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How many ways can the eight vertices of a three-dimensional cube be colored red and blue such that no two points connected by an edge are both red? Rotations and reflections of a given coloring are considered distinct. | 35 | 2/8 |
Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if $ x_1 > a$ , then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$ , then the sequence is not monotonic. | 8^{1/8} | 1/8 |
Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$. | 60 | 6/8 |
Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$ . Find the length of the segment that connects the midpoints of the bases of the trapezium. | 5 | 4/8 |
In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win? | \frac{5}{6} | 3/8 |
Given two moving points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \) on the parabola \( x^{2}=4 y \) (where \( y_{1} + y_{2} = 2 \) and \( y_{1} \neq y_{2} \))), if the perpendicular bisector of line segment \( AB \) intersects the \( y \)-axis at point \( C \), then the maximum value of the area of triangle \( \triangle ABC \) is ________ | \frac{16 \sqrt{6}}{9} | 6/8 |
Let \( R \) be the region in the first quadrant bounded by the x-axis, the line \( 2y = x \), and the ellipse \( \frac{x^2}{9} + y^2 = 1 \). Let \( R' \) be the region in the first quadrant bounded by the y-axis, the line \( y = mx \), and the ellipse. Find \( m \) such that \( R \) and \( R' \) have the same area. | \frac{2}{9} | 4/8 |
Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$. | \frac{1}{4} | 5/8 |
A regular tetrahedron has an edge length of 1. Using each of its edges as the diameter, create spheres. Let \( S \) be the intersection of the six spheres. Prove that there are two points in \( S \) whose distance is \( \frac{\sqrt{6}}{6} \). | \frac{\sqrt{6}}{6} | 1/8 |
The sequence is defined by the recurrence relations \( a_{1} = 1 \),
\[
a_{2n} = \left\{
\begin{array}{ll}
a_{n}, & \text{if } n \text{ is even,} \\
2a_{n}, & \text{if } n \text{ is odd;}
\end{array}
\right. \quad
a_{2n+1} = \left\{
\begin{array}{ll}
2a_{n}+1, & \text{if } n \text{ is even,} \\
a_{n}, & \text{if } n \text{ is odd.}
\end{array}
\right.
\]
Find the smallest natural number \( n \) for which \( a_{n} = a_{2017} \). | 5 | 1/8 |
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$. | 695 | 1/8 |
Given the functions \( f_{0}(x)=|x| \), \( f_{1}(x)=\left|f_{0}(x)-1\right| \), and \( f_{2}(x)=\left|f_{1}(x)-2\right| \), find the area of the closed region formed by the graph of the function \( y=f_{2}(x) \) and the x-axis. | 7 | 3/8 |
Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \(n\) shots are all for 10 points, she will secure her victory regardless of her opponent's scoring in the remaining shots. Find the minimum value for \(n\). | 52 | 7/8 |
A lock code is made up of four digits that satisfy the following rules: - At least one digit is a 4, but neither the second digit nor the fourth digit is a 4. - Exactly one digit is a 2, but the first digit is not 2. - Exactly one digit is a 7. - The code includes a 1, or the code includes a 6, or the code includes two 4s. How many codes are possible? | 22 | 2/8 |
Points \( K, L, M, N, P \) are positioned sequentially on a circle with radius \( 2\sqrt{2} \). Find the area of triangle \( KLM \) if \( LM \parallel KN \), \( KM \parallel NP \), \( MN \parallel LP \), and the angle \( \angle LOM \) is \( 45^\circ \), where \( O \) is the intersection point of chords \( LN \) and \( MP \). | 4 | 1/8 |
Solve the inequality \( n^{3} - n < n! \) for positive integers \( n \). (Here, \( n! \) denotes the factorial of \( n \), which is the product of all positive integers from 1 to \( n \)). | 1orn\ge6 | 7/8 |
A certain country wishes to interconnect 2021 cities with flight routes, which are always two-way, in the following manner:
- There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights.
- For every pair \((A, B)\) of cities that are connected by a direct flight, there is another city \(C\) such that \((A, C)\) and \((B, C)\) are connected by direct flights.
Show that at least 3030 flight routes are needed to satisfy the two requirements. | 3030 | 2/8 |
For which values of the parameter \(a\) does the equation \(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\) have exactly three distinct solutions? | 48 | 3/8 |
In the triangular pyramid $A-BCD$, where $AB=AC=BD=CD=BC=4$, the plane $\alpha$ passes through the midpoint $E$ of $AC$ and is perpendicular to $BC$, calculate the maximum value of the area of the section cut by plane $\alpha$. | \frac{3}{2} | 3/8 |
ABCDEF is a six-digit number. All its digits are different and arranged in ascending order from left to right. This number is a perfect square.
Determine what this number is. | 134689 | 1/8 |
Find the maximum value of the natural number \( a \) such that the inequality \(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n+1}>2a-5\) holds for all natural numbers \( n \). | 3 | 5/8 |
Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\angle CAD = 30^{\circ}$, find $\sin(2\angle BAD)$. | \frac{1}{2} | 7/8 |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells? | 25 | 1/8 |
Given the Fibonacci sequence defined as follows: \( F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n} \) (n ≥ 1), find \( \left(F_{2017}, F_{99}F_{101}+1\right) \). | 1 | 2/8 |
In the tetrahedron \(ABCD\), given that \(AB = 1\), \(CD = \sqrt{3}\), the distance between the lines \(AB\) and \(CD\) is 2, and the angle between them is \(\frac{\pi}{3}\), find the volume of the tetrahedron \(ABCD\). | \frac{1}{2} | 7/8 |
In the Cartesian coordinate system $xOy$, the equation of circle $M$ is given by $x^2+y^2-4x\cos \alpha-2y\sin \alpha+3\cos^2\alpha=0$ (where $\alpha$ is a parameter), and the parametric equation of line $l$ is $\begin{cases}x=\tan \theta \\ y=1+t\sin \theta\end{cases}$ (where $t$ is a parameter).
$(I)$ Find the parametric equation of the trajectory $C$ of the center of circle $M$, and explain what curve it represents;
$(II)$ Find the maximum chord length cut by the trajectory $C$ on line $l$. | \dfrac {4 \sqrt {3}}{3} | 1/8 |
A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 5,$ and
\[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 5.$ Evaluate $a_1 a_2 \dotsm a_{2011} - \sum_{i = 1}^{2011} a_i^2.$ | -1941 | 7/8 |
For every positive integeer $n>1$ , let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$ .
Find $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$ | 1 | 3/8 |
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$ | \textbf{(A)}\frac{7}{36} | 1/8 |
Let $ABC$ be an isosceles triangle with the apex at $A$. Let $M$ be the midpoint of segment $[BC]$. Let $D$ be the symmetric point of $M$ with respect to the segment $[AC]$. Let $x$ be the angle $\widehat{BAC}$. Determine, as a function of $x$, the value of the angle $\widehat{MDC}$. | \frac{x}{2} | 4/8 |
Suppose \( a \) and \( b \) are the roots of \( x^{2}+x \sin \alpha+1=0 \) while \( c \) and \( d \) are the roots of the equation \( x^{2}+x \cos \alpha-1=0 \). Find the value of \( \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \). | 1 | 6/8 |
The number \( N \) is a perfect square and does not end in zero. After removing the last two digits of this number, the resulting number is again a perfect square. Find the largest number \( N \) with this property. | 1681 | 4/8 |
Find the smallest positive integer \( n \) such that the set \( \{1, 2, 3, \cdots, 3n-1, 3n\} \) can be divided into \( n \) disjoint triples \( \{x, y, z\} \) where \( x + y = 3z \). | 5 | 1/8 |
Find a necessary and sufficient condition on the positive integer $n$ that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root.
| 1 | 1/8 |
In acute $\triangle A B C$ with centroid $G, A B=22$ and $A C=19$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $A C$ and $A B$ respectively. Let $G^{\prime}$ be the reflection of $G$ over $B C$. If $E, F, G$, and $G^{\prime}$ lie on a circle, compute $B C$. | 13 | 1/8 |
A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \_\_\_\_\_\_. | \frac{17}{8} | 6/8 |
Given two similar triangles $\triangle ABC\sim\triangle FGH$, where $BC = 24 \text{ cm}$ and $FG = 15 \text{ cm}$. If the length of $AC$ is $18 \text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth. | 11.3 | 2/8 |
Given the function \( f(x) \) presented in Table 1, \( a_{k} (0 \leqslant k \leqslant 4) \) represents the number of times \( k \) appears among \( a_{0}, a_{1}, \cdots, a_{4} \). Then \( a_{0}+a_{1}+a_{2}+a_{3} = \qquad \). | 5 | 2/8 |
A finite set of polygons on a plane is considered properly placed if, for any two polygons in the set, there exists a line passing through the origin that intersects both.
Find the smallest natural number \( m \) such that for any properly placed set of polygons, one can draw \( m \) lines passing through the origin such that each polygon in the set intersects at least one of these \( m \) lines. | 2 | 2/8 |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$. | 2889 | 7/8 |
In the following diagram, \(\angle ACB = 90^\circ\), \(DE \perp BC\), \(BE = AC\), \(BD = \frac{1}{2} \mathrm{~cm}\), and \(DE + BC = 1 \mathrm{~cm}\). Suppose \(\angle ABC = x^\circ\). Find the value of \(x\). | 30 | 4/8 |
The $8\times18$ [rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$?
$\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 7\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 10$ | \textbf{(A)}6 | 1/8 |
Find all positive integers \( N \) such that \( N \) contains only the prime factors 2 and 5, and \( N + 25 \) is a perfect square. | 2000 | 5/8 |
Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$. | 680 | 6/8 |
Suppose a sequence of positive numbers $\left\{a_{n}\right\}$ satisfies: $a_{0} = 1, a_{n} = a_{n+1} + a_{n+2},$ for $n = 0, 1, 2, \ldots$. Find $a_{1}$. | \frac{\sqrt{5} - 1}{2} | 5/8 |
In the spatial quadrilateral $ABCD$, it is known that $AB=2$, $BC=3$, $CD=4$, and $DA=5$. Find the dot product $\overrightarrow{AC} \cdot \overrightarrow{BD}$. | 7 | 6/8 |
In the Cartesian coordinate system, with the origin O as the pole and the positive x-axis as the polar axis, a polar coordinate system is established. The polar coordinate of point P is $(1, \pi)$. Given the curve $C: \rho=2\sqrt{2}a\sin(\theta+ \frac{\pi}{4}) (a>0)$, and a line $l$ passes through point P, whose parametric equation is:
$$
\begin{cases}
x=m+ \frac{1}{2}t \\
y= \frac{\sqrt{3}}{2}t
\end{cases}
$$
($t$ is the parameter), and the line $l$ intersects the curve $C$ at points M and N.
(1) Write the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;
(2) If $|PM|+|PN|=5$, find the value of $a$. | 2\sqrt{3}-2 | 4/8 |
The Fibonacci sequence is defined as $f_1=f_2=1$ , $f_{n+2}=f_{n+1}+f_n$ ( $n\in\mathbb{N}$ ). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$ . Show that $b\ge f_{n+1}$ . | b\gef_{n+1} | 3/8 |
In the Cartesian coordinate plane \(xOy\), points \(A\) and \(B\) are on the parabola \(y^2 = 4x\) and satisfy \(\overrightarrow{OA} \cdot \overrightarrow{OB} = -4\). \(F\) is the focus of the parabola. Find \(S_{\triangle OP} \cdot S_{\triangle ORP} =\). | 2 | 1/8 |
There are a batch of wooden strips with lengths of \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many different triangles can be formed? | 36 | 6/8 |
A rectangle with a [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of length $x$ is twice as long as it is wide. What is the area of the rectangle?
$\textbf{(A) } \frac{1}{4}x^2\qquad \textbf{(B) } \frac{2}{5}x^2\qquad \textbf{(C) } \frac{1}{2}x^2\qquad \textbf{(D) } x^2\qquad \textbf{(E) } \frac{3}{2}x^2$ | \textbf{(B)}\frac{2}{5}x^2 | 1/8 |
Given a square with side length \(a\), its corners have been cut off to form a regular octagon. Determine the area of this octagon. | 2a^2(\sqrt{2}-1) | 7/8 |
How many ordered pairs of integers $(x, y)$ satisfy the equation \[x^{2020}+y^2=2y?\]
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$ | \textbf{(D)}4 | 1/8 |
Given that \( n \) is a fixed positive integer, for \( -1 \leq x_{i} \leq 1 \) (where \( i = 1, 2, \cdots, 2n \)), find the maximum value of
\[
\sum_{1 \leq r < s \leq 2n} (s - r - n) x_{r} x_{s}.
\] | n(n-1) | 1/8 |
Define a regular \(n\)-pointed star as described in the original problem, but with a modification: the vertex connection rule skips by \(m\) steps where \(m\) is coprime with \(n\) and \(m\) is not a multiple of \(3\). How many non-similar regular 120-pointed stars adhere to this new rule? | 15 | 6/8 |
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$ . Prove that the sequence $\{x_{n}\}$ converges and find its limit. | 1 | 3/8 |
After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in? | 5 | 6/8 |
Inside rectangle \(ABCD\), point \(M\) is chosen such that \(\angle BMC + \angle AMD = 180^\circ\). Find the measure of \(\angle BCM + \angle DAM\). | 90 | 1/8 |
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$. | f(x) = x + 1 | 1/8 |
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction). | 168 | 7/8 |
Given that the positive numbers \(x, y, z\) satisfy the system:
$$
\begin{cases}
x^{2}+y^{2}+x y=529 \\
x^{2}+z^{2}+\sqrt{3} x z=441 \\
z^{2}+y^{2}=144
\end{cases}
$$
Find the value of the expression \(\sqrt{3} x y + 2 y z + x z\). | 224\sqrt{5} | 1/8 |
Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points. | 72 | 1/8 |
Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of $n$ . | 1915 | 1/8 |
One day, School A bought 56 kilograms of fruit candy at 8.06 yuan per kilogram. A few days later, School B also needed to buy the same 56 kilograms of fruit candy, but it happened that there was a promotional event, and the price of fruit candy was reduced by 0.56 yuan per kilogram. Additionally, they received 5% extra fruit candy for free. How much less did School B spend compared to School A? | 51.36 | 1/8 |
A total area of \( 2500 \, \mathrm{m}^2 \) will be used to build identical houses. The construction cost for a house with an area \( a \, \mathrm{m}^2 \) is the sum of the material cost \( 100 p_{1} a^{\frac{3}{2}} \) yuan, labor cost \( 100 p_{2} a \) yuan, and other costs \( 100 p_{3} a^{\frac{1}{2}} \) yuan, where \( p_{1} \), \( p_{2} \), and \( p_{3} \) are consecutive terms of a geometric sequence. The sum of these terms is 21 and their product is 64. Given that building 63 of these houses would result in the material cost being less than the sum of the labor cost and the other costs, find the maximum number of houses that can be built to minimize the total construction cost. | 156 | 2/8 |
Teams A and B each have 7 members who appear in a predetermined order to compete in a Go game relay. The first player of each team competes first; the loser is eliminated, and the winner advances to compete against the next player of the losing team. This process continues until all the players of one team are eliminated. The other team wins, forming a competition sequence. What is the total number of possible competition sequences? $\qquad$ . | 3432 | 4/8 |
In how many different ways can four couples sit around a circular table such that no couple sits next to each other? | 1488 | 3/8 |
$15\times 36$ -checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$
Tiles are supposed to cover whole squares of the board and be non-overlapping.
What is the maximum number of squares to be covered? | 540 | 2/8 |
In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the least $k$, such that, there exists a strategy that allows to identify the two false coins using the eletronic device, at most, $k$ times. | 21 | 4/8 |
In triangle $ABC,$ $\angle B = 60^\circ$ and $\angle C = 45^\circ.$ The point $D$ divides $\overline{BC}$ in the ratio $1:3$. Find
\[\frac{\sin \angle BAD}{\sin \angle CAD}.\] | \frac{\sqrt{6}}{6} | 7/8 |
Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers \( n \) satisfy \( f(n) = f(2005) \) ? | 225 | 2/8 |
Given real numbers \(\alpha\) and \(\beta\) satisfying:
\[
\alpha^{3} - 3 \alpha^{2} + 5 \alpha = 1, \quad \beta^{3} - 3 \beta^{2} + 5 \beta = 5,
\]
find \(\alpha + \beta\). | 2 | 7/8 |
Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$. | -\frac{3}{5} | 7/8 |
\(ABCD\) is a square with a side length of \(20 \sqrt{5} x\). \(P\) and \(Q\) are the midpoints of \(DC\) and \(BC\) respectively.
1. If \(AP = ax\), find \(a\).
2. If \(PQ = b \sqrt{10} x\), find \(b\).
3. If the distance from \(A\) to \(PQ\) is \(c \sqrt{10} x\), find \(c\).
4. If \(\sin \theta = \frac{d}{100}\), find \(d\). | 60 | 3/8 |
The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is:
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$ | \textbf{(C)}\3 | 1/8 |
A sequence $s_{0}, s_{1}, s_{2}, s_{3}, \ldots$ is defined by $s_{0}=s_{1}=1$ and, for every positive integer $n, s_{2 n}=s_{n}, s_{4 n+1}=s_{2 n+1}, s_{4 n-1}=s_{2 n-1}+s_{2 n-1}^{2} / s_{n-1}$. What is the value of $s_{1000}$? | 720 | 5/8 |
For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$ . Find the smallest $n$ for which
\[ P(P(P(n))) > 10^{12} \] | 6 | 3/8 |
In an isosceles triangle, the base and the lateral side are equal to 5 and 20 respectively. Find the angle bisector of the angle at the base of the triangle. | 6 | 7/8 |
Xiaoying goes home at noon to cook noodles by herself, which involves the following steps: ① Wash the pot and fill it with water, taking 2 minutes; ② Wash the vegetables, taking 3 minutes; ③ Prepare the noodles and seasonings, taking 2 minutes; ④ Boil the water in the pot, taking 7 minutes; ⑤ Use the boiling water to cook the noodles and vegetables, taking 3 minutes. Except for step ④, each step can only be performed one at a time. The minimum time Xiaoying needs to cook the noodles is minutes. | 12 | 5/8 |
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more? | 30 \% | 1/8 |
Let \(\alpha\) and \(\beta\) be the two real roots of the quadratic equation \(x^{2} - 2kx + k + 20 = 0\). Find the minimum value of \((\alpha+1)^{2} + (\beta+1)^{2}\), and determine the value of \(k\) for which this minimum value is achieved. | 18 | 1/8 |
The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$. | \frac{125}{3} | 7/8 |
Circle $\Gamma$ is the incircle of $\triangle ABC$ and is also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A=40^\circ$, $\angle B=60^\circ$, and $\angle C=80^\circ$, what is the measure of $\angle AYX$? | 120^\circ | 3/8 |
Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal. | 14 | 1/8 |
Find the smallest \( n > 4 \) for which we can find a graph on \( n \) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them. | 16 | 2/8 |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimeter of $\triangle APR$. | 50 | 7/8 |
Solve
\[(x^3 + 3x^2 \sqrt{2} + 6x + 2 \sqrt{2}) + (x + \sqrt{2}) = 0.\]Enter all the solutions, separated by commas. | -\sqrt{2}, -\sqrt{2} + i, -\sqrt{2} - i | 1/8 |
Given the function $f(x) = x + a\ln x$ has its tangent line at $x = 1$ perpendicular to the line $x + 2y = 0$, and the function $g(x) = f(x) + \frac{1}{2}x^2 - bx$,
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Let $x_1$ and $x_2$ ($x_1 < x_2$) be two extreme points of the function $g(x)$. If $b \geq \frac{7}{2}$, find the minimum value of $g(x_1) - g(x_2)$. | \frac{15}{8} - 2\ln 2 | 3/8 |
Let a positive integer \( n \) be called a cubic square if there exist positive integers \( a, b \) with \( n = \gcd(a^2, b^3) \). Count the number of cubic squares between 1 and 100 inclusive. | 13 | 1/8 |
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 12$ | \textbf{(C)}\6 | 1/8 |
If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid. | 36 | 7/8 |
Given that D is a point on the hypotenuse BC of right triangle ABC, and $AC= \sqrt {3}DC$, $BD=2DC$. If $AD=2 \sqrt {3}$, then $DC=\_\_\_\_\_\_$. | \sqrt {6} | 5/8 |
Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$ | 90 | 6/8 |
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$. | 961 | 1/8 |
There are $2n$ students in a school $(n \in \mathbb{N}, n \geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen? | 6 | 1/8 |
2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle? | 666 | 1/8 |
Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step? | 974 | 6/8 |
In an acute-angled triangle \( A B C \) on a plane, circles with diameters \( A B \) and \( A C \) intersect the altitude from \( C \) at points \( M \) and \( N \), and intersect the altitude from \( B \) at points \( P \) and \( Q \). Prove that the points \( M \), \( N \), \( P \), and \( Q \) are concyclic. | M,N,P,Q | 1/8 |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,1));
draw((6,0)--(6,1));
draw((6,2)--(7,2)--(7,-4)--(6,-4)--cycle);
draw((6,-1)--(7,-1));
draw((6,-2)--(7,-2));
draw((6,-3)--(7,-3));
draw((3,0)--(4,0)--(4,-3)--(3,-3)--cycle);
draw((3,-1)--(4,-1));
draw((3,-2)--(4,-2));
label("21",(0.5,0.8),S);
label("14",(3.5,-1.2),S);
label("18",(3.5,-2.2),S);
label("$N$",(6.5,1.8),S);
label("-17",(6.5,-3.2),S);
[/asy] | -7 | 1/8 |
Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$ , and $$ f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)} $$ for all integers $m, n \ge 1$ . Prove that $f(99, 99)<1.99$ .
*Proposed by Navilarekallu Tejaswi* | f(99,99)<1.99 | 1/8 |
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