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Ellipse $\frac{x^{2}}{4} + y^{2} = 1$ has any two points $P$ and $Q$ on it, and $O$ is the origin of coordinates. If $OP \perp OQ$, then the minimum area of triangle $POQ$ is $\qquad$. | \frac{4}{5} | 7/8 |
Given real numbers $x$ and $y$ that satisfy the system of inequalities $\begin{cases} x - 2y - 2 \leqslant 0 \\ x + y - 2 \leqslant 0 \\ 2x - y + 2 \geqslant 0 \end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determine the minimum value of $\frac{2}{a} + \frac{3}{b}$. | \frac{10 + 4\sqrt{6}}{3} | 4/8 |
Let $ ABCD $ be a rectangle of area $ S $ , and $ P $ be a point inside it. We denote by $ a, b, c, d $ the distances from $ P $ to the vertices $ A, B, C, D $ respectively. Prove that $ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2\ge 2S $ . When there is equality?
| ^2+b^2+^2+^2\ge2S | 2/8 |
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t\,$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t\,$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator? | 163 | 1/8 |
Two players play a game, starting with a pile of \( N \) tokens. On each player's turn, they must remove \( 2^{n} \) tokens from the pile for some nonnegative integer \( n \). If a player cannot make a move, they lose. For how many \( N \) between 1 and 2019 (inclusive) does the first player have a winning strategy? | 1346 | 7/8 |
Throw 6 dice at a time, find the probability, in the lowest form, such that there will be exactly four kinds of the outcome. | 325/648 | 5/8 |
Let a triangle have an altitude $m$ from one of its vertices, and let the segment of this altitude from the vertex to the orthocenter be $m_{1}$. Calculate the value of the product $m m_{1}$ in terms of the sides. | \frac{b^2+^2-^2}{2} | 7/8 |
The sequence $\left\{a_{n}\right\}$ is: $1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,\cdots$. The sequence is formed as follows: start with $a_{1}=1$ and copy this term to obtain $a_{2}=1$. Then add the number 2 as $a_{3}=2$. Next, copy all terms $1,1,2$ and place them as $a_{4}, a_{5}, a_{6}$, then add the number 3 as $a_{7}=3$, and so on. What is $a_{2021}$? | 1 | 1/8 |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then $\angle BDC =$ | 55^{\circ} | 7/8 |
Two chords \(AB\) and \(CD\) of a circle with center \(O\) each have a length of 10. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\), with \(DP = 3\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL : LC\). | 3/13 | 1/8 |
The pentagon $ABCDE$ is inscribed around a circle. The angles at its vertices $A$, $C$, and $E$ are $100^{\circ}$. Find the angle $ACE$. | 40 | 1/8 |
Let \( f(x) \) be an odd function defined on \( \mathbf{R} \), such that \( f(x) = x^2 \) for \( x \geqslant 0 \). If for any \( x \in [a, a+2] \), \( f(x+a) \geqslant 2 f(x) \), then determine the range of the real number \( a \). | [\sqrt{2},+\infty) | 2/8 |
A deck of cards contains 52 cards (excluding two jokers). The cards are dealt to 4 people, each person receiving 13 cards. In how many ways can one person receive exactly 13 cards with all four suits represented? (Express the answer using combinations.) | \binom{52}{13}-4\binom{39}{13}+6\binom{26}{13}-4\binom{13}{13} | 1/8 |
Given \( 0 \leqslant a_{k} \leqslant 1 \) for \( k = 1, 2, \cdots, 2002 \), and let \( a_{2003} = a_{1} \) and \( a_{2004} = a_{2} \), find the maximum value of \( \sum_{k=1}^{20002} \left( a_{k} - a_{k+1} a_{k+2} \right) \). | 1001 | 4/8 |
Given the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)\) and the moving circle \(x^2 + y^2 = R^2 (b < R < a)\), where point \(A\) is on the ellipse and point \(B\) is on the moving circle such that the line \(AB\) is tangent to both the ellipse and the circle, find the maximum distance \(|AB|\) between points \(A\) and \(B\). | b | 5/8 |
The product of two 2-digit numbers is $4536$. What is the smaller of the two numbers? | 54 | 7/8 |
Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.0001, 0.001).$ | 2865 | 5/8 |
Given that the curves $y=x^2-1$ and $y=1+x^3$ have perpendicular tangents at $x=x_0$, find the value of $x_0$. | -\frac{1}{\sqrt[3]{6}} | 1/8 |
The height of Cylinder A is equal to its diameter. The height and diameter of Cylinder B are each twice those of Cylinder A. The height of Cylinder C is equal to its diameter. The volume of Cylinder C is the sum of the volumes of Cylinders A and B. What is the ratio of the diameter of Cylinder C to the diameter of Cylinder A? | \sqrt[3]{9}:1 | 1/8 |
The value of \( 6\left(\frac{3}{2} + \frac{2}{3}\right) \) is:
(A) 13
(B) 6
(C) \(\frac{13}{6}\)
(D) \(\frac{29}{3}\)
(E) 5 | 13 | 7/8 |
The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to:
$\textbf{(A)}\ \infty \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt[4]{2} \qquad \textbf{(D)}\ \sqrt{2} \qquad \textbf{(E)}\ \text{None of these}$ | \textbf{(D)}\\sqrt{2} | 1/8 |
There is a prize in one of three boxes, while the other two boxes are empty. You do not know which box contains the prize, but the host does. You must initially choose one of the boxes, where you think the prize might be. After your choice, the host opens one of the two remaining boxes, revealing it to be empty. Since the host doesn't want to give the prize immediately, he always opens an empty box. After this, you are given a final choice to either stick with your initial choice or switch to the other unopened box. Can you win the prize with a probability greater than $1 / 2$? | Yes | 1/8 |
There are two prime numbers $p$ so that $5 p$ can be expressed in the form $\left\lfloor\frac{n^{2}}{5}\right\rfloor$ for some positive integer $n$. What is the sum of these two prime numbers? | 52 | 4/8 |
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter.
| 676 | 7/8 |
In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory has more than 4 people but less than 8 people. The total number of students in the class participating in the study tour is ______. | 46 | 7/8 |
A car with mail departed from $A$ to $B$. Twenty minutes later, a second car left on the same route at a speed of 45 km/h. After catching up with the first car, the driver of the second car handed over a package and immediately returned to $A$ at the same speed (time spent for stop and turnaround is not considered). By the time the first car arrived at $B$, the second car had only reached halfway from the meeting point back to $A$. Find the speed of the first car, given that the distance between $A$ and $B$ is 40 km. | 30\, | 1/8 |
Chris and Paul each rent a different room of a hotel from rooms $1-60$. However, the hotel manager mistakes them for one person and gives "Chris Paul" a room with Chris's and Paul's room concatenated. For example, if Chris had 15 and Paul had 9, "Chris Paul" has 159. If there are 360 rooms in the hotel, what is the probability that "Chris Paul" has a valid room? | \frac{153}{1180} | 1/8 |
Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample? | \frac{7}{4} | 1/8 |
Given that points $C$ and $F$ are on line segment $AB$, with $AB=12$ and $AC=6$. Point $D$ is any point on the circle centered at $A$ with radius $AC$. The perpendicular bisector of segment $FD$ intersects line $AD$ at point $P$. If the locus of point $P$ is a hyperbola, determine the range of values for the eccentricity of this hyperbola. | (1,2] | 1/8 |
For the positive integer \( n \), if the expansion of \( (xy - 5x + 3y - 15)^n \) is combined and simplified, and \( x^i y^j \) (where \( i, j = 0, 1, \ldots, n \)) has at least 2021 terms, what is the minimum value of \( n \)? | 44 | 7/8 |
Given vectors $\overrightarrow{a} = (\cos x, -\sqrt{3}\cos x)$ and $\overrightarrow{b} = (\cos x, \sin x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$.
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;
(Ⅱ) If $f(\theta) = \frac{5}{6}$, where $\theta \in \left( \frac{\pi}{3}, \frac{2\pi}{3} \right)$, find the value of $\sin 2\theta$. | \frac{2\sqrt{3} - \sqrt{5}}{6} | 7/8 |
Let the real numbers \( a \) and \( b \) satisfy \( a = x_1 + x_2 + x_3 = x_1 x_2 x_3 \) and \( ab = x_1 x_2 + x_2 x_3 + x_3 x_1 \), where \( x_1, x_2, x_3 > 0 \). Then the maximum value of \( P = \frac{a^2 + 6b + 1}{a^2 + a} \) is ______. | \frac{9+\sqrt{3}}{9} | 1/8 |
The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. | 12 | 1/8 |
If snow falls at a rate of 1 mm every 6 minutes, then how many hours will it take for 1 m of snow to fall?
(A) 33
(B) 60
(C) 26
(D) 10
(E) 100 | 100 | 1/8 |
On side \(AB\) of an acute triangle \(ABC\), a point \(M\) is marked. Inside the triangle, a point \(D\) is chosen. Circles \(\omega_{A}\) and \(\omega_{B}\) are the circumcircles of triangles \(AMD\) and \(BMD\), respectively. Side \(AC\) intersects circle \(\omega_{A}\) again at point \(P\), and side \(BC\) intersects circle \(\omega_{B}\) again at point \(Q\). Ray \(PD\) intersects circle \(\omega_{B}\) again at point \(R\), and ray \(QD\) intersects circle \(\omega_{A}\) again at point \(S\). Find the ratio of the areas of triangles \(ACR\) and \(BCS\). | 1 | 4/8 |
On the Island of Truth and Lies, there are knights who always tell the truth and liars who always lie. One day, 15 islanders lined up in order of height (from tallest to shortest, with the tallest standing first) for a game. Each person had to say one of the following phrases: "There is a liar below me" or "There is a knight above me." Those standing in positions four through eight said the first phrase, and the rest said the second phrase. How many knights were among these 15 people, given that all the residents are of different heights? | 11 | 2/8 |
Find $$ \inf_{\substack{ n\ge 1 a_1,\ldots ,a_n >0 a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) . $$ | -\pi | 3/8 |
Alice conducted a survey among a group of students regarding their understanding of snakes. She found that $92.3\%$ of the students surveyed believed that snakes are venomous. Of the students who believed this, $38.4\%$ erroneously thought that all snakes are venomous. Knowing that only 31 students held this incorrect belief, calculate the total number of students Alice surveyed. | 88 | 5/8 |
The circle $k_{1}$ with radius $R$ is externally tangent to the circle $k_{2}$ with radius $2R$ at point $E_{3}$, and the circles $k_{1}$ and $k_{2}$ are both externally tangent to the circle $k_{3}$ with radius $3R$. The point of tangency between circles $k_{2}$ and $k_{3}$ is $E_{1}$, and the point of tangency between circles $k_{3}$ and $k_{1}$ is $E_{2}$. Prove that the circumcircle of triangle $E_{1} E_{2} E_{3}$ is congruent to circle $k_{1}$. | R | 7/8 |
Let $A B C$ be a triangle and $D$ a point on $B C$ such that $A B=\sqrt{2}, A C=\sqrt{3}, \angle B A D=30^{\circ}$, and $\angle C A D=45^{\circ}$. Find $A D$. | \frac{\sqrt{6}}{2} | 7/8 |
A trapezoid \(ABCD\) (\(AD \parallel BC\)) and a rectangle \(A_{1}B_{1}C_{1}D_{1}\) are inscribed in a circle \(\Omega\) with a radius of 13 such that \(AC \parallel B_{1}D_{1}\) and \(BD \parallel A_{1}C_{1}\). Find the ratio of the areas of \(ABCD\) and \(A_{1}B_{1}C_{1}D_{1}\), given that \(AD = 24\) and \(BC = 10\). | \frac{1}{2} | 1/8 |
Given the inequality about $x$, $2\log_2^2x - 5\log_2x + 2 \leq 0$, the solution set is $B$.
1. Find set $B$.
2. If $x \in B$, find the maximum and minimum values of $f(x) = \log_2 \frac{x}{8} \cdot \log_2 (2x)$. | -4 | 7/8 |
A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is:
$\textbf{(A)}\ 784\text{ ft.}\qquad\textbf{(B)}\ 342\text{ ft.}\qquad\textbf{(C)}\ 1568\text{ ft.}\qquad\textbf{(D)}\ 156.8\text{ ft.}\qquad\textbf{(E)}\ \text{none of these}$ | \textbf{(A)}\784 | 1/8 |
It is known that there exists a natural number \( N \) such that \( (\sqrt{3}-1)^{N} = 4817152 - 2781184 \cdot \sqrt{3} \). Find \( N \). | 16 | 2/8 |
There is a basket of apples. If Class A shares the apples such that each person gets 3 apples, 10 apples remain. If Class B shares the apples such that each person gets 4 apples, 11 apples remain. If Class C shares the apples such that each person gets 5 apples, 12 apples remain. How many apples are there in the basket at least? | 67 | 7/8 |
Given a sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.
If $a_1 = 1$ and
\[a_n = \frac{2S_n^2}{2S_n - 1}\]for all $n \ge 2,$ then find $a_{100}.$ | -\frac{2}{39203} | 7/8 |
The right triangular prism \(ABC-A_1B_1C_1\) has a base \(\triangle ABC\) which is an equilateral triangle. Points \(P\) and \(E\) are movable points (including endpoints) on \(BB_1\) and \(CC_1\) respectively. \(D\) is the midpoint of side \(BC\), and \(PD \perp PE\). Find the angle between lines \(AP\) and \(PE\). | 90 | 6/8 |
In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$? | 12 \sqrt[4]{3} | 1/8 |
Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \). | 137 | 5/8 |
All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list? | 156 | 7/8 |
Given \(0 < a \neq 1\) and \( f(x) = \log_{a}(6ax^2 - 2x + 3) \) is monotonically increasing on \(\left[\frac{3}{2}, 2\right]\), find the range of values for \(a\). | (\frac{1}{24},\frac{1}{12}]\cup(1,\infty) | 4/8 |
How many divisors does $68^{2008}$ have?
Exercise: $E$ is a set that contains $n$ elements. How many pairs $(A, B)$ of two sets exist such that $A \subset B \subset E$? | 3^n | 3/8 |
Consider a regular polygon with $2^n$ sides, for $n \ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ | \frac{2}{\pi} | 7/8 |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
| 67 | 3/8 |
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least $6$ miles away," Bob replied, "We are at most $5$ miles away." Charlie then remarked, "Actually the nearest town is at most $4$ miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty)$ | \textbf{(D)}(5,6) | 1/8 |
Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? | 4 | 2/8 |
Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 3000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$? | 750 | 7/8 |
The eccentricity of the ellipse given by the parametric equations $\begin{cases} x=3\cos\theta \\ y=4\sin\theta\end{cases}$ is $\frac{\sqrt{7}}{\sqrt{3^2+4^2}}$, calculate this value. | \frac { \sqrt {7}}{4} | 1/8 |
How many points of intersection are there for the diagonals of a convex n-gon if no three diagonals intersect at a single point? | \frac{n(n-1)(n-2)(n-3)}{24} | 6/8 |
Calculate the probability that all the rational terms are not adjacent to each other when rearranging the terms of the expansion $( \sqrt {x}+ \dfrac {1}{2 \sqrt[4]{x}})^{8}$ in a list. | \frac{5}{12} | 7/8 |
A natural number $k$ is said $n$ -squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$ , the smallest natural $k$ that is $n$ -squared. | 2n^2-n+1 | 1/8 |
The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,
$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is _______.
$(2)$ If three seeds are sown per hole, the probability of having seedlings in this hole is _______; the probability of having strong seedlings in this hole is _______. | 0.936 | 5/8 |
In $\triangle ABC$, $\overrightarrow {AD}=3 \overrightarrow {DC}$, $\overrightarrow {BP}=2 \overrightarrow {PD}$, if $\overrightarrow {AP}=λ \overrightarrow {BA}+μ \overrightarrow {BC}$, then $λ+μ=\_\_\_\_\_\_$. | - \frac {1}{3} | 7/8 |
In a circle, two mutually perpendicular chords $AB$ and $CD$ are given. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places. | 4.24\, | 1/8 |
Given sets \( A \) and \( B \) that are composed of positive integers, with \( |A|=20 \) and \( |B|=16 \). Set \( A \) satisfies the condition that if \( a, b, m, n \in A \) and \( a+b=m+n \), then \( \{a, b\}=\{m, n\} \). Define \( A+B=\{a+b \mid a \in A, b \in B\} \). Determine the minimum value of \( |A+B| \). | 200 | 2/8 |
Given three positive numbers \(a, b, c\) satisfying \(a \leqslant b+c \leqslant 3a\), \(3b^{2} \leqslant a(a+c) \leqslant 5b^{2}\), find the minimum value of \(\frac{b-2c}{a}\). | -\frac{18}{5} | 5/8 |
How many ways are there to arrange the numbers $1$ through $8$ into a $2$ by $4$ grid such that the sum of the numbers in each of the two rows are all multiples of $6,$ and the sum of the numbers in each of the four columns are all multiples of $3$ ? | 288 | 1/8 |
Find the tenth digit of \( 11^{12^{13}} \) (where \( 11^{12^{13}} \) represents 11 raised to the power of \( 12^{13} \)). | 2 | 1/8 |
The circles $k_{1}$ and $k_{2}$, both with unit radius, touch each other at point $P$. One of their common tangents that does not pass through $P$ is the line $e$. For $i>2$, let $k_{i}$ be the circle different from $k_{i-2}$ that touches $k_{1}$, $k_{i-1}$, and $e$. Determine the radius of $k_{1999}$. | \frac{1}{1998^2} | 5/8 |
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left focus $F$ and a chord perpendicular to the major axis of length $6\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the midpoint of $AB$. Find the maximum distance from a point $M$ on ellipse $C$ to focus $F$. | 6\sqrt{2} + 6 | 2/8 |
In the interval [1, 6], three different integers are randomly selected. The probability that these three numbers are the side lengths of an obtuse triangle is ___. | \frac{1}{4} | 7/8 |
100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box? | 12 | 1/8 |
Given the hyperbola $\frac{x^{2}}{a-3} + \frac{y^{2}}{2-a} = 1$, with foci on the $y$-axis and a focal distance of $4$, determine the value of $a$. The options are:
A) $\frac{3}{2}$
B) $5$
C) $7$
D) $\frac{1}{2}$ | \frac{1}{2} | 1/8 |
There is an exam with 7 questions, each asking about the answers to these 7 questions, and the answers can only be 1, 2, 3, or 4. The questions are as follows:
1. How many questions have the answer 4?
2. How many questions have an answer that is neither 2 nor 3?
3. What is the average of the answers to questions (5) and (6)?
4. What is the difference between the answers to questions (1) and (2)?
5. What is the sum of the answers to questions (1) and (7)?
6. Which question is the first to have the answer 2?
7. How many different answers are given by only one question?
What is the sum of the answers to all 7 questions? | 16 | 1/8 |
Let $ABCD$ be a square with side $4$ . Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$ , such that they are on the interior but not on the sides, we always have a square with sidr length $1$ , which is inside the square $ABCD$ , such that it contains no points in its interior(they can be on the sides). | 15 | 4/8 |
Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.
Determine the maximum possible number of positive integers among the 2018 integers. | 1009 | 1/8 |
In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.
[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,linewidth(1));
draw((0,0)--(1,1)--(1,2)--(0,0)--cycle,linewidth(1));
dot((1,2));
label("A",(0,0),SW);
label("B",(2,0),SE);
label("C",(1,2),N);
label("D",(1,1),NE);
label("E",(1,0),S);
[/asy] | \frac{1}{2} | 2/8 |
Let \( x_{i} > 0 \) for \( i = 1, 2, \ldots, n \) and \( n > 2 \). What is the minimum value of \( S = \sum_{i, j, k \text{ mutually distinct}} \frac{x_{i}}{x_{j} + x_{k}} \), where \( 1 \leq i, j, k \leq n \)? | \frac{n(n-1)(n-2)}{2} | 1/8 |
Determine the smallest positive integer $n$ whose prime factors are all greater than $18$ , and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$ . | 1843 | 1/8 |
On the board, 26 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 26 minutes? | 325 | 7/8 |
In the polar coordinate system, curve $C$: $\rho =2a\cos \theta (a > 0)$, line $l$: $\rho \cos \left( \theta -\frac{\pi }{3} \right)=\frac{3}{2}$, $C$ and $l$ have exactly one common point. $O$ is the pole, $A$ and $B$ are two points on $C$, and $\angle AOB=\frac{\pi }{3}$, then the maximum value of $|OA|+|OB|$ is __________. | 2 \sqrt{3} | 7/8 |
The radii of five concentric circles \(\omega_{0}, \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\) form a geometric progression with common ratio \(q\). What is the largest value of \(q\) for which it is possible to draw an open broken line \(A_{0} A_{1} A_{2} A_{3} A_{4}\), consisting of four segments of equal length, where \(A_{i}\) lies on \(\omega_{i}\) for all \(i = 0, 1, 2, 3, 4\)? | \frac{\sqrt{5}+1}{2} | 1/8 |
Let \( f(x) = \frac{1}{x^3 + 3x^2 + 2x} \). Determine the smallest positive integer \( n \) such that
\[ f(1) + f(2) + f(3) + \cdots + f(n) > \frac{503}{2014}. \] | 44 | 6/8 |
A frustum \(ABCD\) is bisected by a circle with a diameter \(ab\). This circle is parallel to the top and bottom bases of the frustum. What is the diameter \(ab\) of this circle? | 2\sqrt[3]{\frac{R^3+r^3}{2}} | 1/8 |
Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5. | 1350 | 1/8 |
In an opaque bag, there are four small balls labeled with the Chinese characters "阳", "过", "阳", and "康" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.<br/>$(1)$ If one ball is randomly drawn from the bag, the probability that the character on the ball is exactly "康" is ________; <br/>$(2)$ If two balls are drawn from the bag by person A, using a list or a tree diagram, find the probability $P$ that one ball has the character "阳" and the other ball has the character "康". | \frac{1}{3} | 4/8 |
Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer. | 1680 | 7/8 |
Point $(x, y)$ is randomly selected from the rectangular region defined by vertices $(0, 0), (3013, 0), (3013, 3014),$ and $(0, 3014)$. What is the probability that $x > 8y$? | \frac{3013}{48224} | 3/8 |
Without using any tables, find the exact value of the product:
\[ P = \cos \frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos \frac{6\pi}{15} \cos \frac{7\pi}{15}. \] | 1/128 | 6/8 |
On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions:
(1) Each block must have at most $50$ states.
(2) Every pair of states must be together in at least one block.
Find the minimum number of blocks that must be formed. | 6 | 5/8 |
A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo.
A psychologist noticed the following:
1. The number of Native Americans drawn is twice the number of Eskimos.
2. The number of Eskimos with teepees is equal to the number of Native Americans with igloos.
3. Each teepee drawn with an Eskimo is matched with three igloos.
Based on this information, determine the proportion of Native Americans among the inhabitants of teepees. | 7/8 | 5/8 |
What is the value of $\dfrac{\sqrt[5]{11}}{\sqrt[7]{11}}$ expressed as 11 raised to what power? | \frac{2}{35} | 7/8 |
Given $n$ points on a plane, no three of which are collinear. A line is drawn through every pair of points. What is the minimum number of pairwise non-parallel lines that can be among them? | n | 1/8 |
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$ , tens digit $y$ , and units digit $z$ , and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$ . How many three-digit numbers $\underline{abc}$ , none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$ ? | 120 | 1/8 |
Find all positive integers \( x \) for which \( p(x) = x^2 - 10x - 22 \), where \( p(x) \) denotes the product of the digits of \( x \). | 12 | 6/8 |
Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute $\lim_{n\to \infty} P_n$ | e | 7/8 |
A certain product is manufactured by two factories. The production volume of the second factory is three times that of the first. The defect rate of the first factory is $2\%$, whereas the defect rate of the second factory is $1\%$. Products made by both factories over the same period are mixed together and sent for sale. What is the probability that a purchased product is from the second factory, given that it is defective? | 0.6 | 2/8 |
Given an equilateral triangle $ABC$ with side length 11, points $A_1$, $B_1$, $C_1$ are taken on sides $BC$, $CA$, $AB$ respectively such that $AC_1 = BA_1 = CB_1 = 5$. Find the ratio of the area of triangle $ABC$ to the area of the triangle formed by the lines $A A_1$, $B B_1$, $C C_1$. | 91 | 2/8 |
Find the functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all \( n \), the following equation holds:
\[ f(n) + f(f(n)) + f(f(f(n))) = 3n \] | f(n)=n | 1/8 |
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM? | 10:25 PM | 1/8 |
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