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Given a quadratic function in terms of \\(x\\), \\(f(x)=ax^{2}-4bx+1\\).
\\((1)\\) Let set \\(P=\\{1,2,3\\}\\) and \\(Q=\\{-1,1,2,3,4\\}\\), randomly pick a number from set \\(P\\) as \\(a\\) and from set \\(Q\\) as \\(b\\), calculate the probability that the function \\(y=f(x)\\) is increasing in the interval \\([1,+∞)\\).
\\((2)\\) Suppose point \\((a,b)\\) is a random point within the region defined by \\( \\begin{cases} x+y-8\\leqslant 0 \\\\ x > 0 \\\\ y > 0\\end{cases}\\), denote \\(A=\\{y=f(x)\\) has two zeros, one greater than \\(1\\) and the other less than \\(1\\}\\), calculate the probability of event \\(A\\) occurring. | \dfrac{961}{1280} | 2/8 |
Two circles with radii of 4 and 5 are externally tangent to each other and are both circumscribed by a third circle. Find the area of the shaded region outside these two smaller circles but within the larger circle. Express your answer in terms of $\pi$. Assume the configuration of tangency and containment is similar to the original problem, with no additional objects obstructing. | 40\pi | 6/8 |
Billy's basketball team scored the following points over the course of the first $11$ games of the season. If his team scores $40$ in the $12^{th}$ game, which of the following statistics will show an increase?
\[42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73\]
$\textbf{(A) } \text{range} \qquad \textbf{(B) } \text{median} \qquad \textbf{(C) } \text{mean} \qquad \textbf{(D) } \text{mode} \qquad \textbf{(E) } \text{mid-range}$ | \textbf{(A)} | 1/8 |
Let \(A B C\) be a triangle and \(M\) be any point in the plane. Show that \(\frac{M A}{B C}+\frac{M B}{C A}+\frac{M C}{A B} \geqslant \sqrt{3}\). | \sqrt{3} | 2/8 |
Fill in each blank with a digit not equal to 1, such that the equation holds true. How many different ways can this be done?
$$
[\mathrm{A} \times(\overline{1 \mathrm{~B}}+\mathrm{C})]^{2}=\overline{9 \mathrm{DE} 5}
$$ | 8 | 7/8 |
In a toy store, there are large and small plush kangaroos. In total, there are 100 of them. Some of the large kangaroos are female kangaroos with pouches. Each female kangaroo has three small kangaroos in her pouch, and the other kangaroos have empty pouches. Find out how many large kangaroos are in the store, given that there are 77 kangaroos with empty pouches. | 31 | 7/8 |
Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain? | 2118 | 3/8 |
Please write an irrational number that is greater than -3 and less than -2. | -\sqrt{5} | 7/8 |
Let the function \( f(x) \) (for \( x \in \mathbf{R}, x \neq 0 \)) satisfy the condition \( f(x_{1} x_{2}) = f(x_{1}) + f(x_{2}) \) for any nonzero real numbers \( x_{1} \) and \( x_{2} \), and \( f(x) \) is an increasing function over \( (0, +\infty) \). Then the solution to the inequality \( f(x) + f\left(x - \frac{1}{2}\right) \leqslant 0 \) is ______. | [\frac{1-\sqrt{17}}{4},0)\cup(0,\frac{1}{2})\cup(\frac{1}{2},\frac{1+\sqrt{17}}{4}] | 5/8 |
Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number \( k \), she calls a placement of nonzero real numbers on the \( 2^{2019} \) vertices of the hypercube \( k \)-harmonic if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to \( k \) times the number on this vertex. Let \( S \) be the set of all possible values of \( k \) such that there exists a \( k \)-harmonic placement. Find \( \sum_{k \in S}|k| \). | 2040200 | 7/8 |
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6 \mathrm{\ and \ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations.
| 462 | 4/8 |
The side length of the base of a regular triangular pyramid is \(a\). The lateral edge forms an angle of \(60^{\circ}\) with the plane of the base. Find the distance between opposite edges of the pyramid. | \frac{3a}{4} | 5/8 |
Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt 2$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers?
$\textbf{(A)}\ 30\ \text{and}\ 31 \qquad\textbf{(B)}\ 31\ \text{and}\ 32 \qquad\textbf{(C)}\ 32\ \text{and}\ 33 \qquad\textbf{(D)}\ 33\ \text{and}\ 34 \qquad\textbf{(E)}\ 34\ \text{and}\ 35$ | \textbf{(B)}\31\\32 | 1/8 |
Gilda has a bag of marbles. She gives $20\%$ of them to her friend Pedro. Then Gilda gives $10\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?
$\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54$ | \textbf{(E)}\54 | 1/8 |
Find all pairs of positive integers \( m, n \geqslant 3 \) such that there exist infinitely many positive integers \( a \) for which
$$
\frac{a^{m}+a-1}{a^{n}+a^{2}-1}
$$
is an integer. | (5,3) | 1/8 |
Let the set \( I = \{1, 2, \cdots, n\} (n \geqslant 3) \). If two non-empty proper subsets \( A \) and \( B \) of \( I \) satisfy \( A \cap B = \varnothing \) and \( A \cup B = I \), then \( A \) and \( B \) are called a partition of \( I \). If for any partition \( A \) and \( B \) of the set \( I \), there exist two numbers in \( A \) or \( B \) such that their sum is a perfect square, then \( n \) must be at least \(\qquad\). | 15 | 1/8 |
The triangle \( ABC \) has \( AC = 1 \), \(\angle ACB = 90^\circ\), and \(\angle BAC = \varphi\). \( D \) is the point between \( A \) and \( B \) such that \( AD = 1 \). \( E \) is the point between \( B \) and \( C \) such that \(\angle EDC = \varphi\). The perpendicular to \( BC \) at \( E \) meets \( AB \) at \( F \). Find \(\lim_{\varphi \to 0} EF\). | \frac{1}{3} | 3/8 |
It is known that the quadratic equations $2017 x^{2} + px + q = 0$ and $up x^{2} + q x + 2017 = 0$ (where $p$ and $q$ are given real numbers) have one common root. List all possible values of this common root and prove that there are no others. | 1 | 1/8 |
Elisenda has a piece of paper in the shape of a triangle with vertices $A, B$, and $C$ such that $A B=42$. She chooses a point $D$ on segment $A C$, and she folds the paper along line $B D$ so that $A$ lands at a point $E$ on segment $B C$. Then, she folds the paper along line $D E$. When she does this, $B$ lands at the midpoint of segment $D C$. Compute the perimeter of the original unfolded triangle. | 168+48 \sqrt{7} | 1/8 |
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$ . Find the value of $2^{-(1+\log_23)x}$ | 216 | 1/8 |
In 1993, several number theory problems were proposed by F. Smarandache from the United States, garnering attention from scholars both domestically and internationally. One of these is the well-known Smarandache function. The Smarandache function of a positive integer \( n \) is defined as
$$
S(n) = \min \left\{ m \mid m \in \mathbf{Z}_{+},\ n \mid m! \right\},
$$
For example, \( S(2) = 2 \), \( S(3) = 3 \), and \( S(6) = 3 \).
(1) Find the values of \( S(16) \) and \( S(2016) \).
(2) If \( S(n) = 7 \), find the maximum value of the positive integer \( n \).
(3) Prove that there are infinitely many composite numbers \( n \) such that \( S(n) = p \), where \( p \) is the largest prime factor of \( n \). | 5040 | 7/8 |
The bisector of angle $B A D$ of the right trapezoid $A B C D$ (with bases $A D$ and $B C$, and $\angle B A D=90^{\circ}$) intersects the lateral side $C D$ at point $E$. Find the ratio $C E: E D$ if $A D+B C=A B$. | 1:1 | 4/8 |
If $a$ and $b$ are digits for which
$\begin{array}{ccc}& 2 & a\ \times & b & 3\ \hline & 6 & 9\ 9 & 2 & \ \hline 9 & 8 & 9\end{array}$
then $a+b =$ | 7 | 7/8 |
A number is said to be TOP if it has 5 digits and when the product of the 1st and 5th digits is equal to the sum of the 2nd, 3rd, and 4th digits. For example, 12,338 is TOP because it has 5 digits and $1 \cdot 8 = 2 + 3 + 3$.
a) What is the value of $a$ such that $23,4a8$ is TOP?
b) How many TOP numbers end with 2 and start with 1?
c) How many TOP numbers start with 9? | 112 | 7/8 |
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $ , $ a $ , $ b $ , $ c $ , $ d \neq 3 $ , and $ d \le 50 $ . Find the maximum value of the difference between two of these prime numbers. | 2067 | 1/8 |
Given the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$, evaluate $f(-2)$ using the Qin Jiushao algorithm to find the value of $v_2$. | -4 | 7/8 |
A chord is drawn on a circle by choosing two points uniformly at random along its circumference. This is done two more times to obtain three total random chords. The circle is cut along these three lines, splitting it into pieces. The probability that one of the pieces is a triangle is $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$. | 115 | 1/8 |
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000? | 458 | 1/8 |
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have? | 62 | 5/8 |
A circle $\omega$ is circumscribed around triangle $ABC$. Circle $\omega_{1}$ is tangent to line $AB$ at point $A$ and passes through point $C$, while circle $\omega_{2}$ is tangent to line $AC$ at point $A$ and passes through point $B$. A tangent to circle $\omega$ at point $A$ intersects circle $\omega_{1}$ again at point $X$ and intersects circle $\omega_{2}$ again at point $Y$. Find the ratio $\frac{A X}{X Y}$. | \frac{1}{2} | 3/8 |
Students from three different schools worked on a summer project. Six students from Atlas school worked for $4$ days, five students from Beacon school worked for $6$ days, and three students from Cedar school worked for $10$ days. The total payment for the students' work was $774. Given that each student received the same amount for a day's work, determine how much the students from Cedar school earned altogether. | 276.43 | 1/8 |
In triangle \\(ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(A < B < C\\) and \\(C = 2A\\).
\\((1)\\) If \\(c = \sqrt{3}a\\), find the measure of angle \\(A\\).
\\((2)\\) If \\(a\\), \\(b\\), and \\(c\\) are three consecutive positive integers, find the area of \\(\triangle ABC\\). | \dfrac{15\sqrt{7}}{4} | 3/8 |
Write the canonical equations of the line.
$$
\begin{aligned}
& x+y+z-2=0 \\
& x-y-2z+2=0
\end{aligned}
$$ | \frac{x}{-1}=\frac{y-2}{3}=\frac{z}{-2} | 7/8 |
Given the ellipse C₁: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) with one focus coinciding with the focus of the parabola C₂: y<sup>2</sup>\=4 $$\sqrt {2}$$x, and the eccentricity of the ellipse is e= $$\frac { \sqrt {6}}{3}$$.
(I) Find the equation of C₁.
(II) A moving line l passes through the point P(0, 2) and intersects the ellipse C₁ at points A and B. O is the origin. Find the maximum area of △OAB. | \frac{\sqrt{3}}{2} | 7/8 |
For certain natural numbers $n$, the numbers $2^{n}$ and $5^{n}$ share the same leading digit. What are these possible leading digits?
(The 14th All-Russian Mathematical Olympiad, 1988) | 3 | 4/8 |
Using the digits 0, 1, 2, 3, 4, and 5, form six-digit numbers without repeating any digit.
(1) How many such six-digit odd numbers are there?
(2) How many such six-digit numbers are there where the digit 5 is not in the unit place?
(3) How many such six-digit numbers are there where the digits 1 and 2 are not adjacent? | 408 | 5/8 |
The orthocenter of triangle $ABC$ divides altitude $\overline{CF}$ into segments with lengths $HF = 6$ and $HC = 15.$ Calculate $\tan A \tan B.$
[asy]
unitsize (1 cm);
pair A, B, C, D, E, F, H;
A = (0,0);
B = (5,0);
C = (4,4);
D = (A + reflect(B,C)*(A))/2;
E = (B + reflect(C,A)*(B))/2;
F = (C + reflect(A,B)*(C))/2;
H = extension(A,D,B,E);
draw(A--B--C--cycle);
draw(C--F);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$F$", F, S);
dot("$H$", H, W);
[/asy] | \frac{7}{2} | 7/8 |
For each positive integer \( n \), let \( a_n \) be the smallest nonnegative integer such that there is only one positive integer at most \( n \) that is relatively prime to all of \( n, n+1, \ldots, n+a_n \). If \( n < 100 \), compute the largest possible value of \( n - a_n \). | 16 | 1/8 |
A company needs to transport two types of products, $A$ and $B$, with the following volumes and masses per unit as shown in the table:
| | Volume $(m^{3}/$unit) | Mass (tons$/$unit) |
|----------|-----------------------|--------------------|
| $A$ type | $0.8$ | $0.5$ |
| $B$ type | $2$ | $1$ |
1. Given a batch of products containing both $A$ and $B$ types with a total volume of $20m^{3}$ and a total mass of $10.5$ tons, find the number of units for each type of product.
2. A logistics company has trucks with a rated load of $3.5$ tons and a capacity of $6m^{3}$. The company offers two payment options:
- Charging per truck: $600$ yuan per truck to transport goods to the destination.
- Charging per ton: $200$ yuan per ton to transport goods to the destination.
Determine how the company should choose to transport the products from part (1) in one or multiple shipments to minimize the shipping cost, and calculate the cost under the chosen method. | 2100 | 7/8 |
Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (and the cat disappears) as soon as the number of beans Alice has is greater than 2008 and has last two digits 42. What is the minimum number of cents Alice can spend to win the game, assuming she starts with 0 beans? | 35 | 1/8 |
For values of $x$ less than $1$ but greater than $-4$, the expression
$\frac{x^2 - 2x + 2}{2x - 2}$
has:
$\textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{+1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{+1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{-1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{-1}$ | \textbf{(E)}\ | 1/8 |
Construct the cross-section of the triangular prism \( A B C A_1 B_1 C_1 \) with a plane passing through points \( A_1 \) and \( C \) that is parallel to line \( B C_1 \). In what ratio does this plane divide edge \( A B \)? | 1:1 | 6/8 |
Given the sequence of numbers with only even digits in their decimal representation, determine the $2014^\text{th}$ number in the sequence. | 62048 | 1/8 |
A cube with a side length of 10 is divided into 1000 smaller cubes with an edge length of 1. A number is written in each small cube, such that the sum of the numbers in every column of 10 cubes (in any of the three directions) is equal to 0. One of the small cubes (denoted by $A$) contains the number one. Three layers, each parallel to a face of the larger cube and passing through cube $A$, have a thickness of 1. Find the sum of all numbers in the cubes that do not lie within these layers. | -1 | 7/8 |
In a circle with radius \( R \), two chords \( AB \) and \( AC \) are drawn. A point \( M \) is taken on \( AB \) or its extension such that the distance from \( M \) to the line \( AC \) is equal to \( |AC| \). Similarly, a point \( N \) is taken on \( AC \) or its extension such that the distance from \( N \) to the line \( AB \) is equal to \( |AB| \). Find the distance \( |MN| \). | 2R | 1/8 |
Given \( x_{0} > 0 \), \( x_{0} \neq \sqrt{3} \), a point \( Q\left( x_{0}, 0 \right) \), and a point \( P(0, 4) \), the line \( PQ \) intersects the hyperbola \( x^{2} - \frac{y^{2}}{3} = 1 \) at points \( A \) and \( B \). If \( \overrightarrow{PQ} = t \overrightarrow{QA} = (2-t) \overrightarrow{QB} \), then \( x_{0} = \) _______. | \frac{\sqrt{2}}{2} | 1/8 |
In circle \(\omega\), two perpendicular chords intersect at a point \( P \). The two chords have midpoints \( M_1 \) and \( M_2 \) respectively, such that \( P M_1 = 15 \) and \( P M_2 = 20 \). Line \( M_1 M_2 \) intersects \(\omega\) at points \( A \) and \( B \), with \( M_1 \) between \( A \) and \( M_2 \). Compute the largest possible value of \( B M_2 - A M_1 \). | 7 | 1/8 |
A round robin tournament is held with $2016$ participants. Each player plays each other player once and no games result in ties. We say a pair of players $A$ and $B$ is a *dominant pair* if all other players either defeat $A$ and $B$ or are defeated by both $A$ and $B$ . Find the maximum number dominant pairs.
[i]Proposed by Nathan Ramesh | 2015 | 1/8 |
Positive real numbers \( x, y, z \) satisfy: \( x^{4} + y^{4} + z^{4} = 1 \). Find the minimum value of the algebraic expression \( \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} \). | \frac{9 \sqrt[4]{3}}{8} | 4/8 |
Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$? | 21 | 5/8 |
Consider a \( 2 \times 2 \) grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid? | 88 | 7/8 |
Find the largest constant \( c \) such that for all real numbers \( x \) and \( y \) satisfying \( x > 0, y > 0, x^2 + y^2 = 1 \), the inequality \( x^6 + y^6 \geq cx y \) always holds. | \frac{1}{2} | 7/8 |
A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \mathrm{~m} /\mathrm{s}$? | 0.8 | 1/8 |
\( A_i \) (\( i = 1, 2, 3, 4 \)) are subsets of the set \( S = \{ 1, 2, \cdots, 2005 \} \). \( F \) is the set of all ordered quadruples \( (A_1, A_2, A_3, A_1) \). Find the value of \( \sum_{(A_1, A_2, A_3, A_4) \in F} |A_1 \cup A_2 \cup A_3 \cup A_4| \). | 2^{8016}\times2005\times15 | 1/8 |
Let \( f_{n+1} = \left\{ \begin{array}{ll} f_n + 3 & \text{if } n \text{ is even} \\ f_n - 2 & \text{if } n \text{ is odd} \end{array} \right. \).
If \( f_1 = 60 \), determine the smallest possible value of \( n \) satisfying \( f_m \geq 63 \) for all \( m \geq n \). | 11 | 3/8 |
In a certain football invitational tournament, 16 cities participate, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that aside from Team A from city $A$, the number of matches already played by each of the other teams was different. Find the number of matches already played by Team B from city $A$. | 15 | 1/8 |
Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$ . Find the angle between the planes $DOB$ and $DOC.$ | 90 | 1/8 |
Given the expansion of $\left(x-\frac{a}{x}\right)^{5}$, find the maximum value among the coefficients in the expansion. | 10 | 1/8 |
Given \( f(x) = \frac{1}{x^{2}} + \frac{1}{x^{4}} \), if \( f(a-2) < f(2a+1) \), then the range of values for \( a \) is | (-3,-\frac{1}{2})\cup(-\frac{1}{2},\frac{1}{3}) | 3/8 |
Given four sequences of real numbers \(\left\{a_{n}\right\}, \left\{b_{n}\right\}, \left\{c_{n}\right\}, \left\{d_{n}\right\}\), for any \(n\), the following relationships hold:
$$
\begin{array}{l}
a_{n+1}=a_{n}+b_{n} \\
b_{n+1}=b_{n}+c_{n} \\
c_{n+1}=c_{n}+d_{n} \\
d_{n+1}=d_{n}+a_{n}
\end{array}
$$
Prove: If for some \(k, m \geq 1\), we have:
$$
a_{k+m}=a_{m}, \quad b_{k+m}=b_{m}, \quad c_{k+m}=c_{m}, \quad d_{k+m}=d_{m}
$$
then \(a_{2}=b_{2}=c_{2}=d_{2}=0\). | a_2=b_2=c_2=d_2=0 | 1/8 |
What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)? | 64 | 1/8 |
Place the arithmetic operation signs and parentheses between the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ so that the resulting expression equals 100. | 100 | 1/8 |
The quadratic polynomial $P(x),$ with real coefficients, satisfies
\[P(x^3 + x) \ge P(x^2 + 1)\]for all real numbers $x.$ Find the sum of the roots of $P(x).$ | 4 | 6/8 |
In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum value of $k$ such that regardless of the initial order of the shirts, it is possible to remove $k$ white shirts and $k$ purple shirts so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively. | 10 | 2/8 |
Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$ . On the side $AB$ construct the rhombus $BAFC$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$ . If the area of $BAFE$ is equal to $65$ , calculate the area of $ABCD$ . | 120 | 4/8 |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{1}{2}} \frac{\sqrt[3]{\frac{x}{4}}-\frac{1}{2}}{\sqrt{\frac{1}{2}+x}-\sqrt{2x}} \] | -\frac{2}{3} | 7/8 |
Let \( a \in \mathbf{R} \), and the function \( f(x) = ax^2 + x - a \) where \( |x| \leq 1 \).
1. If \( |a| \leq 1 \), prove: \( |f(x)| \leq \frac{5}{4} \).
2. Find the value of \( a \) such that the function \( f(x) \) has a maximum value of \( \frac{17}{8} \). | -2 | 7/8 |
The difference between two positive integers is 8 and their product is 56. What is the sum of these integers? | 12\sqrt{2} | 4/8 |
Let $n$ be the answer to this problem. Given $n>0$, find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$. | 48 | 1/8 |
Two digits of a number were swapped, and as a result, it increased by more than 3 times. The resulting number is 8453719. Find the original number. | 1453789 | 4/8 |
A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow center itself takes up 4% of the area of the flag, what percent of the area of the flag is green? | 45\% | 4/8 |
When \( s \) and \( t \) range over all real numbers, the expression
$$
(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}
$$
achieves a minimum value of \(\qquad\). | 2 | 4/8 |
Given the set \( K(n, 0) = \varnothing \). For any non-negative integers \( m \) and \( n \), define \( K(n, m+1) = \{ k \mid 1 \leq k \leq n, K(k, m) \cap K(n-k, m) = \varnothing \} \). Find the number of elements in the set \( K(2004, 2004) \). | 127 | 1/8 |
The smaller square in the figure below has a perimeter of $4$ cm, and the larger square has an area of $16$ $\text{cm}^2$. What is the distance from point $A$ to point $B$? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(12,0));
draw((2,0)--(2,10));
draw((0,0)--(0,2));
draw((0,2)--(2,2));
draw((0,2)--(12,10));
draw((12,0)--(12,10));
draw((2,10)--(12,10));
label("B",(0,2),W);
label("A",(12,10),E);
[/asy] | 5.8 | 2/8 |
Eliminate every second number in a clockwise direction from numbers $1, 2, 3, \cdots, 2001$ that have been placed on a circle, starting with the number 2, until only one number remains. What is the last remaining number? | 1955 | 2/8 |
Twelve pencils are sharpened so that all of them have different lengths. Masha wants to arrange the pencils in a box in two rows of 6 pencils each, such that the lengths of the pencils in each row decrease from left to right, and each pencil in the second row lies on a longer pencil. How many ways can she do this? | 132 | 7/8 |
Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible? | 96 | 4/8 |
Several balls are divided into $n$ heaps and are now reorganized into $n+k$ heaps, where $n$ and $k$ are given positive integers, and each heap contains at least 1 ball. Prove that there exist $k+1$ balls such that the number of balls in their original heap is greater than the number of balls in their current heap. | k+1 | 4/8 |
If the maximum value and the minimum value of the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \) sum up to 4, then find \( a + b \). | 3 | 1/8 |
Given the function $f(x) = \sqrt{3}\cos x\sin x - \frac{1}{2}\cos 2x$.
(1) Find the smallest positive period of $f(x)$.
(2) Find the maximum and minimum values of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding values of $x$. | -\frac{1}{2} | 1/8 |
Alice chooses three primes \( p, q, r \) independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of \( p x^{2} + q x + r \). What is the probability that at least one of her roots is an integer? | \frac{3}{200} | 1/8 |
The digits of the number 123456789 can be rearranged to form a number that is divisible by 11. For example, 123475869, 459267831, and 987453126. How many such numbers are there? | 31680 | 2/8 |
Let \( P_{1}, P_{2}, \ldots, P_{1993} = P_{0} \) be distinct points in the \( xy \)-plane with the following properties:
1. Both coordinates of \( P_{i} \) are integers, for \( i = 1, 2, \ldots, 1993 \).
2. There is no point other than \( P_{i} \) and \( P_{i+1} \) on the line segment joining \( P_{i} \) with \( P_{i+1} \) whose coordinates are both integers, for \( i = 0, 1, \ldots, 1992 \).
Prove that for some \( i \), \( 0 \leq i \leq 1992 \), there exists a point \( Q \) with coordinates \( (q_x, q_y) \) on the line segment joining \( P_{i} \) with \( P_{i+1} \) such that both \( 2 q_x \) and \( 2 q_y \) are odd integers. | 0 | 1/8 |
Consider (3-variable) polynomials
\[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\]
and
\[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\]
Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients. | 1 | 2/8 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? | 12 | 5/8 |
In triangle \( \triangle ABC \), \( AC = BC \), and \( D \) is a point on side \( AB \). Consider the incircle of \( \triangle ACD \) and the excircle of \( \triangle BCD \) that is tangent to side \( BD \). Suppose both circles have a radius equal to \( r \). Prove that the altitude of \( \triangle ABC \) on the equal sides is equal to \( 4r \). | 4r | 2/8 |
Matrices $A$ , $B$ are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0 1 & 2 & 0 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 2 & 4 & 0 0 & 0 & 12\end{pmatrix}\]
Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ . | \frac{\pi}{3} | 5/8 |
For some positive integers $a$ and $b$, the product \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1) \cdot\log_{b-1} b\]contains exactly $1000$ terms, and its value is $3.$ Compute $a+b.$ | 1010 | 1/8 |
Given a right trapezoid \( ABCD \) with the upper base \( AB \) of length 1 and the lower base \( DC \) of length 7. Connecting point \( E \) on side \( AB \) and point \( F \) on side \( DC \), the segment \( EF \) is parallel to \( AD \) and \( BC \). If segment \( EF \) divides the area of the right trapezoid into two equal parts, what is the length of segment \( EF \)? | 5 | 5/8 |
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire length of the pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe, if Gavrila's step is 80 cm? Round your answer to the nearest whole meter. The speed of the tractor is constant. | 108 | 4/8 |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row? | 62 | 6/8 |
A point \( P \) is chosen uniformly at random inside a square of side length 2. If \( P_{1}, P_{2}, P_{3}, \) and \( P_{4} \) are the reflections of \( P \) over each of the four sides of the square, find the expected value of the area of quadrilateral \( P_{1} P_{2} P_{3} P_{4} \). | 8 | 1/8 |
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$.
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set
$\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing
order are coded as $VYZ, VYX, VVW$, respectively. What is the base-$10$ expression for the integer coded as $XYZ$? | 108 | 7/8 |
Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| 17 | 7/8 |
Given the ellipse \( C \) passing through the point \( M(1,2) \) with foci at \((0, \pm \sqrt{6})\) and the origin \( O \) as the center, a line \( l \) parallel to \( OM \) intersects the ellipse \( C \) at points \( A \) and \( B \). Find the maximum area of \( \triangle OAB \). | 2 | 7/8 |
Given a circle \( C: x^{2} + y^{2} = 24 \) and a line \( l: \frac{x}{12} + \frac{y}{8} = 1 \), let \( P \) be a point on \( l \). The ray \( OP \) intersects the circle at point \( R \). Also, point \( Q \) lies on \( OP \) and satisfies the condition \( |OQ| \cdot |OP| = |OR|^2 \). As point \( P \) moves along \( l \), find the equation of the locus of point \( Q \), and describe what kind of curve this locus is. | (x-1)^2+(y-\frac{3}{2})^2=\frac{13}{4} | 7/8 |
Find the four-digit number that is 4 times smaller than the number formed by reversing its digits. | 2178 | 7/8 |
For any positive integer \( x \), define \( \text{Accident}(x) \) to be the set of ordered pairs \( (s, t) \) with \( s \in \{0, 2, 4, 5, 7, 9, 11\} \) and \( t \in \{1, 3, 6, 8, 10\} \) such that \( x + s - t \) is divisible by 12. For any nonnegative integer \( i \), let \( a_i \) denote the number of \( x \in \{0, 1, \ldots, 11\} \) for which \( |\text{Accident}(x)| = i \). Find \( a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} \). | 26 | 4/8 |
A T-tetromino is formed by adjoining three unit squares to form a \(1 \times 3\) rectangle, and adjoining on top of the middle square a fourth unit square. Determine the minimum number of unit squares that must be removed from a \(202 \times 202\) grid so that it can be tiled with T-tetrominoes. | 4 | 3/8 |
Calculate the winning rate per game;
List all possible outcomes;
Calculate the probability of satisfying the condition "$a+b+c+d \leqslant 2$". | \dfrac{11}{16} | 6/8 |
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