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Given a line \( l \) intersects an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) at points \( A \) and \( B \), and \( O \) is the origin of the coordinates.
1. Find the maximum area of triangle \(\triangle AOB\) and the equation of the line \( l \) when this maximum area is achieved.
2. Let \( L=\{l \mid \) the line \( l \) gives the maximum area of \(\triangle AOB \) \}. If \( l_{1}, l_{2}, l_{3}, l_{4} \in L \) such that \( l_{1} \parallel l_{2}, l_{3} \parallel l_{4} \), and \( k_{1}, k_{2}, k_{3}, k_{4} \) are the slopes of the corresponding lines such that \( k_{1} + k_{2} + k_{3} + k_{4} = 0 \), find the minimum area of the quadrilateral formed by these four lines. | 2\sqrt{2} | 1/8 |
Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\square$ is unclearly printed.<br/>$(1)$ She guessed $\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find the value of the expression when $x=-2$;<br/>$(2)$ Her father said she guessed wrong, the standard answer's simplification does not contain quadratic terms. Please calculate and determine the value of $\square$ in the original question. | -2 | 3/8 |
Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \). | 1475 | 3/8 |
A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$. | 15 | 5/8 |
A line divides the length of an arc of a circle in the ratio 1:3. In what ratio does it divide the area of the circle? | \frac{\pi - 2}{3\pi + 2} | 2/8 |
Let the two foci of the conic section $C$ be $F_1$ and $F_2$, respectively. If there exists a point $P$ on curve $C$ such that the ratio $|PF_1| : |F_1F_2| : |PF_2| = 4 : 3 : 2$, determine the eccentricity of curve $C$. | \frac{3}{2} | 1/8 |
Liu and Li, each with one child, go to the park together to play. After buying tickets, they line up to enter the park. For safety reasons, the first and last positions must be occupied by fathers, and the two children must stand together. The number of ways for these 6 people to line up is \_\_\_\_\_\_. | 24 | 3/8 |
Let $r$ be a fixed positive real number. It is known that for some positive integer $n$ the following statement is true: for any positive real numbers $a_1,...,a_n$ satisfying the equation $a_1+...+a_n=r(\frac{1}{a_1}+...+\frac{1}{a_n})$ they also satisfy the equation $\frac{1}{\sqrt{r}-a_1}+...+\frac{1}{\sqrt{r}-a_n}=\frac{1}{\sqrt{r}}$ ( $a_i\neq\sqrt{r}$ ). Find $n$ . | 2 | 6/8 |
A clock has an hour, minute, and second hand, all of length $1$ . Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$ ?
*Proposed by Dylan Toh* | \frac{3}{2\pi} | 1/8 |
Two regular triangular pyramids with the same base are inscribed in the same sphere. Knowing that the side length of the base of the triangular pyramids is \(a\) and the radius of the sphere is \(R\), and that the angles between the lateral faces and the base of the two pyramids are \(\alpha\) and \(\beta\) respectively, find the value of \(\tan (\alpha+\beta)\). | -\frac{4\sqrt{3}R}{3a} | 1/8 |
An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
$\textbf{(A)}\ 190 \qquad\textbf{(B)}\ 191 \qquad\textbf{(C)}\ 192 \qquad\textbf{(D)}\ 195 \qquad\textbf{(E)}\ 196$ | \textbf{(B)}\191 | 1/8 |
Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$
Show that $m_n \to \frac{1}{2}$ , as $n \to \infty.$ | \frac{1}{2} | 2/8 |
Through the point \( P \), which lies on the common chord \( AB \) of two intersecting circles, chord \( KM \) of the first circle and chord \( LN \) of the second circle are drawn. Prove that the quadrilateral \( KLMN \) is cyclic. | KLMN | 3/8 |
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?
$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 110 \qquad \text{(D)}\ 120 \qquad \text{(E)}\ 130$ | (E)\130 | 1/8 |
In the right triangle \( \triangle ABC \),
\[
\angle A = 90^\circ, \, AB = AC
\]
\( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively. \( D \) is an arbitrary point on the segment \( MN \) (excluding points \( M \) and \( N \)). The extensions of \( BD \) and \( CD \) intersect \( AC \) and \( AB \) at points \( F \) and \( E \) respectively. If
\[
\frac{1}{BE} + \frac{1}{CF} = \frac{3}{4}
\]
then find the length of \( BC \). | 4\sqrt{2} | 7/8 |
Let $P$ be a point outside a circle $\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\Gamma$ . Let segment $PO$ intersect circle $\Gamma$ at $C$ . A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$ , respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$ , compute the area of $\triangle{AOC}$ .
*2020 CCA Math Bonanza Individual Round #6* | 12 \sqrt{3} | 7/8 |
An \( n \)-digit number \( x \) has the following property: if the last digit of \( x \) is moved to the front, the result is \( 2x \). Find the smallest possible value of \( n \). | 18 | 5/8 |
Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Find the position of $\frac{2018}{2187}$ in the sequence. | 1552 | 7/8 |
Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass? | 1 | 3/8 |
How many three-digit numbers remain if we exclude all three-digit numbers in which all digits are the same or the middle digit is different from the two identical end digits? | 810 | 7/8 |
A circle is tangent to both branches of the hyperbola \( x^{2} - 20y^{2} = 24 \) as well as the \( x \)-axis. Compute the area of this circle. | 504\pi | 7/8 |
The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is: | 1 : 9 | 2/8 |
Consider a regular \( n \)-gon with \( n > 3 \), and call a line acceptable if it passes through the interior of this \( n \)-gon. Draw \( m \) different acceptable lines, so that the \( n \)-gon is divided into several smaller polygons.
(a) Prove that there exists an \( m \), depending only on \( n \), such that any collection of \( m \) acceptable lines results in one of the smaller polygons having 3 or 4 sides.
(b) Find the smallest possible \( m \) which guarantees that at least one of the smaller polygons will have 3 or 4 sides. | n-4 | 1/8 |
The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number? | 78 | 1/8 |
Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$. | 134 | 1/8 |
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$ | 2 | 4/8 |
On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.
The first person said: "Exactly 1 percent of the people present in this room are liars."
The second person said: "Exactly 2 percent of the people present in this room are liars."
and so on.
The person with number $n$ said: "Exactly $n$ percent of the people present in this room are liars."
How many people could be in the room, given that it is known that at least one of them is a knight? | 100 | 6/8 |
The sum of ten numbers is zero. The sum of all their pairwise products is also zero. Find the sum of their fourth powers. | 0 | 7/8 |
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$ | 480 | 6/8 |
A running competition on an unpredictable distance is conducted as follows. On a circular track with a length of 1 kilometer, two points \( A \) and \( B \) are randomly selected (using a spinning arrow). The athletes then run from \( A \) to \( B \) along the shorter arc. Find the median value of the length of this arc, that is, a value \( m \) such that the length of the arc exceeds \( m \) with a probability of exactly 50%. | 0.25 | 4/8 |
The Wolf and the three little pigs wrote a detective story "The Three Little Pigs-2", and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remaining 2100 gold coins to the Wolf. The fee for each book is divided equally among its authors. How much money should the Wolf take for himself? | 700 | 1/8 |
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine $n$. | 4 | 1/8 |
Calculate the limit of the function:
$$\lim _{x \rightarrow \pi}\left(\operatorname{ctg}\left(\frac{x}{4}\right)\right)^{1 / \cos \left(\frac{x}{2}\right)}$$ | e | 5/8 |
In triangle \( KLM \), with all sides distinct, the bisector of angle \( KLM \) intersects side \( KM \) at point \( N \). A line through point \( N \) intersects side \( LM \) at point \( A \), such that \( MN = AM \). Given that \( LN = a \) and \( KL + KN = b \), find \( AL \). | \frac{^2}{b} | 1/8 |
Tanya wrote a certain two-digit number on a piece of paper; to Sveta, who was sitting opposite her, the written number appeared different and was 75 less. What number did Tanya write? | 91 | 7/8 |
In a $k$ -player tournament for $k > 1$ , every player plays every other player exactly once. Find with proof the smallest value of $k$ such that it is possible that for any two players, there was a third player who beat both of them. | 7 | 4/8 |
Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle? | 3 | 7/8 |
Our Slovak grandmother shopped at a store where they had only apples, bananas, and pears. Apples were 50 cents each, pears were 60 cents each, and bananas were cheaper than pears. Grandmother bought five pieces of fruit, with exactly one banana among them, and paid 2 euros and 75 cents.
How many cents could one banana cost? Determine all possibilities. | 35,45,55 | 1/8 |
The hourly teaching fees of Teacher Huanhuan and Teacher Xixi are in the ratio of 5:4. The company decided to accelerate the training for these two assistant teachers, and increased their hourly teaching fees by 20 yuan each. After the increase, the ratio of their hourly teaching fees became 6:5. After the increase, the sum of their hourly teaching fees is $\qquad$ yuan. | 220\, | 1/8 |
Let \( n \) be a natural number. Decompose \( n \) into sums of powers of \( p \) (where \( p \) is a positive integer greater than 1), in such a way that each power \( p^k \) appears at most \( p^2 - 1 \) times. Denote by \( C(n, p) \) the total number of such decompositions. For example, for \( n = 8 \) and \( p = 2 \):
\[ 8 = 4 + 4 = 4 + 2 + 2 = 4 + 2 + 1 + 1 = 2 + 2 + 2 + 1 + 1 = 8 \]
Thus \( C(8, 2) = 5 \). Note that \( 8 = 4 + 1 + 1 + 1 + 1 \) is not counted because \( 1 = 2^0 \) appears 4 times, which exceeds \( 2^2 - 1 = 3 \). Then determine \( C(2002, 17) \). | 118 | 3/8 |
Let \( P \) be a point on the ellipse \(\frac{y^2}{4} + \frac{x^2}{3} = 1\). Given points \( A(1,1) \) and \( B(0,-1) \), find the maximum value of \( |PA| + |PB| \). | 5 | 2/8 |
Given that $\overrightarrow {a}|=4$, $\overrightarrow {e}$ is a unit vector, and the angle between $\overrightarrow {a}$ and $\overrightarrow {e}$ is $\frac {2π}{3}$, find the projection of $\overrightarrow {a}+ \overrightarrow {e}$ on $\overrightarrow {a}- \overrightarrow {e}$. | \frac {5 \sqrt {21}}{7} | 2/8 |
Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\). | \frac{}{2} | 2/8 |
A circle of radius 1 is tangent to a circle of radius 2. The sides of $\triangle ABC$ are tangent to the circles as shown, and the sides $\overline{AB}$ and $\overline{AC}$ are congruent. What is the area of $\triangle ABC$?
[asy]
unitsize(0.7cm);
pair A,B,C;
A=(0,8);
B=(-2.8,0);
C=(2.8,0);
draw(A--B--C--cycle,linewidth(0.7));
draw(Circle((0,2),2),linewidth(0.7));
draw(Circle((0,5),1),linewidth(0.7));
draw((0,2)--(2,2));
draw((0,5)--(1,5));
label("2",(1,2),N);
label("1",(0.5,5),N);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
[/asy] | 16\sqrt{2} | 7/8 |
Cagney can frost a cupcake every 15 seconds and Lacey can frost a cupcake every 45 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes. | 53 | 5/8 |
Given sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) satisfy:
\[
b_{n}=\begin{cases} a_{\frac{n+1}{2}}, & n \text{ is odd;} \\
\sqrt{a_{n+1}}, & n \text{ is even.}
\end{cases}
\]
If \(\{b_{n}\}\) is a geometric sequence and \(a_{2}+b_{2}=108\), find the general term formula for the sequence \(\{a_{n}\}\). | a_n=9^n | 1/8 |
Suppose \( S = \{1, 2, \cdots, 2005\} \). If any subset of \( S \) containing \( n \) pairwise coprime numbers always includes at least one prime number, find the minimum value of \( n \). | 16 | 6/8 |
In how many ways can we place 8 digits equal to 1 and 8 digits equal to 0 on a 4x4 board such that the sums of the numbers written in each row and column are the same? | 90 | 4/8 |
Let $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $C$ in the line $y = 2x$. Let the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) of $C^*$ be written in the form
\[12x^2 + bxy + cy^2 + d = 0.\]
Find the product $bc$. | 84 | 6/8 |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ²(3+sin²θ)=12.
1. Find the general equation of line l and the rectangular coordinate equation of curve C.
2. If line l intersects curve C at points A and B, and point P is defined as (2,1), find the value of $$\frac {|PB|}{|PA|}+ \frac {|PA|}{|PB|}$$. | \frac{86}{7} | 7/8 |
Raashan, Sylvia, and Ted play the following game. Each starts with $$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $$1$? (For example, Raashan and Ted may each decide to give $$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $$0$, Sylvia will have $$2$, and Ted will have $$1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $$1$ to, and the holdings will be the same at the end of the second round.)
$\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$ | \textbf{(B)}\frac{1}{4} | 1/8 |
There are a certain number of red balls, green balls, and blue balls in a bag. Of the balls in the bag, $\frac{1}{3}$ are red and $\frac{2}{7}$ are blue. The number of green balls in the bag is 8 less than twice the number of blue balls. The number of green balls in the bag is
(A) 12
(B) 16
(C) 20
(D) 24
(E) 28 | 16 | 1/8 |
A company occupies three floors in a tower: the 13th, 14th, and 25th floors. A cleaning crew is hired to clean these spaces. They work for four hours on the 13th and 14th floors. Then, the crew splits in half: one half cleans the 25th floor while the other half continues cleaning the 13th and 14th floors. After four more hours, the cleaning of the lower two floors is complete, but the 25th floor requires an additional cleaner who will finish it in 8 more hours the next day. Assuming that the cleaning time is directly proportional to the area being cleaned, how many cleaners were in the crew initially? | 8 | 2/8 |
Given that the equation \( x^{4} - px^{3} + q = 0 \) has an integer root, find the prime numbers \( p \) and \( q \). | 3,2 | 1/8 |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0. | 2014 | 4/8 |
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\mathcal{T}_{12}$. | 1705 | 6/8 |
Let \(\mathbb{Z}_{\geq 0}\) be the set of non-negative integers, and let \(f: \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0} \rightarrow \mathbb{Z}_{\geq 0}\) be a bijection such that whenever \(f(x_1, y_1) > f(x_2, y_2)\), we have \(f(x_1+1, y_1) > f(x_2+1, y_2)\) and \(f(x_1, y_1+1) > f(x_2, y_2+1)\).
Let \(N\) be the number of pairs of integers \((x, y)\), with \(0 \leq x, y < 100\), such that \(f(x, y)\) is odd. Find the smallest and largest possible values of \(N\). | 5000 | 2/8 |
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}-4x=0$. The parameter equation of curve $C_{2}$ is $\left\{\begin{array}{l}x=\cos\beta\\ y=1+\sin\beta\end{array}\right.$ ($\beta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive $x$-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of curves $C_{1}$ and $C_{2}$;<br/>$(2)$ If the ray $\theta =\alpha (\rho \geqslant 0$, $0＜\alpha＜\frac{π}{2})$ intersects curve $C_{1}$ at point $P$, the line $\theta=\alpha+\frac{π}{2}(\rho∈R)$ intersects curves $C_{1}$ and $C_{2}$ at points $M$ and $N$ respectively, and points $P$, $M$, $N$ are all different from point $O$, find the maximum value of the area of $\triangle MPN$. | 2\sqrt{5} + 2 | 1/8 |
A store distributes 9999 cards among its customers. Each card has a 4-digit number, ranging from 0001 to 9999. If the sum of the first 2 digits is equal to the sum of the last 2 digits, the card is a winning card. For example, card 0743 is a winning card. Prove that the sum of the numbers of all the winning cards is divisible by 101. | 101 | 5/8 |
Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
[/asy] | 8 | 6/8 |
(1) The definite integral $\int_{-1}^{1}(x^{2}+\sin x)dx=$ ______.
(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.
(3) Given the function $f(x)=\begin{cases}1-\log_{a}(x+2), & x\geqslant 0 \\ g(x), & x < 0\end{cases}$ is an odd function, then the root of the equation $g(x)=2$ is ______.
(4) Given the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(0 < b < a < \sqrt{2}b)$, its foci are $F_{1}$ and $F_{2}$ respectively. Circle $N$ has $F_{2}$ as its center, and its minor axis length as the diameter. A tangent line to circle $N$ passing through point $F_{1}$ touches it at points $A$ and $B$. If the area of quadrilateral $F_{1}AF_{2}B$ is $S= \frac{2}{3}a^{2}$, then the eccentricity of ellipse $M$ is ______. | \frac{\sqrt{3}}{3} | 6/8 |
What is the minimum number of points in which 5 different non-parallel lines, not passing through a single point, can intersect? | 10 | 1/8 |
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$ . Determine the measure of the angle $CBF$ . | 15 | 6/8 |
Prove that if the polynomial \( x^4 + ax^3 + bx + c \) has all real roots, then \( ab \leq 0 \). | \le0 | 5/8 |
A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car not rented out, the maintenance cost is 50 yuan per month. To maximize the monthly revenue of the rental company, the monthly rent for each car should be set at ______. | 4050 | 7/8 |
Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the number of possible values of $s$? | 22 | 1/8 |
Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time. | 1.25 | 3/8 |
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares. | 6 | 3/8 |
A fishing vessel illegally fishes in a foreign country's territorial waters, resulting in an identical loss of value for the foreign country with each cast of the net. The probability that the vessel will be detained by the foreign coast guard during each cast is $1 / k$, where $k$ is a natural number. Assume that the event of the vessel being detained or not during each cast is independent of the previous fishing activities. If the vessel is detained by the foreign coast guard, all previously caught fish are confiscated, and it can no longer fish in these waters. The captain plans to leave the foreign territorial waters after casting the net for the $n$-th time. Since the possibility that the vessel might be detained by the foreign coast guard cannot be disregarded, the profit from fishing is a random variable. Find the number $n$ that maximizes the expected value of the fishing profit. | k-1 | 3/8 |
Determine all composite positive integers \( n \) with the following property: If \( 1 = d_1 < d_2 < \ldots < d_k = n \) are all the positive divisors of \( n \), then
\[
\left(d_2 - d_1\right) : \left(d_3 - d_2\right) : \cdots : \left(d_k - d_{k-1}\right) = 1 : 2 : \cdots : (k-1).
\] | 4 | 1/8 |
An infinite tree graph has edges that connect lattice points such that the distance between two connected points is at most 1998. Every lattice point in the plane is a vertex of the graph. Prove that there exists a pair of points in the plane at a unit distance from each other that are connected in the tree graph by a path of length at least \( 10^{1998} \). | 10^{1998} | 2/8 |
I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day. | \frac{1}{63} | 5/8 |
Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $\dots$ by $a_1 = 1$ and $a_{n + 1}^3 = 99a_n^3$ for all $n \geq 1$. Then $a_{100}$ equals
$\textbf{(A)}\ 33^{33} \qquad \textbf{(B)}\ 33^{99} \qquad \textbf{(C)}\ 99^{33} \qquad \textbf{(D)}\ 99^{99} \qquad \textbf{(E)}\ \text{none of these}$ | \textbf{(C)}\99^{33} | 1/8 |
The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible. | 127 | 1/8 |
Given \\(|a|=1\\), \\(|b|= \sqrt{2}\\), and \\(a \perp (a-b)\\), the angle between vector \\(a\\) and vector \\(b\\) is ______. | \frac{\pi}{4} | 3/8 |
How many different real numbers $x$ satisfy the equation \[(x^{2}-5)^{2}=16?\]
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$ | \textbf{(D)}4 | 1/8 |
A plane is at a distance $a$ from the center of a unit sphere. Find the edge length of a cube, one face of which lies in this plane, with the vertices of the opposite face on the sphere. | \frac{\sqrt{6-2a^2}-2a}{3} | 6/8 |
.4 + .02 + .006 = | .426 | 7/8 |
What is the smallest number, \( n \), which is the product of 3 distinct primes where the mean of all its factors is not an integer? | 130 | 6/8 |
Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head. | 48 | 5/8 |
Let $p$ be an odd prime number, and define $d_{p}(n)$ as the remainder of the Euclidean division of $n$ by $p$. A sequence $\left(a_{n}\right)_{n \geqslant 0}$ is called a $p$-sequence if for all $n \geq 0$, $a_{n+1} = a_{n} + d_{p}(a_{n})$.
- Does there exist an infinite number of prime numbers $p$ for which there exist two $p$-sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ such that $a_{n} > b_{n}$ for infinitely many $n$, and $a_{n} < b_{n}$ for infinitely many $n$?
- Does there exist an infinite number of prime numbers $p$ for which there exist two $p$-sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ such that $a_{0} < b_{0}$ but $a_{n} > b_{n}$ for all $n \geq 1$? | Yes | 1/8 |
Given a parallelogram \(A B C D\) with \(\angle B = 111^\circ\) and \(B C = B D\). A point \(H\) is marked on the segment \(B C\) such that \(\angle B H D = 90^\circ\). Point \(M\) is the midpoint of side \(A B\). Find the angle \(A M H\). Provide the answer in degrees. | 132 | 1/8 |
For what values of the real parameter \( t \) does the system of inequalities
\[ x \geq y^2 + t y \geq x^2 + t \]
have exactly one solution in the set of real number pairs? | \frac{1}{4} | 1/8 |
Given a geometric sequence $\{a_n\}$ with a common ratio $q=-5$, and $S_n$ denotes the sum of the first $n$ terms of the sequence, the value of $\frac{S_{n+1}}{S_n}=\boxed{?}$. | -4 | 1/8 |
Call a natural number $n{}$ *interesting* if any natural number not exceeding $n{}$ can be represented as the sum of several (possibly one) pairwise distinct positive divisors of $n{}$ .
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other than the powers of two.
[/list]
*Proposed by N. Agakhanov* | 992 | 3/8 |
Let point \(P\) be any point on the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) (where \(a > 0\) and \(b > 0\)). A line passing through point \(P\) intersects the asymptotes \(l_{1}: y = \frac{b}{a} x\) and \(l_{2}: y = -\frac{b}{a} x\) at points \(P_{1}\) and \(P_{2}\), respectively. Let \(\lambda = \frac{P_{1} P}{P P_{2}}\). Prove that the area of the triangle \(\triangle O P_{1} P_{2}\) is \( \frac{(1 + \lambda)^{2}}{4 |\lambda|} a b \). | \frac{(1+\lambda)^2}{4|\lambda|} | 3/8 |
In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.)
The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count. | 792 | 6/8 |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 2/8 |
The table below displays the grade distribution of the $30$ students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2. What percent of the students received the same grade on both tests? | 40\% | 1/8 |
Let the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfy the condition
\[
f\left(x^{3}+y^{3}\right)=(x+y)\left((f(x))^{2}-f(x) \cdot f(y)+(f(y))^{2}\right), \quad x, y \in \mathbf{R}.
\]
Prove that for all \( x \in \mathbf{R} \),
\[
f(1996 x) = 1996 f(x).
\] | f(1996x)=1996f(x) | 3/8 |
We colored the faces of a regular dodecahedron with red, blue, yellow, and green so that any two adjacent faces have different colors. How many edges of the dodecahedron have one face colored blue and the other green? | 5 | 3/8 |
Two sons inherit a herd of cows after their father dies. They sell the herd, receiving as many dollars per head as there were heads in the herd. With the money from the sale, the sons buy sheep at $10 each and one lamb that costs less than $10. They then divide the sheep and the lamb such that each son gets an equal number of animals. How much should the son who received only sheep pay his brother to ensure each has an equal share? | 2 | 2/8 |
The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere? | 100\pi | 4/8 |
Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that
(i) $\frac{p+1}{2}$ is even but is not a power of $2$ , and
(ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$ .
*Proposed by Evan Chen* | 431 | 1/8 |
Let $x_{1} > 0$ be a root of the equation $a x^{2} + b x + c = 0$.
Prove: there exists a root $x_{2}$ of the equation $c x^{2} + b x + a = 0$ such that $x_{1} + x_{2} \geq 2$. | x_1+x_2\ge2 | 5/8 |
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$ , find $p+q$ .
*Proposed by Michael Kural* | 9901 | 2/8 |
Given a convex quadrilateral \( EFGH \) with vertices \( E \), \( F \), \( G \), and \( H \) respectively on the edges \( AB \), \( BC \), \( CD \), and \( DA \) of another convex quadrilateral \( ABCD \), such that \(\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1\). Additionally, the points \( A \), \( B \), \( C \), and \( D \) lie on the edges \( H_1E_1 \), \( E_1F_1 \), \( F_1G_1 \), and \( G_1H_1 \) of yet another convex quadrilateral \( E_1F_1G_1H_1 \), such that \( E_1F_1 \parallel EF \), \( F_1G_1 \parallel FG \), \( G_1H_1 \parallel GH \), and \( H_1E_1 \parallel HE \). Given \(\frac{E_1A}{AH_1} = \lambda\), find the value of \(\frac{F_1C}{CG_1}\). | \lambda | 3/8 |
The number in the parentheses that satisfies the following equation is equal to $\qquad$
$$
\frac{\frac{(}{128}+1 \frac{2}{7}}{5-4 \frac{2}{21} \times 0.75} \div \frac{\frac{1}{3}+\frac{5}{7} \times 1.4}{\left(4-2 \frac{2}{3}\right) \times 3}=4.5
$$ | \frac{1440}{7} | 4/8 |
Xiao Xiao did an addition problem, but he mistook the second addend 420 for 240, and the result he got was 390. The correct result is ______. | 570 | 7/8 |
Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$?
$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$ | \textbf{(A)}45 | 1/8 |
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