^{2} q^{2}$. Then the maximum value of $p+q$ is | 5 |
40,170 | What is the largest $n$ such that there exists a non-degenerate convex $n$-gon such that each of its angles are an integer number of degrees, and are all distinct? | 26 |
40,194 | The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD.
Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$. | \frac{p^2 - 4}{4} |
40,201 | 4. Given $\left\{\begin{array}{l}\frac{x+9 y}{9 x+7 y}=\frac{m}{n}, \\ \frac{x+9 y}{9 x+8 y}=\frac{m+a n}{b m+c n} .\end{array}\right.$
If all real numbers $x, y, m, n$ that satisfy (1) also satisfy (2), then the value of $a+b+c$ is $\qquad$. | \frac{41}{37} |
40,230 | 12. $A, B, C, D$ are the 4 vertices of a regular tetrahedron, with each edge length being $1 \, \text{m}$. A gecko starts from point $A$ and crawls along the edges, following these rules: it does not change direction midway along any edge, and when it reaches each vertex, it has an equal probability of choosing any of the three edges that meet at that vertex to continue crawling. What is the probability that the gecko, after crawling $7 \, \text{m}$, returns to vertex $A$? | \frac{182}{729} |
40,250 | 5. Let the set of all permutations $X=(x_{1}, x_{2}, \cdots, x_{9})$ of $1,2, \cdots, 9$ be $A$. For any $X \in A$, let
\[
\begin{array}{l}
f(X)=x_{1}+2 x_{2}+\cdots+9 x_{9}, \\
M=\{f(X) \mid X \in A\} .
\end{array}
\]
Find $|M|$ (where $|M|$ denotes the number of elements in the set $M$).
(Xiong Bin) | 121 |
40,263 | An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle. | n(n+1) |
40,265 | There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2^{x})(3^{y})(5^{z})$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 9$. At step i of a 1000-step process, the $i$-th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$-th switch. After step 1000 has been completed, how many switches will be in position $A$? | 650 |
40,293 | Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$. | 17 |
40,302 | Example 4 Find the value of the finite continued fraction $\langle-2,1,2 / 3,2,1 / 2,3\rangle$. | -\frac{143}{95} |
40,336 | A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called [b]diagonal[/b] if the two cells in it [i]aren't[/i] in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image. | 2892 |
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