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43,827 | Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.
[i]Proposed by tkhalid[/i] | 5 |
43,835 | Let $n$ be a natural number, $\alpha_{n} \backslash \beta_{n}\left(\alpha_{n}>\beta_{n}\right)$ are the integer parts of the roots of the quadratic equation $x^{2}-2(n+2) x+3(n+1)=0$. Find the value of $\frac{\alpha_{1}}{\beta_{1}}+\frac{\alpha_{2}}{\beta_{2}}+\cdots+\frac{\alpha_{99}}{\beta_{99}}$. | 10098 |
43,842 | 11. Given the function
$$
f(x)=\left\{\begin{array}{ll}
2^{x}-1, & x \leqslant 0 ; \\
f(x-1)+1, & x>0 .
\end{array}\right.
$$
Let the sum of all real roots of the equation $f(x)=x$ in the interval $(0, n]$ be $S_{n}$. Then the sum of the first $n$ terms of the sequence $\left\{\frac{1}{S_{n}}\right\}$ is $T_{n}=$ $\qquad$ | \frac{2 n}{n+1} |
43,845 | Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. | x_i = 2 |
43,952 | 17. 14 people participate in a Japanese chess round-robin tournament, where each person plays against the other 13 people, and there are no draws in the matches. Find the maximum number of "triangles" (here, a "triangle" refers to a set of three people where each person has one win and one loss).
(2002, Japan Mathematical Olympiad (First Round)) | 112 |
43,965 | ## Problem 3
Three vertices of a regular $2 n+1$ sided polygon are chosen at random. Find the probability that the center of the polygon lies inside the resulting triangle.
| \frac{n+1}{4n-2} |
43,971 | Example 2 In $\triangle A B C$, $\angle A B C=60^{\circ}, O, H$ are the circumcenter and orthocenter of $\triangle A B C$ respectively. Points $D, E$ lie on sides $B C, A B$ respectively, such that $B D=B H, B E=B O$. Given that $B O=1$. Find the area of $\triangle B D E$. ${ }^{[1]}$
(2008, International Youth Math City Invitational Competition) | \frac{\sqrt{3}}{4} |
44,006 | In triangle $ABC$, let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$. Find $OH$.
[i]Proposed by Kevin You[/i] | 10 |
44,050 | 5. Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies:
(1) $f(1)=1$;
(2) When $00$;
(3) For any real numbers $x, y$,
$$
f(x+y)-f(x-y)=2 f(1-x) f(y) \text {. }
$$
Then $f\left(\frac{1}{3}\right)=$ . $\qquad$ | \frac{1}{2} |
44,093 | A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$. | 60 |
44,140 | Example 12 Let $x, y, z$ be real numbers greater than -1. Find the minimum value of
$$\frac{1+x^{2}}{1+y+z^{2}}+\frac{1+y^{2}}{1+z+x^{2}}+\frac{1+z^{2}}{1+x+y^{2}}$$ | 2 |
44,166 | The measures, in degrees, of the angles , $\alpha, \beta$ and $\theta$ are greater than $0$ less than $60$. Find the value of $\theta$ knowing, also, that $\alpha + \beta = 2\theta$ and that $$\sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ).$$ | 30^\circ |
44,167 | The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$ | \frac{3^{n+2} + 3}{2} |
44,175 | 12. A. As shown in Figure 5, in rectangle $A B C D$, $A B=a$, $B C=b$, and $a<b$. A line through the center $O$ of rectangle $A B C D$ intersects segments $B C$ and $D A$ at points $E$ and $F$, respectively. The quadrilateral $E C D F$ is folded along $E F$ to the plane of quadrilateral $B E F A$, such that point $C$ coincides with point $A$, resulting in quadrilateral EFGA.
(1) Prove that the area of pentagon $A B E F G$ is $\frac{a\left(3 b^{2}-a^{2}\right)}{4 b}$;
(2) If $a=1$ and $b$ is a positive integer, find the minimum value of the area of pentagon $A B E F G$. | \frac{11}{8} |
44,179 | 3. Given the function
$$
f(x)=\sin \omega x+\cos \omega x(\omega>0)(x \in \mathbf{R}) \text {. }
$$
If the function $f(x)$ is monotonically increasing in the interval $(-\omega, \omega)$, and the graph of the function $y=f(x)$ is symmetric about the line $x=\omega$, then $\omega=$ . $\qquad$ | \frac{\sqrt{\pi}}{2} |
44,180 |
Problem 9.1. Let $f(x)=x^{2}+6 a x-a$ where $a$ is a real parameter.
a) Find all values of $a$ for which the equation $f(x)=0$ has at least one real root.
b) If $x_{1}$ and $x_{2}$ are the real roots of $f(x)=0$ (not necessarily distinct) find the least value of the expression
$$
A=\frac{9 a-4 a^{2}}{\left(1+x_{1}\right)\left(1+x_{2}\right)}-\frac{70 a^{3}+1}{\left(1-6 a-x_{1}\right)\left(1-6 a-x_{2}\right)}
$$
| -\frac{89}{81} |
44,184 | Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \] | 2^{n-1} |
44,196 | Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$
| 8 |
44,201 | Four, (20 points) Given $f(x)=2 x-\frac{2}{x^{2}}+\frac{a}{x}$, where the constant $a \in(0,4]$. Find all real numbers $k$, such that for any $x_{1} 、 x_{2} \in \mathbf{R}_{+}$, it always holds that
$$
\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \geqslant k\left|x_{1}-x_{2}\right| .
$$ | \left(-\infty, 2-\frac{a^{3}}{108}\right] |
44,209 | 4. As shown in the upper right figure, the radii of semicircles $A$ and $B$ are equal, they are tangent to each other, and both are internally tangent to the semicircle $O$ with radius 1. $\odot O_{1}$ is tangent to both of them, and $\odot O_{2}$ is tangent to $\odot O_{1}$, semicircle $O$, and semicircle $B$. Then the radius $r$ of $\odot O_{2}$ is $\qquad$ | \frac{1}{6} |
44,210 | The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan Chen[/i] | 336 |
44,222 | Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are positive integers. What is $a+b$?
[i]2021 CCA Math Bonanza Individual Round #12[/i] | 112 |
44,223 | Three, (50 points) If the three sides of a triangle are all rational numbers, and one of its interior angles is also a rational number, find all possible values of this interior angle.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 60^{\circ}, 90^{\circ}, 120^{\circ} |
44,224 | 11. (20 points) Let the function
$$
f(x)=\frac{1+\ln (x+1)}{x} \text {, }
$$
$k$ is a positive integer. When $x>0$, $f(x)>\frac{k}{x+1}$ always holds. Find the maximum value of $k$.
untranslated part:
$k$ is a positive integer. When $x>0$, $f(x)>\frac{k}{x+1}$ always holds. Find the maximum value of $k$.
(Note: The last part was already in English, so it remains unchanged.) | 3 |
44,238 | \section*{Problem 7}
A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets the other two sides at \(\mathrm{X}\) and \(\mathrm{Y}\). What is the maximum length \(\mathrm{XY}\), if the triangle has perimeter \(\mathrm{p}\) ?
| \frac{p}{8} |
44,240 | Example 2-3 Let quadrilateral $A B C D$ be a rectangle with an area of 2, $P$ a point on side $C D$, and $Q$ the point where the incircle of $\triangle P A B$ touches side $A B$. The product $P A \cdot P B$ varies with the changes in rectangle $A B C D$ and point $P$. When $P A \cdot P B$ is minimized,
(1) Prove: $A B \geqslant 2 B C$;
(2) Find the value of $A Q \cdot B Q$.
(2001, China Western Mathematical Olympiad) | 1 |
44,255 | There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery? | 1024 |
44,267 | Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. | 3 |
44,287 | If $ x,y$ are positive real numbers with sum $ 2a$, prove that :
$ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$
When does equality hold ?
Babis | 4a^{10} |
44,289 | Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$. | 251 |
44,293 | 11. (20 points) It is known that a box contains 100 red and 100 blue cards, each color of cards containing one card labeled with each of the numbers $1, 3, 3^2, \cdots, 3^{99}$. The total sum of the numbers on the cards of both colors is denoted as $s$.
For a given positive integer $n$, if it is possible to pick several cards from the box such that the sum of their labels is exactly $n$, then it is called a "scheme for $n$". The number of different schemes for $n$ is denoted as $f(n)$. Try to find the value of $f(1) + f(2) + \cdots + f(s)$. | 2^{200}-1 |
44,304 | Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$. | -1 |
44,307 | II. (25 points) Given the quadratic function
$$
y=x^{2}+b x-c
$$
the graph passes through three points
$$
P(1, a), Q(2,3 a)(a \geqslant 3), R\left(x_{0}, y_{0}\right) .
$$
If the centroid of $\triangle P Q R$ is on the $y$-axis, find the minimum perimeter of $\triangle P Q R$. | 4 \sqrt{2}+5 \sqrt{5}+\sqrt{37} |
44,310 | 1 In $\triangle A B C$, $\angle C=90^{\circ}, \angle B=$ $30^{\circ}, A C=2, M$ is the midpoint of $A B$, and $\triangle A C M$ is folded along $C M$ such that the distance between $A$ and $B$ is $2 \sqrt{2}$. At this moment, the volume of the tetrahedron $A-B C M$ is $\qquad$ (1998, National High School League) | \frac{2 \sqrt{2}}{3} |
44,331 | 3. In the tetrahedron $P A B C$, it is known that
$$
\angle A P B=\angle B P C=\angle C P A=90^{\circ} \text {, }
$$
the sum of the lengths of all edges is $S$. Then the maximum volume of this tetrahedron is $\qquad$ . | \frac{S^{3}}{162(1+\sqrt{2})^{3}} |
44,337 | 1. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the size of the dihedral angle $B-A_{1} C-D$ is $\qquad$
(2016, National High School Mathematics League Fujian Province Preliminary)
Hint Establish a spatial rectangular coordinate system as shown in Figure 11. | 120^{\circ} |
44,344 | Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$. | 11 |
44,353 |
5. Find all real $a$ for which there exists a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x-f(y))=f(x)+a[y]$ for every real $x$ и $y$ ( $[y]$ denotes the integral part of $y$ ).
Answer: $a=-n^{2}$ for arbitrary integer $n$.
| -n^{2} |
44,385 | Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$ | x = y = z = \sqrt{2} |
44,417 | 6. Given that the hyperbola has asymptotes $2 x \pm y=0$, and it passes through the intersection point of the lines $x+y-3=0$ and $2 x-y+3 t=0$, where $-2 \leqslant t \leqslant 5$. Then the maximum possible value of the real axis length of the hyperbola is $\qquad$ . | 4 \sqrt{3} |
44,426 | Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$. Furthermore, each segment $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2450$, that have rational length. | 20 |
44,433 | Example 4 Let $x_{1}, x_{2}, \cdots, x_{n}$ and $a_{1}, a_{2}, \cdots, a_{n}$ be any two sets of real numbers satisfying the conditions:
(1) $\sum_{i=1}^{n} x_{i}=0$;
(2) $\sum_{i=1}^{n}\left|x_{i}\right|=1$;
(3) $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}, a_{1}>a_{n}$
For any $n \geqslant 2$, try to find the range of values for $A$ such that the inequality $\left|\sum_{i=1}^{n} a_{i} x_{i}\right| \leqslant A\left(a_{1}-a_{n}\right)$ always holds. | \frac{1}{2} |
44,449 | 9. (16 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=\frac{1}{3}, \frac{a_{n-1}}{a_{n}}=\frac{2 n a_{n-1}+1}{1-a_{n}}(n \geqslant 2) .
$$
Find the value of $\sum_{n=2}^{\infty} n\left(a_{n}-a_{n+1}\right)$. | \frac{13}{24} |
44,474 | Let $S$ be the [set](https://artofproblemsolving.com/wiki/index.php/Set) of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | 913 |
44,478 | Let $F=\max _{1<x<3}\left|x^{3}-a x^{2}-b x-c\right|$, when $a$, $b$, $c$ take all real numbers, find the minimum value of $F$. | \frac{1}{4} |
44,480 | Let $a=256$. Find the unique real number $x>a^2$ such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
[i]Proposed by James Lin.[/i] | 2^{32} |
44,486 | In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$.
[i]Proposed by Ray Li[/i] | 55 |
44,512 | 1. If the function
$$
f(x)=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)
$$
has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is . $\qquad$ | 2 \sqrt{3} |
44,530 | Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value. | \frac{3}{2}e^{-\frac{1}{3}} |
44,542 | Compute the positive difference between the two real solutions to the equation
$$(x-1)(x-4)(x-2)(x-8)(x-5)(x-7)+48\sqrt 3 = 0.$$ | \sqrt{25 + 8\sqrt{3}} |
44,547 | 1. The smallest positive odd number that cannot be expressed as $7^{x}-3 \times 2^{y}\left(x 、 y \in \mathbf{Z}_{+}\right)$ is $\qquad$ | 3 |
44,563 | 1. Given
$$
x \sqrt{x^{2}+3 x+18}-x \sqrt{x^{2}-6 x+18}=1 \text {. }
$$
then the value of $2 x \sqrt{x^{2}-6 x+18}-9 x^{3}$ is | -1 |
44,577 |
A9 Consider an integer $n \geq 4$ and a sequence of real numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$. An operation consists in eliminating all numbers not having the rank of the form $4 k+3$, thus leaving only the numbers $x_{3}, x_{7}, x_{11}, \ldots$ (for example, the sequence $4,5,9,3,6,6,1,8$ produces the sequence 9,1 . Upon the sequence $1,2,3, \ldots, 1024$ the operation is performed successively for 5 times. Show that at the end only 1 number remains and find this number.
| 683 |
44,580 | One, (40 points) Find the smallest integer $c$, such that there exists a sequence of positive integers $\left\{a_{n}\right\}(n \geqslant 1)$ satisfying:
$$
a_{1}+a_{2}+\cdots+a_{n+1}<c a_{n}
$$
for all $n \geqslant 1$. | 4 |
44,591 | A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is $m-n\sqrt[3]{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m+n+p.$ | 52 |
44,612 | 2. Find the smallest positive real number $k$ such that for any 4 distinct real numbers $a, b, c, d$ not less than $k$, there exists a permutation $p, q, r, s$ of $a, b, c, d$ such that the equation $\left(x^{2}+p x+q\right)\left(x^{2}+r x+s\right)=0$ has 4 distinct real roots. (Feng Zhigang) | 4 |
44,621 | Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$. | f(x) = c |
44,642 | Let $r_1$, $r_2$, $\ldots$, $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$. Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$. Determine the minimum and maximum values of $f(n)$. Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$. | 0 \text{ and } m-1 |
44,653 | 1. Given $\operatorname{ctg} \theta=\sqrt[3]{7}\left(0^{\circ}<\theta<90^{\circ}\right)$. Then, $\frac{\sin ^{2} \theta+\sin \theta \cos \theta+2 \cos ^{2} \theta}{\sin ^{2} \theta+\sin \theta \cos \theta+\cos ^{2} \theta}=$ $\qquad$ . | \frac{13-\sqrt[3]{49}}{6} |
44,694 | Four. (20 points) Let $a_{1} \in \mathbf{R}_{+}, i=1,2, \cdots, 5$. Find
$$
\begin{array}{l}
\frac{a_{1}}{a_{2}+3 a_{3}+5 a_{4}+7 a_{5}}+\frac{a_{2}}{a_{3}+3 a_{4}+5 a_{5}+7 a_{1}}+ \\
\cdots+\frac{a_{5}}{a_{1}+3 a_{2}+5 a_{3}+7 a_{4}}
\end{array}
$$
the minimum value. | \frac{5}{16} |
44,696 | Let $A B C$ be a triangle in which $\angle A B C=60^{\circ}$. Let $I$ and $O$ be the incentre and circumcentre of $A B C$, respectively. Let $M$ be the midpoint of the arc $B C$ of the circumcircle of $A B C$, which does not contain the point $A$. Determine $\angle B A C$ given that $M B=O I$. | 30^{\circ} |
44,705 | Let $a, b, c, d$ be the roots of the quartic polynomial $f(x) = x^4 + 2x + 4$. Find the value of
$$\frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}.$$ | \frac{3}{2} |
44,718 | 9. Let $[x]$ be the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1, \quad n \in N^{*}$, find $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.
| 2015 |
44,721 | 7. Given an acute angle $\alpha$ satisfies the equation
$$
\begin{array}{l}
\sin \left(2 \alpha-20^{\circ}\right) \cdot \sin \left(2 \alpha-10^{\circ}\right) \\
=\cos \left(\alpha-10^{\circ}\right) \cdot \sin 10^{\circ} .
\end{array}
$$
Then $\alpha=$ . $\qquad$ | 20^{\circ} |
44,735 | Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$. | 6 |
44,763 | 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 speci c time instance or no two of them were both in the shop at any time instance. | 45 |
44,771 | Example 3 Let the complex numbers $z_{1}$ and $z_{2}$ satisfy
$$
\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3} \text {. }
$$
Find the value of $\log _{3}\left|\left(z_{1} \bar{z}_{2}\right)^{2000}+\left(\bar{z}_{1} z_{2}\right)^{2000}\right|$.
(1991, National High School Mathematics Competition) | 4000 |
44,786 | Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays. | m = 2^{\frac{n}{2}} |
44,800 | In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$.
A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?
[i]Proposed by Michael Kural[/i] | 17 |
44,802 | What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?
| 3 |
44,857 | 10. For the string of three letters $\hat{\mathrm{i}}$ "aaa" and "bbb", they are transmitted through a circuit, with each string being sent one letter at a time. Due to issues with the line, each of the 6 letters has a $\frac{1}{3}$ chance of being transmitted incorrectly (an $a$ is received as a $b$, or a $b$ is received as an $a$), and the correctness of each letter's reception is independent of the others. Let $S_{\mathrm{a}}$ be the message received when "aaa" is transmitted, and $S_{\mathrm{b}}$ be the message received when "bbb" is transmitted. Let $P$ be the probability that $S_{\mathrm{a}}$ is lexicographically before $S_{\mathrm{b}}$. When $P$ is written as a reduced fraction, what is the numerator? | 532 |
44,896 | 10. Let $S$ be the area of a triangle inscribed in a circle of radius 1. Then the minimum value of $4 S+\frac{9}{S}$ is $\qquad$ . | 7 \sqrt{3} |
44,912 | Let $T = TNFTPP$. As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime.
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separately, [hide=here's the value of T]$T=2161$[/hide]. | 1801 |
44,916 | 7. As shown in Figure $2, \triangle A B C$ has an area of 1, points $D, G, E,$ and $F$ are on sides $A B, A C, B C$ respectively, $B D < D A$, $D G \parallel B C, D E \parallel A C, G F \parallel A B$. Then the maximum possible area of trapezoid $D E F G$ is $\qquad$ | \frac{1}{3} |
44,936 | 244 Given real numbers $a, b, c, d$ satisfy
$$
a+b+c+d=ab+ac+ad+bc+bd+cd=3 \text{. }
$$
Find the maximum real number $k$, such that the inequality
$$
a+b+c+2ab+2bc+2ca \geqslant k d
$$
always holds. | 2 |
44,948 | In $\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\overline{AC}$ such that the [incircles](https://artofproblemsolving.com/wiki/index.php/Incircle) of $\triangle{ABM}$ and $\triangle{BCM}$ have equal [radii](https://artofproblemsolving.com/wiki/index.php/Inradius). Then $\frac{AM}{CM} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 45 |
44,953 | Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$. | 1010 |
44,954 | 9. (15 points) In the tetrahedron $S-ABC$, it is known that $SC \perp$ plane $ABC$, $AB=BC=CA=4\sqrt{2}$, $SC=2$, and $D$, $E$ are the midpoints of $AB$, $BC$ respectively. If point $P$ moves on $SE$, find the minimum value of the area of $\triangle PCD$.
---
The translation preserves the original text's formatting and structure. | 2 \sqrt{2} |
45,009 | In [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5$. The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m+n$. | 61 |
45,018 | 11. (20 points) Given the sequence $\left\{a_{n}\right\}$:
$1,1,2,1,2,3, \cdots, 1,2, \cdots, n, \cdots$
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Find all positive real number pairs $(\alpha, \beta)$ such that
$$
\lim _{n \rightarrow+\infty} \frac{S_{n}}{n^{\alpha}}=\beta .
$$ | \left(\frac{3}{2}, \frac{\sqrt{2}}{3}\right) |
45,066 | 1. Let $f(x)$ be a decreasing function on $\mathbf{R}$, for any $x \in \mathbf{R}$, we have
$$
f(x+2013)=2013 f(x) .
$$
Then a function that satisfies this condition is $\qquad$ | -2013^{\frac{x}{2013}} |
45,068 | Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$. | \frac{14}{11} |
45,073 | 7. Given a regular triangular prism $A B C-A_{1} B_{1} C_{1}$ with all 9 edges of equal length, $P$ is the midpoint of edge $C C_{1}$, and the dihedral angle $B-A_{1} P-B_{1}=\alpha$. Then $\sin \alpha=$ $\qquad$ | \frac{\sqrt{10}}{4} |
45,076 | Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $\sqrt{m}-n$, where $m$ and $n$ are positive integers. Find $m+n$. | 254 |
45,079 | Example 11 As shown in Figure 8, sector $A O B$ is one quarter of a unit circle. The center of the semicircle $\odot O_{1}$, $O_{1}$, is on $O A$ and is internally tangent to $\overparen{A B}$ at point $A$. The center of the semicircle $\odot O_{2}$, $O_{2}$, is on $O B$ and is internally tangent to $\overparen{A B}$ at point $B$. The semicircle $\odot O_{1}$ is tangent to the semicircle $\odot O_{2}$. Let the sum of the radii of the two circles be $x$, and the sum of their areas be $y$.
(1) Try to establish the analytical expression of the function $y$ with $x$ as the independent variable;
(2) Find the minimum value of the function $y$.
(2000, Taiyuan City Junior High School Mathematics Competition) | (3-2 \sqrt{2}) \pi |
45,082 | Example 3 If $x, y, z$ are all positive real numbers, and $x^{2}+y^{2}+$ $z^{2}=1$, then the minimum value of $S=\frac{(z+1)^{2}}{2 x y z}$ is $\qquad$. | 3+2\sqrt{2} |
45,096 | ## Aufgabe 1 - 181241
Man ermittle alle ganzen Zahlen $a$ mit der Eigenschaft, dass zu den Polynomen
$$
\begin{aligned}
& f(x)=x^{12}-x^{11}+3 x^{10}+11 x^{3}-x^{2}+23 x+30 \\
& g(x)=x^{3}+2 x+a
\end{aligned}
$$
ein Polynom $h(x)$ so existiert, dass für alle reellen $x$ die Gleichung $f(x)=g(x) \cdot h(x)$ gilt.
| 3 |
45,112 | Example 5 Let $n \geqslant 3$ be a positive integer, and $a_{1}, a_{2}, \cdots, a_{n}$ be any $n$ distinct real numbers with a positive sum. If a permutation $b_{1}, b_{2}, \cdots, b_{n}$ of these numbers satisfies: for any $k=1,2, \cdots, n$, $b_{1}+b_{2}+\cdots+b_{k}>0$, then this permutation is called good. Find the minimum number of good permutations. (2002
| (n-1)! |
45,113 | 3. In $\triangle A B C$, the lengths of the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. Point $G$ satisfies
$$
\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=0, \overrightarrow{G A} \cdot \overrightarrow{G B}=0 \text {. }
$$
If $(\tan A+\tan B) \tan C=m \tan A \cdot \tan B$, then $m=$ . $\qquad$ | \frac{1}{2} |
45,121 | Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$. | 472 |
45,134 | II. (16 points) As shown in the figure, extend the sides $AB, BC, CD, DA$ of quadrilateral $ABCD$ to $E, F, G, H$ respectively, such that $\frac{BE}{AB} = \frac{CF}{BC} = \frac{DG}{CD} = \frac{AH}{DA} = m$. If $S_{EFGH} = 2 S_{ABCD}$ ($S_{EFGH}$ represents the area of quadrilateral $EFGH$), find the value of $m$.
---
Please note that the figure mentioned in the problem is not provided here. | \frac{\sqrt{3}-1}{2} |
45,143 | For n real numbers $a_{1},\, a_{2},\, \ldots\, , a_{n},$ let $d$ denote the difference between the greatest and smallest of them and $S = \sum_{i<j}\left |a_i-a_j \right|.$ Prove that \[(n-1)d\le S\le\frac{n^{2}}{4}d\] and find when each equality holds. | (n-1)d \le S \le \frac{n^2}{4}d |
45,152 | 1. Let $n(n \geqslant 2)$ be a given positive integer, and $a_{1}, a_{2}$, $\cdots, a_{n} \in(0,1)$. Find the maximum value of $\sum_{i=1}^{n} \sqrt[6]{a_{i}\left(1-a_{i+1}\right)}$, where $a_{n+1}=a_{1}$.
(Provided by Hua-Wei Zhu) | \frac{\sqrt[3]{4} n}{2} |
45,171 | The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$,
$a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$.
Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
| 2 |
45,196 | Sure, here is the translated text:
```
II. Fill-in-the-Blanks (9 points each, total 54 points)
1. Let the three interior angles of $\triangle A B C$ be $\angle A, \angle B, \angle C$ with the corresponding side lengths $a, b, c$. If $a < b < c$, and
$$
\left\{\begin{array}{l}
\frac{b}{a}=\frac{\left|b^{2}+c^{2}-a^{2}\right|}{b c}, \\
\frac{c}{b}=\frac{\left|c^{2}+a^{2}-b^{2}\right|}{c a}, \\
\frac{a}{c}=\frac{\left|a^{2}+b^{2}-c^{2}\right|}{a b},
\end{array}\right.
$$
then the radian measures of $\angle A, \angle B, \angle C$ are in the ratio $\qquad$ .
``` | 1:2:4 |
45,228 | 7. Given $\theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$, the quadratic equation
$$
\left(\tan ^{2} \theta+\sec ^{2} \theta\right) x^{2}+2\left(\tan ^{2} \theta-\sin ^{2} \theta\right) x-\cos 2 \theta=0
$$
has a repeated root. Then the value of $\cos \theta$ is . $\qquad$ | \sqrt{\frac{3-\sqrt{5}}{2}} |
45,256 | 8. Given the function $f(x)=\mathrm{e}^{x}\left(x-a \mathrm{e}^{x}\right)$ has exactly two critical points $x_{1} 、 x_{2}\left(x_{1}<x_{2}\right)$. Then the range of values for $a$ is $\qquad$ | \left(0, \frac{1}{2}\right) |
45,260 | 12. Given real numbers $a, b, c$ satisfy
$$
\frac{a(b-c)}{b(c-a)}=\frac{b(c-a)}{c(b-a)}=k>0,
$$
where $k$ is some constant. Then the greatest integer not greater than $k$ is . $\qquad$ | 0 |
45,262 | 3. In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=$ $\frac{2-\sqrt{3}}{4}$. Then the maximum value of $\cos A \cos B$ is $\qquad$ . | \frac{2+\sqrt{3}}{4} |
45,274 | For $n \in \mathbf{N}_{+}$, calculate
$$
\begin{array}{l}
\mathrm{C}_{4 n+1}^{1}+\mathrm{C}_{4 n+1}^{5}+\cdots+\mathrm{C}_{4 n+1}^{4 n+1} \\
=
\end{array}
$$ | 2^{4 n-1}+(-1)^{n} 2^{2 n-1} |
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