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45,284 | 1. Given that $D$ is any point on side $A B$ of $\triangle A B C$ with area 1, $E$ is any point on side $A C$, and $F$ is any point on segment $D E$. Let $\frac{A D}{A B}=x, \frac{A E}{A C}=y, \frac{D F}{D E}=z$, and $y+z-x=\frac{1}{2}$. Try to find the maximum area of $\triangle B D F$. (2005 Hunan Province Mathematics Competition Problem) | \frac{1}{8} |
45,285 | Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$ | 3 |
45,304 | One, (40 points) Given a set $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ of $n$ positive integers that satisfies: for any two different subsets of set $A$, the sums of their respective elements are not equal. Find the minimum value of $\sum_{i=1}^{n} \sqrt{a_{i}}$.
| (\sqrt{2}+1)\left(\sqrt{2^{n}}-1\right) |
45,308 | 10. (15 points) As shown in Figure 2, given that the side length of square $ABCD$ is 1, a line passing through vertex $C$ intersects the rays $AB$ and $AD$ at points $P$ and $Q$ respectively. Find the maximum value of $\frac{1}{AP} +$ $\frac{1}{AQ} + \frac{1}{PQ}$. | 1+\frac{\sqrt{2}}{4} |
45,309 | We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \[k = \sum_{i=2}^n a_i F_i,\] where $a_i\in\{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i\ge2$). This representation is said to be $\textit{minimal}$ if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation. | 1596 |
45,312 | 7. Given that $O$ is the circumcenter of $\triangle A B C$. If $A B=A C$, $\angle C A B=30^{\circ}$, and $\overrightarrow{C O}=\lambda_{1} \overrightarrow{C A}+\lambda_{2} \overrightarrow{C B}$, then $\lambda_{1} \lambda_{2}=$ $\qquad$ . | 7 \sqrt{3}-12 |
45,317 | 3. Given a sphere that is tangent to all six edges of a regular tetrahedron with edge length $a$. Then the volume of this sphere is $\qquad$ . | \frac{\sqrt{2}}{24} \pi a^{3} |
45,328 | For all positive real numbers $x$ and $y$ let
\[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \]
Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$. | \frac{1}{\sqrt{2}} |
45,339 | Example 10 Let $a, b, c \in \mathbf{R}_{+}$, and $abc=1$. Find
$$
\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}
$$
the minimum value. | 1 |
45,384 | A [right circular cone](https://artofproblemsolving.com/wiki/index.php/Right_cone) has a [base](https://artofproblemsolving.com/wiki/index.php/Base) with [radius](https://artofproblemsolving.com/wiki/index.php/Radius) $600$ and [height](https://artofproblemsolving.com/wiki/index.php/Height) $200\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}.$ Find the least distance that the fly could have crawled. | 625 |
45,397 | A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$. | 113 |
45,401 | $2 \cdot 87$ Let $4^{27}+4^{500}+4^{n}$ be a perfect square (square of an integer), find the maximum value of the integer $n$.
| 972 |
45,445 | For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$.
[i](Swiss Mathematical Olympiad, Final round, problem 3)[/i] | 14 |
45,451 | 6. As shown in Figure 2, given the ellipse $\frac{x^{2}}{2}+y^{2}=1, A$ and $B$ are the intersection points of the ellipse with the $x$-axis, $D A$ $\perp A B, C B \perp A B$, and $|D A|$ $=3 \sqrt{2},|C B|=\sqrt{2}$. A moving point $P$ is on the arc $\overparen{A B}$ above the $x$-axis. Then the minimum value of $S_{\triangle P C D}$ is | 4-\sqrt{6} |
45,459 | 1. Given non-zero vectors $\boldsymbol{a} 、 \boldsymbol{b}$ with an angle of $120^{\circ}$ between them. If vector $\boldsymbol{a}-\boldsymbol{b}$ is perpendicular to $\boldsymbol{a}+2 \boldsymbol{b}$, then $\left|\frac{2 a-b}{2 a+b}\right|=$ | \frac{\sqrt{10+\sqrt{33}}}{3} |
45,471 | Find the largest value of the expression
$$
x y + x \sqrt{1-y^{2}} + y \sqrt{1-x^{2}} - \sqrt{\left(1-x^{2}\right)\left(1-y^{2}\right)}
$$ | \sqrt{2} |
45,503 |
5. There is $|B C|=1$ in a triangle $A B C$ and there is a unique point $D$ on $B C$ such that $|D A|^{2}=|D B| \cdot|D C|$. Find all possible values of the perimeter of $A B C$.
(Patrik Bak)
| 1+\sqrt{2} |
45,514 | 1. Given the set $S=\{1,2, \cdots, 3 n\}, n$ is a positive integer, $T$ is a subset of $S$, satisfying: for any $x, y, z \in T$ (where $x, y, z$ can be the same), we have $x+y+z \notin T$. Find the maximum number of elements in all such sets $T$. | 2n |
45,518 | Example 3 The polynomial $\left(x^{2}+2 x+2\right)^{2001}+\left(x^{2}-\right.$ $3 x-3)^{2001}$, after expansion and combining like terms, the sum of the coefficients of the odd powers of $x$ is
保留源文本的换行和格式，直接输出翻译结果如下：
```
Example 3 The polynomial $\left(x^{2}+2 x+2\right)^{2001}+\left(x^{2}-\right.$ $3 x-3)^{2001}$, after expansion and combining like terms, the sum of the coefficients of the odd powers of $x$ is
``` | -1 |
45,532 | Example 2: With $2009^{12}$ as one of the legs, and all three sides being integers, the number of different right-angled triangles (congruent triangles are considered the same) is $\qquad$.
(2009, International Mathematics Tournament of the Cities for Young Mathematicians) | 612 |
45,556 | Example 2 Let non-negative real numbers $a, b, c$ satisfy $ab + bc + ca = 1$. Find the minimum value of $u = \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}$. | \frac{5}{2} |
45,561 | Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$.
Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$.
In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$.
[i]Proposed by Yang Liu[/i] | 19320 |
45,601 | In the diagram below, $BP$ bisects $\angle ABC$, $CP$ bisects $\angle BCA$, and $PQ$ is perpendicular to $BC$. If $BQ\cdot QC=2PQ^2$, prove that $AB+AC=3BC$.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC8zL2IwZjNjMDAxNWEwMTc1ZGNjMTkwZmZlZmJlMGRlOGRhYjk4NzczLnBuZw==&rn=VlRSTUMgMjAwNi5wbmc=[/img] | AB + AC = 3BC |
45,629 | Find all real solutions to $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$ | x = 3 |
45,636 | Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$.
[i](B. Dejean, 6 points)[/i] | 3 |
45,641 | Find the maximum number of natural numbers $x_{1}, x_{2}, \ldots, x_{m}$ satisfying the conditions:
a) No $x_{i}-x_{j}, 1 \leq i<j \leq m$ is divisible by 11 ; and
b) The sum $x_{2} x_{3} \ldots x_{m}+x_{1} x_{3} \ldots x_{m}+\cdots+x_{1} x_{2} \ldots x_{m-1}$ is divisible by 11. | 10 |
45,646 | Find all solutions of the equation $\quad x^{2 y}+(x+1)^{2 y}=(x+2)^{2 y}$ with $x, y \in N$. | x=3, y=1 |
45,650 | Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?
[i](C. Wu)[/i] | 8 \cdot (8!)^3 |
45,693 | Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$. | 15 |
45,713 | Consider a sequence $F_0=2$, $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$. If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$, what is the positive difference between $r_1$ and $r_2$?
| \frac{\sqrt{17}}{2} |
45,716 | Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A) + \cos(3B) + \cos(3C) = 1$. Two sides of the triangle have lengths $10$ and $13$. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}$. Find $m$. | 399 |
45,725 | 45a * If positive numbers $a, b, c$ and constant $k$ satisfy the inequality $\frac{k a b c}{a+b+c} \leqslant(a+b)^{2}+(a+b+$ $4 c)^{2}$, find the maximum value of the constant $k$.
| 100 |
45,765 | 3. A line $l$ passing through the right focus $F$ of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$, and there are exactly 3 such lines, find $\lambda$.
untranslated text remains unchanged:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4 |
45,809 | 10. Let $a>1$ be a positive real number, and $n \geqslant 2$ be a natural number, and the equation $[a x]=x$ has exactly $n$ distinct solutions, then the range of values for $a$ is $\qquad$ . | \left[1+\frac{1}{n}, 1+\frac{1}{n-1}\right) |
45,815 | Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines. | m = 2 |
45,832 | Example 5 In a certain competition, each player plays exactly one game against each of the other players. A win earns the player 1 point, a loss 0 points, and a draw 0.5 points for each player. After the competition, it is found that exactly half of each player's points were earned in games against the 10 lowest-scoring players (the 10 lowest-scoring players earned half of their points in games against each other). Find the total number of players in the competition.
(3rd American Invitational Mathematics Examination) | 25 |
45,838 | Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$, inclusive, such that, if $$q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}},$$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.
[i]Proposed by Ankit Bisain[/i] | 132 |
45,844 | 12. If the equation with respect to $x$
$$
x^{2}+a x+\frac{1}{x^{2}}+\frac{a}{x}+b+2=0(a, b \in \mathbf{R})
$$
has real roots, then the minimum value of $a^{2}+b^{2}$ is $\qquad$. | \frac{16}{5} |
45,858 | 19 The sequence of positive integers $\left\{a_{n}\right\}$ satisfies: for any positive integers $m, n$, if $m \mid n, m<n$, then $a_{m} \mid a_{n}$, and $a_{m}<a_{n}$. Find the minimum possible value of $a_{2000}$. | 128 |
45,862 | Example 1. If $a+b+c=0, a b c=0$, find the value of $\frac{a^{2}+b^{2}+c^{2}}{a^{3}+b^{3}+c^{3}}+\frac{2}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$. | 0 |
45,868 | 12. (16 points) On the Cartesian plane, a point whose both coordinates are rational numbers is called a rational point. Find the smallest positive integer $k$ such that: for every circle that contains $k$ rational points on its circumference, the circle must contain infinitely many rational points on its circumference. | 3 |
45,874 | 10. Given the function
$$
f(x)=\log _{2}\left[a x^{2}+(a+2) x+(a+2)\right] \text {. }
$$
If $f(x)$ has a maximum or minimum value, then the range of values for $a$ is $\qquad$ . | (-2,0) \cup\left(\frac{2}{3},+\infty\right) |
45,884 | For example, let $8 a$ be a real number greater than zero. It is known that there exists a unique real number $k$ such that the quadratic equation in $x$
$$
x^{2}+\left(k^{2}+a k\right) x+1999+k^{2}+a k=0
$$
has two roots that are both prime numbers. Find the value of $a$. | 2 \sqrt{502} |
45,928 | Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$.
Determine the value of $\frac{BZ}{ZC}$. | 1 |
45,929 | The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$. Let $k=a-1$. If the $k$-th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$, find the highest possible value of $n$. | 2009 |
45,930 | Let the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, and let $t$ be a common tangent of $k_{1}$ and $k_{2}$, that touches $k_{1}$ and $k_{2}$ at $M$ and $N$, respectively. If $t \perp A M$ and $M N=2 A M$, evaluate $\angle N M B$. | 45^{\circ} |
45,932 | Let $n^{}_{}$ be the smallest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) that is a multiple of $75_{}^{}$ and has exactly $75_{}^{}$ positive integral divisors, including $1_{}^{}$ and itself. Find $\frac{n}{75}$. | 432 |
45,965 | 4. As shown in Figure 1, in a regular hexagon $A B C D E F$ with side length 2, a moving circle $\odot Q$ has a radius of 1, and its center moves on line segment $C D$ (including endpoints). $P$ is a moving point on and inside $\odot Q$. Let vector $\overrightarrow{A P}=m \overrightarrow{A B}+n \overrightarrow{A F}(m, n \in \mathbf{R})$. Then the range of $m+n$ is . $\qquad$ | [2,5] |
45,975 | Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$. | 2014 |
45,996 | 6. In tetrahedron $ABCD$, $AD > AB$, $AD \perp AB$, $AD \perp AC$, $\angle BAC = \frac{\pi}{3}$. Let the areas of $\triangle ADB$, $\triangle ADC$, $\triangle ABC$, and $\triangle BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively, and they satisfy $S_{1} + S_{2} = S_{3} + S_{4}$. Then the value of $\frac{S_{3}}{S_{1}} + \frac{S_{3}}{S_{2}}$ is | \frac{3}{2} |
45,998 | 6 . Three circles, each with a radius of 3. The centers are at $(14,92)$, $(17,76)$, and $(19,84)$. Draw a line through the point $(17,76)$ such that the sum of the areas of the parts of the three circles on one side of the line equals the sum of the areas of the parts of the three circles on the other side of the line. Find the absolute value of the slope of this line. | 24 |
46,000 | 6. Let the real number $x$ satisfy $\cos \left(x+30^{\circ}\right) \cdot \cos (x+$ $\left.45^{\circ}\right) \cdot \cos \left(x+105^{\circ}\right)+\cos ^{3} x=0$. Then, $\tan x$ $=$ . $\qquad$ | 2\sqrt{3}-1 |
46,015 | One, (20 points) Find all integer pairs $(x, y) (x > y > 2012)$ that satisfy
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{x y}=\frac{1}{2012},
$$
and $x-y$ is maximized. | (4052168, 2013) |
46,030 | Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$
When does equality hold? | 64 |
46,051 | 12.B. The real numbers $a, b, c$ satisfy $a \leqslant b \leqslant c$, and $ab + bc + ca = 0, abc = 1$. Find the largest real number $k$ such that the inequality $|a+b| \geqslant k|c|$ always holds. | 4 |
46,055 | 492 Select five subsets $A_{1}, A_{2}, \cdots, A_{5}$ from the set $\{1,2, \cdots, 1000\}$ such that $\left|A_{i}\right|=500(i=1,2$, $\cdots, 5)$. Find the maximum value of the number of common elements in any three of these subsets. | 50 |
46,059 | Nine, find the minimum value of $f(x)=\frac{9 x^{2} s \sin ^{2} x+4}{x \sin x}(0<x<\pi)$. | 12 |
46,101 |
2. For a given value $t$, we consider number sequences $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{n+1}=\frac{a_{n}+t}{a_{n}+1}$ for all $n \geqslant 1$.
(a) Suppose that $t=2$. Determine all starting values $a_{1}>0$ such that $\frac{4}{3} \leqslant a_{n} \leqslant \frac{3}{2}$ holds for all $n \geqslant 2$.
(b) Suppose that $t=-3$. Investigate whether $a_{2020}=a_{1}$ for all starting values $a_{1}$ different from -1 and 1 .
| a_{2020}=a_{1} |
46,143 | Example 6 Find the largest positive integer $n$ that satisfies the following condition: $n$ is divisible by all positive integers less than $\sqrt[3]{n}$. (1998, Asia Pacific Mathematical Olympiad) | 420 |
46,144 | 3. Given the set $M=\{(a, b) \mid a \leqslant-1, b \leqslant m\}$. If for any $(a, b) \in M$, it always holds that $a \cdot 2^{b}-b-3 a \geqslant 0$, then the maximum value of the real number $m$ is $\qquad$. | 1 |
46,146 | 12. Let $F_{1}$ and $F_{2}$ be the two foci of the ellipse $C$, and $AB$ be a chord of the ellipse passing through point $F_{2}$. In $\triangle F_{1} A B$,
$$
\left|F_{1} A\right|=3,|A B|=4,\left|B F_{1}\right|=5 .
$$
Then $\tan \angle F_{2} F_{1} B=$ $\qquad$ | \frac{1}{7} |
46,152 | 5. Let $a_{1}, a_{2}, \cdots, a_{6}$ be any permutation of $1,2, \cdots, 6$, and $f$ be a one-to-one mapping from $\{1,2, \cdots, 6\}$ to $\{1,2, \cdots, 6\}$, satisfying
$$
f(i) \neq i, f(f(i))=i(i=1,2, \cdots, 6) .
$$
Consider the number table
$$
A=\left[\begin{array}{cccccc}
a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\
f\left(a_{1}\right) & f\left(a_{2}\right) & f\left(a_{3}\right) & f\left(a_{4}\right) & f\left(a_{5}\right) & f\left(a_{6}\right)
\end{array}\right] .
$$
If the number tables $M$ and $N$ differ in at least one position, then $M$ and $N$ are considered two different number tables. The number of different number tables that satisfy the conditions is $\qquad$ (answer with a number). | 10800 |
46,160 | 4. Given that the radius of $\odot O$ is 1. Then the area of the region formed by the orthocenters of all inscribed triangles in $\odot O$ is $\qquad$ . | 9 \pi |
46,170 | Example 1. The moving circle $\mathrm{M}$ is externally tangent to the fixed circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}=0$, and also tangent to the $\mathrm{y}$-axis. Find the equation of the locus of the center $\mathrm{M}$. | y^2=8x |
46,189 | 9. (16 points) Given the function
$$
f(x)=a \sin x-\frac{1}{2} \cos 2 x+a-\frac{3}{a}+\frac{1}{2} \text {, }
$$
where $a \in \mathbf{R}$, and $a \neq 0$.
(1) If for any $x \in \mathbf{R}$, $f(x) \leqslant 0$, find the range of values for $a$.
(2) If $a \geqslant 2$, and there exists $x \in \mathbf{R}$ such that $f(x) \leqslant 0$, find the range of values for $a$. | [2,3] |
46,201 | 9. (14 points) Let the function $f(x)$ be defined on the closed interval
$[0,1]$, satisfying $f(0)=0, f(1)=1$, and for any $x, y \in[0,1](x \leqslant y)$, we have
$$
f\left(\frac{x+y}{2}\right)=\left(1-a^{2}\right) f(x)+a^{2} f(y),
$$
where the constant $a$ satisfies $0<a<1$. Find the value of $a$. | a=\frac{\sqrt{2}}{2} |
46,205 | Find the maximum positive integer $k$ such that for any positive integers $m,n$ such that $m^3+n^3>(m+n)^2$, we have
$$m^3+n^3\geq (m+n)^2+k$$
[i] Proposed by Dorlir Ahmeti, Albania[/i] | 10 |
46,212 | 5. (Ireland) A conference is attended by $12 k$ people, each of whom has greeted exactly $3 k+6$ others. For any two people, the number of people who have greeted both of them is the same. How many people attended the conference? | 36 |
46,217 | 2. Given vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy
$$
|a|=|b|=a \cdot b=2 \text {, }
$$
and $(a-c) \cdot(b-c)=0$.
Then the minimum value of $|2 b-c|$ is | \sqrt{7}-1 |
46,227 | 4. In tetrahedron $ABCD$, it is known that
$$
\angle ADB = \angle BDC = \angle CDA = \frac{\pi}{3} \text{, }
$$
the areas of $\triangle ADB$, $\triangle BDC$, and $\triangle CDA$ are $\frac{\sqrt{3}}{2}$, $2$, and $1$ respectively. Then the volume of this tetrahedron is $\qquad$ | \frac{2 \sqrt{6}}{9} |
46,232 | 5. Let $x, y, z \in \mathbf{R}_{+}$, satisfying $x+y+z=x y z$. Then the function
$$
\begin{array}{l}
f(x, y, z) \\
=x^{2}(y z-1)+y^{2}(z x-1)+z^{2}(x y-1)
\end{array}
$$
has the minimum value of $\qquad$ | 18 |
46,243 | Three. (Full marks 25 points) On the blackboard, all natural numbers from 1 to 1997 are written. Students $A$ and $B$ take turns to perform the following operations: Student $A$ subtracts the same natural number from each number on the blackboard (the number subtracted can be different in different operations); Student $B$ erases two numbers from the blackboard and writes down their sum. Student $A$ goes first, and the operations continue until only one number remains on the blackboard. If this number is non-negative, find this number. | 1 |
46,268 | Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$. $AN$ and $MB$ intersect at $X$. Given that $\angle AXB$ measures $127^{\circ}$ and the minor arc $AM$ measures $14^{\circ}$, compute the measure of the angle at $P$.
[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(40,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label("$A$",A,plain.NE);label("$B$",B,plain.NW);label("$M$",M,SE);
label("$P$",P,E);label("$N$",N,dir(250));label("$X$",X,plain.N);[/asy] | 39^\circ |
46,270 | 9. Let the function be
$$
2 f(x)+x^{2} f\left(\frac{1}{x}\right)=\frac{3 x^{3}-x^{2}+4 x+3}{x+1} \text {. }
$$
Then $f(x)=$ $\qquad$ . | x^{2}-3 x+6-\frac{5}{x+1} |
46,280 | 4. As shown in Figure 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 $\qquad$ . | 120^{\circ} |
46,296 | problem 1 :A sequence is defined by$ x_1 = 1, x_2 = 4$ and $ x_{n+2} = 4x_{n+1} -x_n$ for $n \geq 1$. Find all natural numbers $m$ such that the number $3x_n^2 + m$ is a perfect square for all natural numbers $n$ | m = 1 |
46,333 | Example 1 Given a positive integer $n(n>3)$, let real numbers $a_{1}$, $a_{2}, \cdots, a_{n}$ satisfy
$$
\begin{array}{l}
a_{1}+a_{2}+\cdots+a_{n} \geqslant n, \\
a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2} \geqslant n^{2} .
\end{array}
$$
Find the minimum value of $\max \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$.
(28th United States of America Mathematical Olympiad) | 2 |
46,352 | 16. (12 points) As shown in Figure 1, given that $D$ is any point on side $AB$ of $\triangle ABC$ with an area of 1, $E$ is any point on side $AC$, and $F$ is any point on segment $DE$. Let $\frac{AD}{AB}=x, \frac{AE}{AC}=y, \frac{DF}{DE}=z$, and $y+z-x=\frac{1}{2}$. Try to find the maximum area of $\triangle BDF$. | \frac{1}{8} |
46,355 | Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$. | x^2 - y^2 = 1 |
46,364 | 2. Given the hyperbola $\frac{x^{2}}{t}-\frac{y^{2}}{t}=1$ with the right focus $F$, any line passing through $F$ intersects the right branch of the hyperbola at $M, N$, and the perpendicular bisector of $M N$ intersects the $x$-axis at $P$. When $t$ takes any positive real number except 0, $\frac{|F P|}{|M N|}=$ $\qquad$ . | \frac{\sqrt{2}}{2} |
46,371 | Let $AB$ be a chord on parabola $y=x^2$ and $AB||Ox$. For each point C on parabola different from $A$ and $B$ we are taking point $C_1$ lying on the circumcircle of $\triangle ABC$ such that $CC_1||Oy$. Find a locus of points $C_1$. | y = 1 + a^2 |
46,407 | Example 11 For a positive integer $n \geqslant 3, x_{\mathrm{F}}, x_{2}, \cdots, x_{n}$ are positive real numbers, $x_{n+j}=x_{j}(1 \leqslant j \leqslant n-\mathrm{F})$, find the minimum value of $\sum_{i=1}^{n} \frac{x_{j}}{x_{j+1}+2 x_{j+2}+\cdots+(n-1) x_{j+n-1}}$. (1995 National Mathematical Olympiad Training Team Question) | \frac{2}{n-1} |
46,428 | II. (16 points) Given that $a$ and $b$ are real numbers, and $\mathrm{i}$ is the imaginary unit, the quadratic equation in $z$ is
$$
4 z^{2}+(2 a+\mathrm{i}) z-8 b(9 a+4)-2(a+2 b) \mathrm{i}=0
$$
has at least one real root. Find the maximum value of this real root. | \frac{3 \sqrt{5}}{5} + 1 |
46,463 | $O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$, and $C$. What is the magnitude of $\angle B AC$ in degrees? | 60^\circ |
46,495 | Example 7 Given that $a, b, c, d$ take certain real values, the equation $x^{4}+a x^{3}+b x^{2}+c x+d=0$ has 4 non-real roots, where the product of 2 of the roots is $13+i$, and the sum of the other 2 roots is $3+4i$, where $i$ is the imaginary unit. Find $b$.
(13th American Invitational Mathematics Examination) | 51 |
46,500 | Given is a triangle $ABC$ with the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on segment $AC$ such that $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively such that $KL \parallel BC$ and $KL$ is tangent to the inscribed circle of $\vartriangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$ | \frac{2}{3} |
46,532 | Bread draws a circle. He then selects four random distinct points on the circumference of the circle to form a convex quadrilateral. Kwu comes by and randomly chooses another 3 distinct points (none of which are the same as Bread's four points) on the circle to form a triangle. Find the probability that Kwu's triangle does not intersect Bread's quadrilateral, where two polygons intersect if they have at least one pair of sides intersecting.
[i]Proposed by Nathan Cho[/i] | \frac{1}{5} |
46,535 | find all functions from the reals to themselves. such that for every real $x,y$.
$$f(y-f(x))=f(x)-2x+f(f(y))$$ | f(x) = x |
46,543 | For integer $n$, let $I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos (2n+1)x}{\sin x}\ dx.$
(1) Find $I_0.$
(2) For each positive integer $n$, find $I_n-I_{n-1}.$
(3) Find $I_5$. | \frac{1}{2} \ln 2 - \frac{8}{15} |
46,557 | Let $ABC$ be a right angled triangle with $\angle A = 90^o$and $BC = a$, $AC = b$, $AB = c$. Let $d$ be a line passing trough the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively.
(a) Prove that $$b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a$$
(b) Find the minimum of $$\left( \frac{PB}{PA}\right)^ 2+\left( \frac{QC}{QA}\right)^ 2$$ | 1 |
46,567 | 7. Given complex numbers $z_{1}, z_{2}, z_{3}$ satisfy
$$
\begin{array}{l}
\left|z_{1}\right| \leqslant 1,\left|z_{2}\right| \leqslant 1, \\
\left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leqslant\left|z_{1}-z_{2}\right| .
\end{array}
$$
Then the maximum value of $\left|z_{3}\right|$ is | \sqrt{2} |
46,582 | 4. Let $v$ and $w$ be two randomly chosen roots of the equation $z^{1997} -1 = 0$ (all roots are equiprobable). Find the probability that $\sqrt{2+\sqrt{3}}\le |u+w|$ | \frac{333}{1997} |
46,606 | Calculate the integral $$\int \frac{dx}{\sin (x - 1) \sin (x - 2)} .$$
Hint: Change $\tan x = t$ . | \frac{1}{\sin 1} \ln \left| \frac{\sin(x-2)}{\sin(x-1)} \right| + C |
46,610 | $ABC$ is a triangle with points $D$, $E$ on $BC$ with $D$ nearer $B$; $F$, $G$ on $AC$, with $F$ nearer $C$; $H$, $K$ on $AB$, with $H$ nearer $A$. Suppose that $AH=AG=1$, $BK=BD=2$, $CE=CF=4$, $\angle B=60^\circ$ and that $D$, $E$, $F$, $G$, $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$. | \sqrt{3} |
46,630 | Let $p,q,r$ be distinct prime numbers and let
\[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \]
Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$.
[i]Ioan Tomescu[/i] | 28 |
46,648 | Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$. | \alpha = 8 |
46,652 | 11. In the sequence $\left\{a_{n}\right\}$, $a_{1}=1$, when $n \geqslant 2$, $a_{n} 、 S_{n} 、 S_{n}-\frac{1}{2}$ form a geometric sequence. Then $\lim _{n \rightarrow \infty} n^{2} a_{n}=$ $\qquad$ . | -\frac{1}{2} |
46,674 | Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$ | -2006 |
46,679 | Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \] | -1 |
46,708 | For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$. | 151 |
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