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42,079 | Seven, (25 points) Let $n$ be a positive integer, $a=[\sqrt{n}]$ (where $[x]$ denotes the greatest integer not exceeding $x$). Find the maximum value of $n$ that satisfies the following conditions:
(1) $n$ is not a perfect square;
(2) $a^{3} \mid n^{2}$.
(Zhang Tongjun
Zhu Yachun, problem contributor) | 24 |
42,097 | Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1106 |
42,100 | 3. As shown in Figure 1, in $\triangle A B C$, $O$ is the midpoint of side $B C$, and a line through $O$ intersects lines $A B$ and $A C$ at two distinct points $M$ and $N$ respectively. If
$$
\begin{array}{l}
\overrightarrow{A B}=m \overrightarrow{A M}, \\
\overrightarrow{A C}=n \overrightarrow{A N},
\end{array}
$$
then $m+n=$ | 2 |
42,113 | What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers? | 7 |
42,160 | 4-70 The sequence of functions $\left\{f_{n}(x)\right\}$ is recursively defined by:
$$\left\{\begin{array}{l}
f_{1}(x)=\sqrt{x^{2}+48} \\
f_{n+1}(x)=\sqrt{x^{2}+6 f_{n}(x)}
\end{array} \text { for each } n \geqslant 1 .\right.$$
Find all real solutions to the equation $f_{n}(x)=2 x$. | 4 |
42,170 | At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number. | 67 |
42,186 | 2. For any point $A(x, y)$ in the plane region $D$:
$$
\left\{\begin{array}{l}
x+y \leqslant 1, \\
2 x-y \geqslant-1, \\
x-2 y \leqslant 1
\end{array}\right.
$$
and a fixed point $B(a, b)$ satisfying $\overrightarrow{O A} \cdot \overrightarrow{O B} \leqslant 1$. Then the maximum value of $a+b$ is $\qquad$ | 2 |
42,191 | 1. Given positive integers $a, b, c (a < b < c)$ form a geometric sequence, and
$$
\log _{2016} a+\log _{2016} b+\log _{2016} c=3 .
$$
Then the maximum value of $a+b+c$ is $\qquad$ | 4066273 |
42,217 | The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$. | 2123 |
42,219 | 39. Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$, find the minimum value of $\frac{x}{1-x^{2}}+\frac{y}{1-y^{2}}+\frac{z}{1-z^{2}}$. (30th IMO Canadian Training Problem) | \frac{3 \sqrt{3}}{2} |
42,247 | 2. Given $x, y \in \mathbf{R}_{+}$, and
$$
x-3 \sqrt{x+1}=3 \sqrt{y+2}-y \text {. }
$$
then the maximum value of $x+y$ is $\qquad$ | 9+3 \sqrt{15} |
42,256 | Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships. | 25 |
42,275 | Four, for what real number $x$ does $y=x^{2}-x+1+$ $\sqrt{2(x+3)^{2}+2\left(x^{2}-5\right)^{2}}$ have a minimum value? What is the minimum value? | 9 |
42,289 | Initial 74. Given isosceles $\triangle A B C$ with vertex angle $A$ being $108^{\circ}, D$ is a point on the extension of $A C$, and $A D=B C, M$ is the midpoint of $B D$. Find the degree measure of $\angle C M A$. | 90^\circ |
42,290 | Four, $\mathrm{AC}$ is in the plane $\mathrm{M}$ of the dihedral angle $\mathrm{M}-\mathrm{EF}-\mathrm{N}$, forming a $45^{\circ}$ angle with $\mathrm{EF}$, and $\mathrm{AC}$ forms a $30^{\circ}$ angle with the plane $N$. Find the degree measure of the dihedral angle $M-E F-N$. | 45^{\circ} |
42,315 | Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms:
$$\sum_{i<j}|x_i -x_j |.$$ | \left\lfloor \frac{n^2}{4} \right\rfloor |
42,335 | A set $S$ is called neighbouring if it has the following two properties:
a) $S$ has exactly four elements
b) for every element $x$ of $S$, at least one of the numbers $x-1$ or $x+1$ belongs to $S$.
Find the number of all neighbouring subsets of the set $\{1,2, \ldots, n\}$. | \frac{(n-3)(n-2)}{2} |
42,337 | Example 5. As shown in the figure, in $\triangle A B C$, $D, E$ are points on $B C, C A$ respectively, $B E, A D$ intersect at $H$, and $\frac{B D}{D C}=\frac{1}{3}, \frac{C E}{E A}=\frac{1}{2}, S_{\triangle A B C}=1$. Find the area of $\triangle A H E$. | \frac{4}{9} |
42,378 | 9. In trapezoid $A B C D$, $A B \| C D$, and $A B$ $=92, B C=50, C D=19, A D=70$. A circle with center $P$ on $A B$ is tangent to sides $B C$ and $A D$. If $A P=\frac{m}{n}$, where $m, n$ are coprime positive integers. Find $m+n_{0} \quad$ | 164 |
42,402 | A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac 12 (a+b)$. The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$. What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
[i]Proposed by Michael Kural[/i] | 9 |
42,411 | 11. In $\triangle A B C$, $B C=12$, the height $h_{a}=8$ on side $B C$, $h_{b}$ and $h_{c}$ are the heights on sides $C A$ and $A B$ respectively. Then the maximum value of the product $h_{b} h_{c}$ is $\qquad$. | \frac{2304}{25} |
42,414 | 5. If the quadratic equation with positive integer coefficients
$$
4 x^{2}+m x+n=0
$$
has two distinct rational roots $p$ and $q$ ($p<q$), and the equations
$$
x^{2}-p x+2 q=0 \text { and } x^{2}-q x+2 p=0
$$
have a common root, find the other root of the equation
$$
x^{2}-p x+2 q=0
$$ | \frac{1}{2} |
42,423 | 4. In space, four lines satisfy that the angle between any two lines is $\theta$. Then $\theta=$ $\qquad$ . | \arccos \frac{1}{3} |
42,445 | To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:
(1) $ f(rs) \equal{} f(r)\plus{}f(s)$
(2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3.
(3) $ f(10) \equal{} 0$.
Find $f(1985)$. Justify your answer. | 0 |
42,453 | Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infi nity, what is the ratio between the number of triangles and the number of squares? | 1:1 |
42,478 | Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors. | 23 |
42,507 | 6. Given the complex number $z=x+y \mathrm{i}(x, y \in \mathbf{R})$, satisfying that the ratio of the real part to the imaginary part of $\frac{z+1}{z+2}$ is $\sqrt{3}$. Then the maximum value of $\frac{y}{x}$ is | \frac{4 \sqrt{2}-3 \sqrt{3}}{5} |
42,528 | For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy \[|A| \cdot |B| = |A \cap B| \cdot |A \cup B|\] | 454 |
42,546 | ## Zadatak B-3.4.
Pravilna uspravna četverostrana piramida kojoj bočni bridovi s ravninom baze zatvaraju kut od $60^{\circ}$ i kocka imaju sukladne baze. Odredite omjer njihovih oplošja.
| \frac{1+\sqrt{7}}{6} |
42,548 | The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of $16$ points and have area of at least six square units?
[asy]
import graph; size(7cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
dot((-2,4),linewidth(6pt) + dotstyle);
dot((-1,4),linewidth(6pt) + dotstyle);
dot((0,4),linewidth(6pt) + dotstyle);
dot((1,4),linewidth(6pt) + dotstyle);
dot((2,4),linewidth(6pt) + dotstyle);
dot((3,4),linewidth(6pt) + dotstyle);
dot((4,4),linewidth(6pt) + dotstyle);
dot((-3,2),linewidth(6pt) + dotstyle);
dot((-2,2),linewidth(6pt) + dotstyle);
dot((-1,2),linewidth(6pt) + dotstyle);
dot((0,2),linewidth(6pt) + dotstyle);
dot((1,2),linewidth(6pt) + dotstyle);
dot((2,2),linewidth(6pt) + dotstyle);
dot((3,2),linewidth(6pt) + dotstyle);
dot((4,2),linewidth(6pt) + dotstyle);
dot((5,2),linewidth(6pt) + dotstyle); [/asy] | 361 |
42,552 | A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$. | \frac{3}{4} \cdot 3^{2014} + \frac{1}{4} |
42,555 | 4. As shown in Figure 2, in the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, the dihedral angle $A-B D_{1}-A_{1}$ is $\alpha, A B: B C: C C_{1}=1: 1: 2$. Then $\tan \alpha=$ $\qquad$ | 2 \sqrt{6} |
42,569 | II. (40 points) Given the sequence of real numbers $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=\frac{1}{3}, a_{n+1}=2 a_{n}-\left[a_{n}\right],
$$
where $[x]$ denotes the greatest integer less than or equal to the real number $x$.
$$
\text { Find } \sum_{i=1}^{2012} a_{i} \text {. }
$$ | 1012036 |
42,619 | Third question: There are $n$ people, and it is known that any two of them make at most one phone call. Any $n-2$ of them have the same total number of phone calls, which is $3^{k}$ times, where $k$ is a natural number. Find all possible values of $n$.
---
The translation maintains the original text's format and line breaks. | 5 |
42,642 | A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D$. | 550 |
42,673 | Seven, (25 points) Let $a$, $b$, and $c$ be the lengths of the three sides of a right triangle, where $c$ is the length of the hypotenuse. Find the maximum value of $k$ such that $\frac{a^{3}+b^{3}+c^{3}}{a b c} \geqslant k$ holds. (Provided by Li Tiehan) | 2+\sqrt{2} |
42,680 | In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when $N$ is divided by $1000$. | 16 |
42,694 | Three, (16 points) Let $n$ be a positive integer, and $d_{1}<d_{2}<$ $d_{3}<d_{4}$ be the 4 smallest consecutive positive integer divisors of $n$. If $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$, find the value of $n$. | 130 |
42,707 | Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called [i]good[/i] if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number. | 667 |
42,717 | 5. 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} |
42,723 | How many solutions of the equation $\tan x = \tan \tan x$ are on the interval $0 \le x \le \tan^{-1} 942$? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.) | 300 |
42,753 | Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that for all $m\in\mathbb{Z}$: [list][*] $f(m+8) \le f(m)+8$, [*] $f(m+11) \ge f(m)+11$.[/list] | f(m) = m + a |
42,754 | 7. For integer $n>1$, let $x=1+\frac{1}{2}+\cdots+\frac{1}{n}$, $y=\lg 2+\lg 3+\cdots+\lg n$. Then the set of all integers $n$ that satisfy $[x]=[y]$ is $\qquad$ ( [a] denotes the greatest integer not exceeding the real number $a$). | \{5,6\} |
42,793 | Three, 13. (20 points) Given the function
$$
f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}),
$$
when $x \in[-1,1]$, $|f(x)| \leqslant 1$.
(1) Prove: $|b| \leqslant 1$;
(2) If $f(0)=-1, f(1)=1$, find the value of $a$. | a=2 |
42,799 | 8. In Rt $\triangle A B C$, the altitude $C D$ on the hypotenuse $A B$ is $3$, extend $D C$ to point $P$, such that $C P=2$, connect $A P$, draw $B F \perp A P$, intersecting $C D$ and $A P$ at points $E$ and $F$ respectively. Then the length of segment $D E$ is $\qquad$ | \frac{9}{5} |
42,804 | Example 2 Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{0}=0, x_{n+1}=$ $x_{n}+a+\sqrt{b^{2}+4 a x_{n}}, n=0,1,2, \cdots$, where $a$ and $b$ are given positive real numbers. Find the general term of this sequence. | x_{n}=a n^{2}+b n |
42,811 | 8. The sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=1, a_{2}=2, \\
a_{n+2}=\frac{2(n+1)}{n+2} a_{n+1}-\frac{n}{n+2} a_{n}(n=1,2, \cdots) .
\end{array}
$$
If $a_{m}>2+\frac{2011}{2012}$, then the smallest positive integer $m$ is . $\qquad$ | 4025 |
42,816 | 2. A person writes the number $2^{x} 9^{y}$ as a four-digit number $2 x 9 y$, by substituting $x$ and $y$ with positive integers less than 10, and it turns out that $2 x 9 y=2^{x} 9^{y}$. What are the values of $x$ and $y$? | x=5, y=2 |
42,836 | 3. In a convex quadrilateral $A B C D$, the midpoints of consecutive sides are marked: $M, N, K, L$. Find the area of quadrilateral $M N K L$, if $|A C|=|B D|=2 a,|M K|+|N L|=2 b$.
| b^{2}-^{2} |
42,847 | II. (12 points) As shown in Figure 3, in $\triangle ABC$, it is given that $AB=9, BC=8, AC=7$, and $AD$ is the angle bisector. A circle is drawn with $AD$ as a chord, tangent to $BC$, and intersecting $AB$ and $AC$ at points $M$ and $N$, respectively. Find the length of $MN$. | 6 |
42,873 | 1. The number of solutions to the equation $\sin |x|=|\cos x|$ in the interval $[-10 \pi, 10 \pi]$ is $\qquad$ | 20 |
42,931 | For a positive integer $n$, written in decimal base, we denote by $p(n)$ the product of its digits.
a) Prove that $p(n) \leq n$;
b) Find all positive integers $n$ such that
\[ 10p(n) = n^2+ 4n - 2005. \]
[i]Eugen Păltănea[/i] | n = 45 |
42,940 | Example 5 If $2 x^{2}+3 x y+2 y^{2}=1$, find the minimum value of $k=x+y+x y$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 5 If $2 x^{2}+3 x y+2 y^{2}=1$, find the minimum value of $k=x+y+x y$. | -\frac{9}{8} |
42,962 | The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region? | 800 |
42,979 | 4. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=99^{\frac{1}{99}}, a_{n}=$ $\left(a_{n-1}\right)^{a_{1}}$. When $a_{n}$ is an integer, the smallest positive integer $n$ is $\qquad$ | 100 |
42,987 |
Problem 2. Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression
$$
A=\frac{2-a^{3}}{a}+\frac{2-b^{3}}{b}+\frac{2-c^{3}}{c}
$$
$19^{\text {th }}$ Junior Balkan Mathematical Olympiad June 24-29, 2015, Belgrade, Serbia
| 3 |
43,006 | The function $g\left(x\right)$ is defined as $\sqrt{\dfrac{x}{2}}$ for all positive $x$.
$ $\\
$$g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right)$$
$ $\\
can be expressed as $\cos(b)$ using degrees, where $0^\circ < b < 90^\circ$ and $b = p/q$ for some relatively prime positive integers $p, q$. Find $p+q$. | 19 |
43,049 | 22 Find all positive integer tuples $(x, y, z, w)$, such that $x!+y!+z!=w!$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (2,2,2,3) |
43,050 | 3. Among the natural numbers from 1 to 144, the number of ways to pick three numbers that form an increasing geometric progression with an integer common ratio is $\qquad$ . | 78 |
43,085 | 5 Let the set $M=\{1,2, \cdots, 10\}$ have $k$ 5-element subsets $A_{1}, A_{2}, \cdots, A_{k}$ that satisfy the condition: any two elements in $M$ appear in at most two subsets $A_{i}$ and $A_{j}(i \neq j)$, find the maximum value of $k$.
| 8 |
43,089 | A square has been divided into $2022$ rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles? | 2025 |
43,092 | Example 6 Given $x, y, z \in \mathbf{R}_{+}$, and $x+y+z=1$. Find
$$\frac{\sqrt{x}}{4 x+1}+\frac{\sqrt{y}}{4 y+1}+\frac{\sqrt{z}}{4 z+1}$$
the maximum value. | \frac{3 \sqrt{3}}{7} |
43,101 | Example: 15 Given positive integers $x, y, z$ satisfy $x^{3}-y^{3}-z^{3}=3 x y z, x^{2}=2(y+z)$. Find the value of $x y+y z+z x$.
---
The translation is provided as requested, maintaining the original text's line breaks and format. | 5 |
43,116 | Given ten points in space, where no four points lie on the same plane. Some points are connected by line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn. ${ }^{[1]}$
(2016, National High School Mathematics Joint Competition) | 15 |
43,152 | 7. As shown in Figure 2, points $A$ and $C$ are both on the graph of the function $y=\frac{3 \sqrt{3}}{x}(x$ $>0)$, points $B$ and $D$ are both on the $x$-axis, and such that $\triangle O A B$ and $\triangle B C D$ are both equilateral triangles. Then the coordinates of point $D$ are $\qquad$. | (2 \sqrt{6}, 0) |
43,166 | 4. Let $a, b \in \mathbf{R}$. Then the minimum value of $M=\frac{\left(a^{2}+a b+b^{2}\right)^{3}}{a^{2} b^{2}(a+b)^{2}}$ is $\qquad$ . | \frac{27}{4} |
43,183 | A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings. | 576 |
43,204 | Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$. | 8 |
43,231 | Example 6.1.2. Let \(x, y, z, t\) be real numbers satisfying \(x y + y z + z t + t x = 1\). Find the minimum of the expression
\[5 x^{2} + 4 y^{2} + 5 z^{2} + t^{2}\] | 2 \sqrt{2} |
43,262 | Example 4 There are $n$ squares arranged in a row, to be painted with red, yellow, and blue. Each square is painted one color, with the requirement that no two adjacent squares are the same color, and the first and last squares are also different colors. How many ways are there to paint them?
(1991, Jiangsu Mathematics Competition) | a_{n}=2^{n}+2(-1)^{n} |
43,271 | In triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $OA = ON = 11$ and $OH = 7.$ Compute $BC^2$. | 288 |
43,292 | 3. As shown in Figure 1, in the regular triangular prism $A B C-A_{1} B_{1} C_{1}$, $D$ and $E$ are points on the side edges $B B_{1}$ and $C C_{1}$, respectively, with $E C=B C=2 B D$. Then the size of the dihedral angle formed by the section $A D E$ and the base $A B C$ is | 45^{\circ} |
43,295 | 9. A hemispherical container with a base contains three small balls that are pairwise externally tangent. If the radii of these three small balls are all 1, and each small ball is tangent to the base and the spherical surface of the hemisphere, then the radius of the hemisphere $R=$
| \frac{3+\sqrt{21}}{3} |
43,331 | Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis.
Find the ratio $\frac{V_2}{V_1}.$ | k |
43,337 | 11. Square $A B C D$ and square $A B E F$ are in planes that form a $120^{\circ}$ angle, $M$ and $N$ are points on the diagonals $A C$ and $B F$ respectively, and $A M = F N$. If $A B=1$, then the range of values for $M N$ is $\qquad$ | \left[\frac{\sqrt{3}}{2}, 1\right] |
43,340 | 5. Let $A$ and $B$ be moving points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and $\overrightarrow{O A} \cdot \overrightarrow{O B}=0$, where $O$ is the origin. Then the distance from $O$ to the line $A B$ is $\qquad$. | \frac{a b}{\sqrt{a^{2}+b^{2}}} |
43,389 | An infinitely long slab of glass is rotated. A light ray is pointed at the slab such that the ray is kept horizontal. If $\theta$ is the angle the slab makes with the vertical axis, then $\theta$ is changing as per the function \[ \theta(t) = t^2, \]where $\theta$ is in radians. Let the $\emph{glassious ray}$ be the ray that represents the path of the refracted light in the glass, as shown in the figure. Let $\alpha$ be the angle the glassious ray makes with the horizontal. When $\theta=30^\circ$, what is the rate of change of $\alpha$, with respect to time? Express your answer in radians per second (rad/s) to 3 significant figures. Assume the index of refraction of glass to be $1.50$. Note: the second figure shows the incoming ray and the glassious ray in cyan.
[asy]
fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7));
draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey);
draw((0,0)--(3+2/sqrt(3),0));
draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1));
arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9);
draw((3,-2)--(3,2), linetype("4 4"));
draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3)));
draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15));
label("$\theta$", (3.2, -1.6), N, fontsize(8));
label("$\alpha$", (6, -0.2), fontsize(8));
[/asy]
[asy]
fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7));
draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey);
draw((0,0)--(3+2/sqrt(3),0), cyan);
draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1), cyan);
arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9, cyan);
draw((3,-2)--(3,2), linetype("4 4"));
draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3)));
draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15));
label("$\theta$", (3.2, -1.6), N, fontsize(8));
label("$\alpha$", (6, -0.2), fontsize(8));
[/asy]
[i]Problem proposed by Ahaan Rungta[/i] | 0.561 |
43,409 | An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\]
the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]2021 CCA Math Bonanza Individual Round #11[/i] | 107 |
43,410 | Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$. | 4 |
43,433 | 7. Let $f(x)=\frac{\sin \pi x}{x^{2}}(x \in(0,1))$. Then
$$
g(x)=f(x)+f(1-x)
$$
the minimum value of $g(x)$ is . $\qquad$ | 8 |
43,461 | On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.
(i) One may give cookies only to people adjacent to himself.
(ii) In order to give a cookie to one's neighbor, one must eat a cookie.
Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning. | 2^n |
43,478 | 10. (20 points) The rules of a card game are as follows: Arrange nine cards labeled $1,2, \cdots, 9$ randomly in a row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is considered one operation. The game stops when no operation can be performed (i.e., the number on the first card is “1”). If a permutation cannot be operated on and is obtained from exactly one other permutation by one operation, it is called a “second-terminal permutation”. Among all possible permutations, find the probability of a second-terminal permutation occurring. | \frac{103}{2520} |
43,523 | 3. Let $n$ and $m$ be positive integers of different parity, and $n > m$. Find all integers $x$ such that $\frac{x^{2^{n}}-1}{x^{2^{m}}-1}$ is a perfect square.
(Pan Chengdu) | x=0 |
43,573 |
1. Find all real numbers $s$ for which the equation
$$
4 x^{4}-20 x^{3}+s x^{2}+22 x-2=0
$$
has four distinct real roots and the product of two of these roots is -2 .
| 17 |
43,579 | For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i] | 2 |
43,590 | Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$. | \frac{1}{2} u |
43,598 | 1. $x, y, z$ are positive real numbers, and satisfy $x^{4}+y^{4}+z^{4}=1$, find the minimum value of $\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}+\frac{z^{3}}{1-z^{8}}$. (1999 Jiangsu Province Mathematical Winter Camp Problem) | \frac{9}{8} \cdot \sqrt[4]{3} |
43,619 | $2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line. | 1000 |
43,641 | Given that $A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,$ find $|A_{19} + A_{20} + \cdots + A_{98}|.$ | 40 |
43,692 | 1. Given the function
$$
f(x)=\frac{1}{4}\left(x^{2}+\ln x\right) \text {. }
$$
When $x \in\left(0, \frac{\pi}{4}\right)$, the size relationship between $\mathrm{e}^{\cos 2 x}$ and $\tan x$ is | \mathrm{e}^{\cos 2 x}>\tan x |
43,700 | A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not lying in $g$) of intersection of semicircles as a function of $n$. | \binom{n}{4} |
43,701 | $$
\begin{array}{l}
a+b+c=5, a^{2}+b^{2}+c^{2}=15, \\
a^{3}+b^{3}+c^{3}=47 . \\
\text { Find }\left(a^{2}+a b+b^{2}\right)\left(b^{2}+b c+c^{2}\right)\left(c^{2}+c a+a^{2}\right)
\end{array}
$$ | 625 |
43,717 | Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$.
[i]Proposed by Andrew Wu[/i]
(Note: wording changed from original to specify that $a \neq 20, 22$.) | 20 |
43,728 | 2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$
(Proposed by the Problem Committee) | 4 |
43,731 | Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$
Determine all integers $m$ such that every term of the sequence is a square. | m = 2 |
43,774 | Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved. | k = 3 |
43,779 | 9. (16 points) Given the function
$$
\begin{array}{l}
f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0), \\
\text { when } 0 \leqslant x \leqslant 1 \text {, }|f^{\prime}(x)| \leqslant 1 .
\end{array}
$$
Try to find the maximum value of $a$. | \frac{8}{3} |
43,796 | 6. Given that all positive integers are in $n$ sets, satisfying that when $|i-j|$ is a prime number, $i$ and $j$ belong to two different sets. Then the minimum value of $n$ is $\qquad$ | 4 |
43,799 | Four. (20 points) Given the function $f_{n}(x)=n^{2} x^{2}(1-$ $x)^{n}, x \in[0,1], n \in \mathbf{N}_{+}$. If the maximum value of $f_{n}(x)$ is denoted as $a_{n}$, try to find the minimum term of the sequence $\left\{a_{n}\right\}$. | \frac{4}{27} |
43,819 | Three. (25 points) Let $p$ be a prime number greater than 2, and $k$ be a positive integer. If the graph of the function $y=x^{2}+p x+(k+1) p-4$ intersects the $x$-axis at two points, at least one of which has an integer coordinate, find the value of $k$.
---
The function is given by:
\[ y = x^2 + px + (k+1)p - 4 \]
To find the points where the graph intersects the $x$-axis, we set $y = 0$:
\[ x^2 + px + (k+1)p - 4 = 0 \]
This is a quadratic equation in the form:
\[ x^2 + px + (k+1)p - 4 = 0 \]
The roots of this quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = p \), and \( c = (k+1)p - 4 \).
Substituting these values into the quadratic formula, we get:
\[ x = \frac{-p \pm \sqrt{p^2 - 4 \cdot 1 \cdot ((k+1)p - 4)}}{2 \cdot 1} \]
\[ x = \frac{-p \pm \sqrt{p^2 - 4((k+1)p - 4)}}{2} \]
\[ x = \frac{-p \pm \sqrt{p^2 - 4(k+1)p + 16}}{2} \]
\[ x = \frac{-p \pm \sqrt{p^2 - 4kp - 4p + 16}}{2} \]
\[ x = \frac{-p \pm \sqrt{p^2 - 4kp - 4p + 16}}{2} \]
\[ x = \frac{-p \pm \sqrt{p^2 - 4p(k + 1) + 16}}{2} \]
For the quadratic equation to have at least one integer root, the discriminant must be a perfect square. Let's denote the discriminant by \( D \):
\[ D = p^2 - 4p(k + 1) + 16 \]
We need \( D \) to be a perfect square. Let \( D = m^2 \) for some integer \( m \):
\[ p^2 - 4p(k + 1) + 16 = m^2 \]
Rearranging the equation, we get:
\[ p^2 - 4p(k + 1) + 16 - m^2 = 0 \]
This is a quadratic equation in \( p \):
\[ p^2 - 4p(k + 1) + (16 - m^2) = 0 \]
For \( p \) to be a prime number greater than 2, the discriminant of this quadratic equation must be a perfect square. The discriminant of this quadratic equation is:
\[ \Delta = (4(k + 1))^2 - 4 \cdot 1 \cdot (16 - m^2) \]
\[ \Delta = 16(k + 1)^2 - 4(16 - m^2) \]
\[ \Delta = 16(k + 1)^2 - 64 + 4m^2 \]
\[ \Delta = 16(k + 1)^2 + 4m^2 - 64 \]
For \( \Delta \) to be a perfect square, we need:
\[ 16(k + 1)^2 + 4m^2 - 64 = n^2 \]
for some integer \( n \).
Simplifying, we get:
\[ 4(4(k + 1)^2 + m^2 - 16) = n^2 \]
\[ 4(k + 1)^2 + m^2 - 16 = \left(\frac{n}{2}\right)^2 \]
Let \( \frac{n}{2} = t \), then:
\[ 4(k + 1)^2 + m^2 - 16 = t^2 \]
We need to find integer solutions for \( k \) and \( m \) such that the above equation holds. Testing small values of \( k \):
For \( k = 1 \):
\[ 4(1 + 1)^2 + m^2 - 16 = t^2 \]
\[ 4 \cdot 4 + m^2 - 16 = t^2 \]
\[ 16 + m^2 - 16 = t^2 \]
\[ m^2 = t^2 \]
This is true for \( m = t \). Therefore, \( k = 1 \) is a solution.
Thus, the value of \( k \) is:
\[ \boxed{1} \] | 1 |
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