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"Xiongdash" $\times$ "Xionger" $=$ "Xiongdibrothers". If the same character represents the same digit from 0 to 9, different characters represent different digits, and "Da" > "Er", then the sum of all three-digit numbers represented by "Xiongdibrothers" that meet these conditions is $\qquad$ | 686 | 1/8 |
Fill the numbers from $1$ to $25$ into a $5 \times 5$ table. Select the maximum number from each row and the minimum number from each column. A total of 10 numbers are selected in this way. Among these 10 selected numbers, there are at least $\qquad$ distinct numbers. | 9 | 1/8 |
Given that the cross-section of a cylinder is a square and the height of the cylinder is equal to the diameter of a sphere, calculate the ratio of the total surface area of the cylinder to the surface area of the sphere. | \frac{3}{2} | 7/8 |
The "One Helmet, One Belt" safety protection campaign is a safety protection campaign launched by the Ministry of Public Security nationwide. It is also an important standard for creating a civilized city and being a civilized citizen. "One helmet" refers to a safety helmet. Drivers and passengers of electric bicycles should wear safety helmets. A certain shopping mall intends to purchase a batch of helmets. It is known that purchasing 8 type A helmets and 6 type B helmets costs $630, and purchasing 6 type A helmets and 8 type B helmets costs $700.
$(1)$ How much does it cost to purchase 1 type A helmet and 1 type B helmet respectively?
$(2)$ If the shopping mall is prepared to purchase 200 helmets of these two types, with a total cost not exceeding $10200, and sell type A helmets for $58 each and type B helmets for $98 each. In order to ensure that the total profit is not less than $6180, how many purchasing plans are there? How many type A and type B helmets are in the plan with the maximum profit? What is the maximum profit? | 6200 | 4/8 |
For any positive integer \( n \), let \( f(n) \) denote the index of the highest power of 2 which divides \( n! \). For example, \( f(10) = 8 \) since \( 10! = 2^8 \times 3^4 \times 5^2 \times 7 \). Find the value of \( f(1) + f(2) + \cdots + f(1023) \). | 518656 | 3/8 |
Given a cube \(ABCD - A_1B_1C_1D_1\) with edge length 1, a sphere is constructed with center at vertex \(A\) and radius \(\frac{2 \sqrt{3}}{3}\). Find the length of the curve formed by the intersection of the sphere's surface with the cube's surface. | \frac{5\sqrt{3}\pi}{6} | 1/8 |
Let \( \triangle ABC \) be a triangle with \( AB = 13 \), \( BC = 14 \), and \( CA = 15 \). Let \( O \) be the circumcenter of \( \triangle ABC \). Find the distance between the circumcenters of \( \triangle AOB \) and \( \triangle AOC \). | \frac{91}{6} | 1/8 |
Calculate the sum of 5739204.742 and -176817.835, and round the result to the nearest integer. | 5562387 | 6/8 |
Given that the lines $l_{1}$: $ax+y+3=0$ and $l_{2}$: $2x+\left(a-1\right)y+a+1=0$ are parallel, find the value of $a$. | -1 | 2/8 |
Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P_{4}$ on $AB$, with the reflection angle equal to the incidence angle. If $P_{4}$ coincides with $P_{0}$, then find $\tan \theta$. | $\frac{1}{2}$ | 7/8 |
Find the least positive integer $k$ for which the equation $\left\lfloor\frac{2002}{n}\right\rfloor=k$ has no integer solutions for $n$. (The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.) | 49 | 6/8 |
Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$ | 27/4 | 2/8 |
Find the positive value of $ k$ for which $ \int_0^{\frac {\pi}{2}} |\cos x \minus{} kx|\ dx$ is minimized. | \frac{2\sqrt{2}}{\pi}\cos(\frac{\pi}{2\sqrt{2}}) | 7/8 |
A group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is between 200 and 300? | 295 | 6/8 |
From 8 female students and 4 male students, 3 students are to be selected to participate in a TV program. Determine the number of different selection methods when the selection is stratified by gender. | 112 | 1/8 |
Given a function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$ which is a power function and is increasing on the interval $(0, \infty)$, find the value of the real number $m$. | -1 | 1/8 |
Given an arithmetic sequence $\{a_n\}$ where $a_1=1$ and $a_n=70$ (for $n\geq3$), find all possible values of $n$ if the common difference is a natural number. | 70 | 7/8 |
In a new sequence, the first term is \(a_1 = 5000\) and the second term is \(a_2 = 5001\). Furthermore, the values of the remaining terms are designed so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \( n \geq 1 \). Determine \(a_{1000}\). | 5666 | 5/8 |
A positive integer \( n \) is said to be good if \( 3n \) is a re-ordering of the digits of \( n \) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11. | 2475 | 6/8 |
Given the positive number sequence $\left\{a_{n}\right\}$ that satisfies $a_{n+1} \geqslant 2 a_{n}+1$ and $a_{n} < 2^{n+1}$ for all $n \in \mathbf{Z}_{+}$, what is the range of values for $a_{1}$? | (0,3] | 3/8 |
Find the sequence \(a_{0}, a_{1}, a_{2}, \ldots\) of positive numbers such that \(a_{0} = 1\) and \(a_{n} - a_{n+1} = a_{n+2}\) for \(n = 0, 1, 2, \ldots\). Show that there is only one such sequence. | {a_n}={(\frac{\sqrt{5}-1}{2})^n} | 6/8 |
Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible. | 5040 | 7/8 |
In a tetrahedron V-ABC with edge length 10, point O is the center of the base ABC. Segment MN has a length of 2, with one endpoint M on segment VO and the other endpoint N inside face ABC. If point T is the midpoint of segment MN, then the area of the trajectory formed by point T is __________. | 2\pi | 2/8 |
Consider all non-empty subsets of the set \( S = \{1, 2, \cdots, 10\} \). A subset is called a "good subset" if the number of even numbers in the subset is not less than the number of odd numbers. How many "good subsets" are there? | 637 | 6/8 |
The minimum value of the function $y = \sin 2 \cos 2x$ is ______. | - \frac{1}{2} | 2/8 |
For any positive integer \( q_{0} \), consider the sequence \( q_{i} = \left( q_{i-1} - 1 \right)^3 + 3 \) for \( i = 1, 2, \cdots, n \). If each \( q_{i} (i = 1, 2, \cdots, n) \) is a prime number, find the largest possible value of \( n \). | 2 | 2/8 |
The line \( K M_{1} \) intersects the extension of \( A B \) at point \( N \). Find the measure of angle \( DNA \). | 90 | 7/8 |
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:
i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$ ;
ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$ .
Suppose that $\frac{E_1A}{AH_1}=\lambda$ . Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$ .
*Xiong Bin* | \lambda | 2/8 |
If a binary number $n=\left(a_{m} a_{m-1} \cdots a_{1} a_{0}\right)_{2}$ satisfies $\left(a_{m} a_{m-1} \cdots a_{1} a_{0}\right)_{2}=$ $\left(a_{0} a_{1} \cdots a_{m-1} a_{m}\right)_{2}$, then $n$ is called a "binary palindrome." How many "binary palindrome" numbers are there among the natural numbers not exceeding 1988? | 92 | 1/8 |
The minimum value of the function \( f(x) = (\sqrt{1+x} + \sqrt{1-x} - 3)\left(\sqrt{1-x^2} + 1\right) \) is \( m \), and the maximum value is \( M \). Then \( \frac{M}{m} = \) . | \frac{3-\sqrt{2}}{2} | 7/8 |
Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$ , 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$ . Determine the largest possible value of $b$ . | \frac{14}{11} | 6/8 |
Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \perp XY.$ Meanwhile, $R$ is a point on line $PY$ such that $XR \perp PY.$ If $XR = 3$, $PQ = 6$, and $XY = 7$, then what is the length of $PY?$ | 14 | 7/8 |
Given a geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\left(1 + \log_{2} a_{n}\right)$ for $n \in \mathbf{N}^{*}$.
(1) Find the sum of the first $n$ terms of the sequence $\{a_{n} b_{n}\}$, denoted as $T_{n}$.
(2) If for any positive integer $n$, the inequality $\frac{1 + b_{1}}{b_{1}} \cdot \frac{1 + b_{2}}{b_{2}} \cdots \frac{1 + b_{n}}{b_{n}} \geq k \sqrt{n + 1}$ holds, determine the maximum value of the real number $k$. | \frac{3 \sqrt{2}}{4} | 2/8 |
Pentagon \( A B C D E \) is inscribed in a circle with radius \( R \). It is known that \( \angle B = 110^\circ \) and \( \angle E = 100^\circ \). Find the side \( C D \). | R | 2/8 |
Given \(\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos \alpha \cos \gamma}=\frac{4}{9}\), find the value of \(\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos (\alpha+\beta+\gamma) \cos \gamma}\). | \frac{4}{5} | 1/8 |
Given that \(\alpha, \beta \in \mathbf{R}\), the intersection point of the lines
\[
\frac{x}{\sin \alpha + \sin \beta} + \frac{y}{\sin \alpha + \cos \beta} = 1
\]
and
\[
\frac{x}{\cos \alpha + \sin \beta} + \frac{y}{\cos \alpha + \cos \beta} = 1
\]
lies on the line \(y = -x\). Determine the value of \(\sin \alpha + \cos \alpha + \sin \beta + \cos \beta\). | 0 | 4/8 |
The distance between the centers of two circles, which lie outside each other, is 65; the length of their common external tangent (between the points of tangency) is 63; the length of their common internal tangent is 25. Find the radii of the circles. | 3822 | 1/8 |
Given the digits 0, 1, 2, 3, 4, 5, how many unique six-digit numbers greater than 300,000 can be formed where the digit in the thousand's place is less than 3? | 216 | 6/8 |
Square $ABCD$ has side length $1$ ; circle $\Gamma$ is centered at $A$ with radius $1$ . Let $M$ be the midpoint of $BC$ , and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$ . Compute $MN$ .
*2018 CCA Math Bonanza Individual Round #11* | \frac{5}{6} | 7/8 |
Given $|a|=3$, $|b-2|=9$, and $a+b > 0$, find the value of $ab$. | -33 | 5/8 |
Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible by 3 when she stops? | \frac{3-\sqrt{3}}{3} | 1/8 |
The smallest number in the list $\{0.40, 0.25, 0.37, 0.05, 0.81\}$ is:
(A) 0.40
(B) 0.25
(C) 0.37
(D) 0.05
(E) 0.81 | 0.05 | 3/8 |
Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$. Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$. | 101 | 5/8 |
Let $ABC$ be a triangle such that $AB=2$ , $CA=3$ , and $BC=4$ . A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$ . Compute the area of the semicircle. | \frac{27\pi}{40} | 2/8 |
For positive integers \( x \), let \( g(x) \) be the number of blocks of consecutive 1's in the binary expansion of \( x \). For example, \( g(19)=2 \) because \( 19=10011_2 \) has a block of one 1 at the beginning and a block of two 1's at the end, and \( g(7)=1 \) because \( 7=111_2 \) only has a single block of three 1's. Compute \( g(1)+g(2)+g(3)+\cdots+g(256) \). | 577 | 2/8 |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$. | 3\sqrt{2} | 3/8 |
How many different seven-digit phone numbers exist (considering that the number cannot start with zero)? | 9\times10^6 | 1/8 |
A school has eight identical copies of a particular book. At any given time, some of these copies are in the school library and some are with students. How many different ways are there for some of the books to be in the library and the rest to be with students if at least one book is in the library and at least one is with students? | 254 | 1/8 |
Which positive numbers $x$ satisfy the equation $(\log_3x)(\log_x5)=\log_35$?
$\textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad \\ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are positive integers} \qquad \\ \textbf{(D)}\ \text{all positive } x \neq 1 \qquad \textbf{(E)}\ \text{none of these}$ | \textbf{(D)}\allpositivex\ne1 | 1/8 |
Horizontal parallel segments \( AB = 10 \) and \( CD = 15 \) are the bases of trapezoid \( ABCD \). Circle \(\gamma\) of radius 6 has its center within the trapezoid and is tangent to sides \( AB \), \( BC \), and \( DA \). If side \( CD \) cuts out an arc of \(\gamma\) measuring \(120^\circ\), find the area of \(ABCD\). | \frac{225}{2} | 3/8 |
Given that \(\tan (3 \alpha - 2 \beta) = \frac{1}{2}\) and \(\tan (5 \alpha - 4 \beta) = \frac{1}{4}\), find the value of \(\tan \alpha\). | \frac{13}{16} | 5/8 |
Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] | 1 | 6/8 |
Given 12 consecutive integers greater than 1000, prove that at least 8 of them are composite numbers. | 8 | 4/8 |
Given the sets $A={1,4,x}$ and $B={1,2x,x^{2}}$, if $A \cap B={4,1}$, find the value of $x$. | -2 | 3/8 |
The center of a sphere with a unit radius is located on the edge of a dihedral angle equal to \( \alpha \). Find the radius of a sphere whose volume equals the volume of the part of the given sphere that lies inside the dihedral angle. | \sqrt[3]{\frac{\alpha}{2\pi}} | 5/8 |
Is it possible to choose 8 numbers from the first 100 natural numbers such that their sum is divisible by each of these numbers? | Yes | 5/8 |
Given $|\overrightarrow {a}|=\sqrt {2}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}-\overrightarrow {b})\bot \overrightarrow {a}$, determine the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac{\pi}{4} | 3/8 |
A clothing store buys 600 pairs of gloves at a price of 12 yuan per pair. They sell 470 pairs at a price of 14 yuan per pair, and the remaining gloves are sold at a price of 11 yuan per pair. What is the total profit made by the clothing store from selling this batch of gloves? | 810 | 7/8 |
The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:
$\textbf{(A)}\ -2\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{undetermined}$ | \textbf{(B)}\0 | 1/8 |
In a right triangle $PQR$ where $\angle R = 90^\circ$, the lengths of sides $PQ = 15$ and $PR = 9$. Find $\sin Q$ and $\cos Q$. | \frac{3}{5} | 1/8 |
A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is | 3n+1 | 1/8 |
Some mice live in three neighboring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram?
A 9
B 11
C 12
D 16
E 19 | 11 | 1/8 |
Let \( \triangle ABC \) be a triangle, \( O \) the center of its circumcircle, and \( I \) the center of its incircle. Denote by \( E \) and \( F \) the orthogonal projections of \( I \) onto \( AB \) and \( AC \), respectively. Let \( T \) be the intersection point of \( EI \) with \( OC \), and let \( Z \) be the intersection point of \( FI \) with \( OB \). Define \( S \) as the intersection point of the tangents to the circumcircle of \( \triangle ABC \) at points \( B \) and \( C \).
Prove that the line \( SI \) is perpendicular to the line \( ZT \). | (SI)\perp(ZT) | 1/8 |
Consider a square of side length 1. Draw four lines that each connect a midpoint of a side with a corner not on that side, such that each midpoint and each corner is touched by only one line. Find the area of the region completely bounded by these lines. | \frac{1}{5} | 5/8 |
Let the greatest common divisor of the integers \( a, b, c, d \in \mathbf{Z} \) be 1. Is it true that any prime divisor of the number \( ad - bc \) is a divisor of the numbers \( a \) and \( c \) if and only if, for every \( n \in \mathbf{Z} \), the numbers \( an + b \) and \( cn + d \) are coprime? | True | 7/8 |
How many functions \( f: \{0,1\}^{3} \rightarrow \{0,1\} \) satisfy the property that, for all ordered triples \(\left(a_{1}, a_{2}, a_{3}\right)\) and \(\left(b_{1}, b_{2}, b_{3}\right)\) such that \( a_{i} \geq b_{i} \) for all \( i \), \( f\left(a_{1}, a_{2}, a_{3}\right) \geq f\left(b_{1}, b_{2}, b_{3}\right) \)? | 20 | 1/8 |
If the system of inequalities about $x$ is $\left\{{\begin{array}{l}{-2({x-2})-x＜2}\\{\frac{{k-x}}{2}≥-\frac{1}{2}+x}\end{array}}\right.$ has at most $2$ integer solutions, and the solution to the one-variable linear equation about $y$ is $3\left(y-1\right)-2\left(y-k\right)=7$, determine the sum of all integers $k$ that satisfy the conditions. | 18 | 1/8 |
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | 750 | 3/8 |
If $a$ and $b$ are elements of the set ${ 1,2,3,4,5,6 }$ and $|a-b| \leqslant 1$, calculate the probability that any two people playing this game form a "friendly pair". | \dfrac{4}{9} | 4/8 |
Find the maximum and minimum values of \( \cos x + \cos y + \cos z \), where \( x, y, z \) are non-negative reals with sum \( \frac{4\pi}{3} \). | 0 | 1/8 |
Given a parallelogram \(ABCD\) with \(AB = 3\), \(AD = \sqrt{3} + 1\), and \(\angle BAD = 60^\circ\). A point \(K\) is taken on side \(AB\) such that \(AK : KB = 2 : 1\). A line parallel to \(AD\) is drawn through point \(K\). On this line, inside the parallelogram, point \(L\) is selected, and on side \(AD\), point \(M\) is chosen such that \(AM = KL\). Lines \(BM\) and \(CL\) intersect at point \(N\). Find the angle \(BKN\). | 105 | 1/8 |
Let \( a \star b = ab + a + b \) for all integers \( a \) and \( b \). Evaluate \( 1 \star (2 \star (3 \star (4 \star \ldots (99 \star 100) \ldots))) \). | 101! - 1 | 7/8 |
The lateral face of a regular quadrangular pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the opposite lateral faces. | 90 | 6/8 |
In a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \), it is known that \( \overrightarrow{AP} = \frac{1}{2} \overrightarrow{AB} \), \( \overrightarrow{AQ} = \frac{1}{3} \overrightarrow{AC} \), and \( \overrightarrow{AR} = \frac{1}{4} \overrightarrow{AD} \). If point \( K \) is the centroid of \( \triangle BCD \), then what is the volume of the tetrahedron \( KPQR \)? | \frac{1}{36} | 1/8 |
Calculate $\sqrt[3]{\frac{x}{2015+2016}}$, where $x$ is the harmonic mean of the numbers
$a=\frac{2016+2015}{2016^{2}+2016 \cdot 2015+2015^{2}}$ and $b=\frac{2016-2015}{2016^{2}-2016 \cdot 2015+2015^{2}}$.
The harmonic mean of two positive numbers $a$ and $b$ is the number $c$ such that $\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)$. | \frac{1}{2016} | 3/8 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac{b}{a}+\sin({A-B})=\sin C$. Find:<br/>
$(1)$ the value of angle $A$;<br/>
$(2)$ if $a=2$, find the maximum value of $\sqrt{2}b+2c$ and the area of triangle $\triangle ABC$. | \frac{12}{5} | 7/8 |
A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$? | \frac{\pi}{3}+1-\sqrt{3} | 1/8 |
Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that $1$ , $2$ , $3$ , $4$ , $5$ , and $6$ dots are on separate faces. In a fit of mischief, James places his $21$ dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct con figuration. Regardless of the confi guration of dots, both dice are unweighted and have equal probability of showing each face after being rolled.
Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability. | \frac{17}{32} | 1/8 |
Let \( n \) be the answer to this problem. We define the digit sum of a date as the sum of its 4 digits when expressed in mmdd format (e.g. the digit sum of 13 May is \( 0+5+1+3=9 \)). Find the number of dates in the year 2021 with digit sum equal to the positive integer \( n \). | 15 | 1/8 |
Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ \[f(3x-2)\leq f(x)\leq f(2x-1).\]
*Proposed by A. Golovanov* | f(x)= | 6/8 |
A function $g$ is defined by $g(z) = (3 + 2i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$? | 2\sqrt{2} | 6/8 |
Points \( M \) and \( N \) are taken on the diagonals \( AB_1 \) and \( BC_1 \) of the faces of the parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), and the segments \( MN \) and \( A_1 C \) are parallel. Find the ratio of these segments. | 1:3 | 1/8 |
Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart. | \frac{2}{11} | 7/8 |
Given the line $y=a (0 < a < 1)$ and the function $f(x)=\sin \omega x$ intersect at 12 points on the right side of the $y$-axis. These points are denoted as $(x\_1)$, $(x\_2)$, $(x\_3)$, ..., $(x\_{12})$ in order. It is known that $x\_1= \dfrac {\pi}{4}$, $x\_2= \dfrac {3\pi}{4}$, and $x\_3= \dfrac {9\pi}{4}$. Calculate the sum $x\_1+x\_2+x\_3+...+x\_{12}$. | 66\pi | 6/8 |
Given the hexagons grow by adding subsequent layers of hexagonal bands of dots, with each new layer having a side length equal to the number of the layer, calculate how many dots are in the hexagon that adds the fifth layer, assuming the first hexagon has only 1 dot. | 61 | 6/8 |
Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;
$(2)$ If in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=\sqrt{7}$, $f(B)=0$, $\sin A=3\sin C$, find the values of $a$, $c$ and the area of $\triangle ABC$. | \frac{3\sqrt{3}}{4} | 7/8 |
Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form an isosceles triangle. | \frac{7}{18} | 7/8 |
Find the maximum value of real number $k$ such that
\[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\]
holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$ . | 4 | 2/8 |
Simplify first, then evaluate: $\left(\frac{2}{m-3}+1\right) \div \frac{2m-2}{m^2-6m+9}$, and then choose a suitable number from $1$, $2$, $3$, $4$ to substitute and evaluate. | -\frac{1}{2} | 7/8 |
The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction. | \frac{60}{7} | 5/8 |
A circle contains the points \((0, 11)\) and \((0, -11)\) on its circumference and contains all points \((x, y)\) with \(x^{2} + y^{2} < 1\) in its interior. Compute the largest possible radius of the circle. | 61 | 7/8 |
Let $T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2}$ where $x,y,z$ are real numbers such that $1 \leq x,y,z \leq 4$ and $x-y+z=4$ .
Find the smallest value of $10 \times T$ . | 23 | 3/8 |
There are 10 different balls: 2 red balls, 5 yellow balls, and 3 white balls. If taking 1 red ball earns 5 points, taking 1 yellow ball earns 1 point, and taking 1 white ball earns 2 points, how many ways are there to draw 5 balls such that the total score is greater than 10 points but less than 15 points?
A. 90
B. 100
C. 110
D. 120 | 110 | 1/8 |
In a store, there are four types of nuts: hazelnuts, almonds, cashews, and pistachios. Stepan wants to buy 1 kilogram of nuts of one type and 1 kilogram of nuts of another type. He has calculated the cost of such a purchase depending on which two types of nuts he chooses. Five of Stepan's six possible purchases would cost 1900, 2070, 2110, 2330, and 2500 rubles. How many rubles is the cost of the sixth possible purchase? | 2290 | 6/8 |
In $\triangle ABC$, where $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, and $AC=1$, let $M$ be the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between points $A$ and $B$ is $\sqrt{2}$. What is the distance from point $A$ to plane $BCM$? | \frac{\sqrt{6}}{3} | 3/8 |
How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) | 36 | 2/8 |
There are 30 people standing in a row, each of them is either a knight who always tells the truth or a liar who always lies. They are numbered from left to right. Each person with an odd number says: "All people with numbers greater than mine are liars", and each person with an even number says: "All people with numbers less than mine are liars".
How many liars could there be? If there are multiple correct answers, list them in ascending order separated by semicolons. | 28 | 1/8 |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$. | -5 | 3/8 |
Given $\overrightarrow{a}=(\sin \pi x,1)$, $\overrightarrow{b}=( \sqrt {3},\cos \pi x)$, and $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$:
(I) If $x\in[0,2]$, find the interval(s) where $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ is monotonically increasing.
(II) Let $P$ be the coordinates of the first highest point and $Q$ be the coordinates of the first lowest point on the graph of $y=f(x)$ to the right of the $y$-axis. Calculate the cosine value of $\angle POQ$. | -\frac{16\sqrt{481}}{481} | 7/8 |
Denote by $\mathbb Z^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n\geq 0$ , let $P_n$ be the subset of $\mathbb Z^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n$ . Determine, as a function of $n$ , the number of four-point subsets of $P_n$ whose elements are the vertices of a square. | 5n+1 | 1/8 |
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