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Person A departs from location A to location B, while persons B and C depart from location B to location A. After A has traveled 50 kilometers, B and C start simultaneously from location B. A and B meet at location C, and A and C meet at location D. It is known that A's speed is three times that of C and 1.5 times that of B. The distance between locations C and D is 12 kilometers. Determine the distance between locations A and B in kilometers. | 130 | 2/8 |
How many three-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 6? | 216 | 7/8 |
Three cells in a grid contain numbers, and all other cells are empty. It is allowed to choose two numbers from different non-empty cells and write their sum in an empty cell. It is also allowed to choose numbers \(a, b, c\) from three different non-empty cells and write the number \(ab + c^2\) in an empty cell. Prove that using a series of such operations, it is possible to write in one of the cells the square of the sum of the three initial numbers, no matter what these numbers are. | ()^2 | 1/8 |
The sine of the dihedral angle at the lateral edge of a regular quadrilateral pyramid is $\frac{15}{17}$. Find the lateral surface area of the pyramid if the area of its diagonal section is $3 \sqrt{34}$. | 68 | 3/8 |
In trapezoid \(ABCD\), diagonal \(AC\) is equal to 1 and is also its height. Perpendiculars \(AE\) and \(CF\) are drawn from points \(A\) and \(C\) to sides \(CD\) and \(AB\) respectively. Find \(AD\) if \(AD=CF\) and \(BC=CE\). | \sqrt{\sqrt{2}-1} | 1/8 |
The concept of negative numbers first appeared in the ancient Chinese mathematical book "Nine Chapters on the Mathematical Art." If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as $-5$ yuan. | -5 | 7/8 |
A small rubber ball moves between two massive vertical walls, colliding with them. One wall is stationary, while the other moves away from it at a constant speed of \( u = 100 \ \text{cm/s} \). Assuming the ball's motion is always horizontal and the collisions are perfectly elastic, find its final speed if the initial speed of the ball was \( v_{0} = 2017 \ \text{cm/s} \). | 17\, | 1/8 |
On the extensions of the medians \(A K\), \(B L\), and \(C M\) of triangle \(A B C\), points \(P\), \(Q\), and \(R\) are taken such that \(K P = \frac{1}{2} A K\), \(L Q = \frac{1}{2} B L\), and \(M R = \frac{1}{2} C M\). Find the area of triangle \(P Q R\) if the area of triangle \(A B C\) is 1. | 25/16 | 5/8 |
Let \( m \) be an integer greater than 1, and let's define a sequence \( \{a_{n}\} \) as follows:
\[
\begin{array}{l}
a_{0}=m, \\
a_{1}=\varphi(m), \\
a_{2}=\varphi^{(2)}(m)=\varphi(\varphi(m)), \\
\vdots \\
a_{n}=\varphi^{(n)}(m)=\varphi\left(\varphi^{(n-1)}(m)\right),
\end{array}
\]
where \( \varphi(m) \) is the Euler's totient function.
If for any non-negative integer \( k \), \( a_{k+1} \) always divides \( a_{k} \), find the greatest positive integer \( m \) not exceeding 2016. | 1944 | 1/8 |
Given a finite number of points in the plane such that no three points are collinear, what is the maximum number of points for which the line segments connecting them can be colored with three colors so that the following conditions are satisfied:
a) No point can have three segments of different colors emanating from it;
b) It is not possible to find three points such that the three segments they determine are of the same color;
c) All three colors are used. | 4 | 1/8 |
We define the weight $W$ of a positive integer as follows: $W(1) = 0$ , $W(2) = 1$ , $W(p) = 1 + W(p + 1)$ for every odd prime $p$ , $W(c) = 1 + W(d)$ for every composite $c$ , where $d$ is the greatest proper factor of $c$ . Compute the greatest possible weight of a positive integer less than 100. | 12 | 3/8 |
One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$.
| 181 | 7/8 |
Given $H$ is the orthocenter of $\triangle ABC$, $\angle A = 75^\circ$, $BC = 2$. Find the area of the circumcircle of $\triangle ABH$. | 4\pi(2-\sqrt{3}) | 1/8 |
Given a triangle \(ABC\). Points \(K, L,\) and \(M\) are placed on the plane such that triangles \(KAM, CLM,\) and \(KLB\) are all congruent to triangle \(KLM\). What inequality sign should be placed between the perimeter of triangle \(KLM\) and the semiperimeter of triangle \(ABC\)?
The vertices of the triangles are listed arbitrarily: for example, one cannot assert that in the congruent triangles \(KAM\) and \(CLM\), point \(K\) corresponds to point \(C\). | \ge | 1/8 |
In an arcade game, the "monster" is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $60^\circ$. What is the perimeter of the monster in cm? | \frac{5}{3}\pi + 2 | 1/8 |
Let \([x]\) be the largest integer not greater than \(x\). If \( A = \left\lfloor \frac{2008 \times 80 + 2009 \times 130 + 2010 \times 180}{2008 \times 15 + 2009 \times 25 + 2010 \times 35} \right\rfloor \), find the value of \(A\). | 5 | 7/8 |
The number 2017 has 7 ones and 4 zeros in its binary representation. When will the next closest year occur in which the binary representation of its number will have no more ones than zeros? Enter the year. Score for the problem: 8 points. | 2048 | 6/8 |
Given that six balls are numbered 1, 2, 3, 4, 5, and 6, and the requirement that ball number 1 must be adjacent to ball number 2, and ball number 5 must not be adjacent to ball number 6, calculate the total number of different arrangements. | 144 | 4/8 |
Triangles $MAB$ and $MCD$ are similar but have opposite orientations. Let $O_{1}$ be the center of rotation by the angle $2 \angle(\overrightarrow{AB}, \overrightarrow{BM})$ that maps $A$ to $C$, and $O_{2}$ be the center of rotation by the angle $2 \angle(\overrightarrow{AB}, \overrightarrow{AM})$ that maps $B$ to $D$. Prove that $O_{1} = O_{2}$. | O_1=O_2 | 5/8 |
The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers? | 103 | 1/8 |
32 volleyball teams participate in a tournament with the following rules. In each round, all remaining teams are randomly paired; if there is an odd number of teams, one team skips the round. In each pair, one team wins and the other loses; there are no ties in volleyball. After three losses, a team is eliminated from the tournament. The tournament ends when all teams except one have been eliminated, and this remaining team is declared the winner.
What is the minimum number of rounds the tournament can last? | 9 | 1/8 |
In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$. | \frac{1}{2} | 4/8 |
The factorial of a number \( n \) is defined as the product of all integers from 1 to \( n \) inclusive. Find all three-digit numbers that are equal to the sum of the factorials of their digits. | 145 | 2/8 |
Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops? | \frac{837}{208} | 1/8 |
Over all real numbers \( x \) and \( y \) such that
\[ x^3 = 3x + y \quad \text{and} \quad y^3 = 3y + x \]
compute the sum of all possible values of \( x^2 + y^2 \). | 15 | 7/8 |
The degree measure of angle $A$ is | 30 | 1/8 |
Find the particular solution of the equation
$$
\left(1+e^{x}\right) y y^{\prime}=e^{x}
$$
satisfying the initial condition $\left.y\right|_{x=0}=1$. | \sqrt{1+2\ln(\frac{1+e^x}{2})} | 7/8 |
Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. | 21 | 4/8 |
Four pirates divided a loot of 100 coins. It is known that among them, there are exactly two liars (who always lie) and exactly two knights (who always tell the truth).
They said:
First pirate: "We divided the coins equally."
Second pirate: "Everyone has a different amount of coins, but everyone got at least 15 coins."
Third pirate: "Everyone's amount of coins is divisible by 5."
Fourth pirate: "Everyone has a different amount of coins, but everyone got at most 35 coins."
What is the maximum number of coins that one pirate could receive? | 40 | 2/8 |
The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves
the room. What is the average age of the four remaining people?
$\mathrm{(A)}\ 25 \qquad\mathrm{(B)}\ 26 \qquad\mathrm{(C)}\ 29 \qquad\mathrm{(D)}\ 33 \qquad\mathrm{(E)}\ 36$ | \textbf{(D)}\33 | 1/8 |
Given the real numbers \( x_1, x_2, \ldots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left|x_k - x_{k+1}\right| = 2001 \). Let \( y_k = \frac{1}{k} \left( x_1 + x_2 + \cdots + x_k \right) \) for \( k = 1, 2, \ldots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_k - y_{k+1} \right| \). | 2000 | 1/8 |
Given the points \( A(1,1,1) \) and \( P(1,1,0) \) in a 3D coordinate system. Rotate point \( P \) around the ray \( OA \) by \( 60^\circ \) in the positive direction. Determine the coordinates of the rotated point. | (\frac{1}{3},\frac{4}{3},\frac{1}{3}) | 6/8 |
The secret object is a rectangle of $200 \times 300$ meters. Outside the object, each of the four corners has a guard. An intruder approaches the perimeter of the secret object from the outside, and all the guards run towards the intruder by the shortest paths along the outer perimeter (the intruder remains in place). The sum of the distances run by three guards to reach the intruder is 850 meters. How many meters did the fourth guard run to reach the intruder? | 150 | 4/8 |
Given the function $f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+\pi)=f(x)+\sin x$, and $f(x)=0$ when $0 \leqslant x < \pi$, determine the value of $f(\frac{23\pi}{6})$. | \frac{1}{2} | 7/8 |
A class of 30 students wrote a history test. Of these students, 25 achieved an average of 75%. The other 5 students achieved an average of 40%. The class average on the history test was closest to:
(A) 46
(B) 69
(C) 63
(D) 58
(E) 71 | 69 | 1/8 |
In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\). | 10 | 1/8 |
Natural numbers \( x, y, z \) are such that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \cdot \operatorname{LCM}(\operatorname{GCD}(x, y), z) = 1400 \).
What is the maximum value that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \) can take? | 10 | 1/8 |
The figure shows a square of side $y$ units divided into a square of side $x$ units and four congruent rectangles. What is the perimeter, in units, of one of the four congruent rectangles? Express your answer in terms of $y$. [asy]
size(4cm);
defaultpen(linewidth(1pt)+fontsize(12pt));
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((1,0)--(1,3));
draw((0,3)--(3,3));
draw((3,4)--(3,1));
draw((1,1)--(4,1));
label("$x$",(1,2),E);
label("$y$",(2,4),N);
pair a,b;
a = (0,4.31);
b = a + (4,0);
draw(a--a+(1.8,0));
draw(a+(2.2,0)--b);
draw(a+(0,.09)--a-(0,.09));
draw(b+(0,.09)--b-(0,.09));
[/asy] | 2y | 5/8 |
The values of $a$ in the equation: $\log_{10}(a^2 - 15a) = 2$ are:
$\textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, - 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$
$\textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$ | \textbf{(B)}\20,-5 | 1/8 |
Let $ABCDEFGHIJ$ be a regular $10$ -sided polygon that has all its vertices in one circle with center $O$ and radius $5$ . The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$ . Calculate the measure of the segment $PQ$ . | PQ=5 | 3/8 |
Gavrila found that the front tires of the car last for 21,000 km, and the rear tires last for 28,000 km. Therefore, he decided to swap them at some point so that the car would travel the maximum possible distance. Find this maximum distance (in km). | 24000 | 7/8 |
Calculate:
\[ 1 \times 15 + 2 \times 14 + 3 \times 13 + 4 \times 12 + 5 \times 11 + 6 \times 10 + 7 \times 9 + 8 \times 8 \] | 372 | 5/8 |
On an exam there are 5 questions, each with 4 possible answers. 2000 students went on the exam and each of them chose one answer to each of the questions. Find the least possible value of $n$ , for which it is possible for the answers that the students gave to have the following property: From every $n$ students there are 4, among each, every 2 of them have no more than 3 identical answers. | 25 | 1/8 |
Prove that for each prime $p>2$ there exists exactly one positive integer $n$ , such that $n^2+np$ is a perfect square. | (\frac{p-1}{2})^2 | 4/8 |
The function $f$ is not defined for $x = 0,$ but for all non-zero real numbers $x,$
\[f(x) + 2f \left( \frac{1}{x} \right) = 3x.\]Find the real solutions to $f(x) = f(-x).$ Enter the real solutions, separated by commas. | \sqrt{2},-\sqrt{2} | 3/8 |
According to statistical data, the daily output of a factory does not exceed 200,000 pieces, and the daily defect rate $p$ is approximately related to the daily output $x$ (in 10,000 pieces) by the following relationship:
$$
p= \begin{cases}
\frac{x^{2}+60}{540} & (0<x\leq 12) \\
\frac{1}{2} & (12<x\leq 20)
\end{cases}
$$
It is known that for each non-defective product produced, a profit of 2 yuan can be made, while producing a defective product results in a loss of 1 yuan. (The factory's daily profit $y$ = daily profit from non-defective products - daily loss from defective products).
(1) Express the daily profit $y$ (in 10,000 yuan) as a function of the daily output $x$ (in 10,000 pieces);
(2) At what daily output (in 10,000 pieces) is the daily profit maximized? What is the maximum daily profit in yuan? | \frac{100}{9} | 1/8 |
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$. | \frac{1}{6} | 7/8 |
In a game, \(N\) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \(m\) be the maximum possible number such that it is possible to win the game by writing down \(m\). Find the smallest possible value of \(N\) for which it is possible to win the game by writing down \(m\) in a room of \(N\) people. | 34 | 7/8 |
The perpendicular bisectors of the sides of triangle $ABC$ meet its circumcircle at points $A',$ $B',$ and $C',$ as shown. If the perimeter of triangle $ABC$ is 35 and the radius of the circumcircle is 8, then find the area of hexagon $AB'CA'BC'.$
[asy]
unitsize(2 cm);
pair A, B, C, Ap, Bp, Cp, O;
O = (0,0);
A = dir(210);
B = dir(60);
C = dir(330);
Ap = dir(15);
Bp = dir(270);
Cp = dir(135);
draw(Circle(O,1));
draw(A--B--C--cycle);
draw((B + C)/2--Ap);
draw((A + C)/2--Bp);
draw((A + B)/2--Cp);
label("$A$", A, A);
label("$B$", B, B);
label("$C$", C, C);
label("$A'$", Ap, Ap);
label("$B'$", Bp, Bp);
label("$C'$", Cp, Cp);
[/asy] | 140 | 3/8 |
Given that Lauren has 4 sisters and 7 brothers, and her brother Lucas has S sisters and B brothers. Find the product of S and B. | 35 | 2/8 |
Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table. | 6851 | 2/8 |
On the faces \(BCD, ACD, ABD,\) and \(ABC\) of a tetrahedron \(ABCD\), points \(A_1, B_1, C_1,\) and \(D_1\) are marked, respectively. It is known that the lines \(AA_1, BB_1, CC_1,\) and \(DD_1\) intersect at point \(P\) with \(\frac{AP}{A_1P} = \frac{BP}{B_1P} = \frac{CP}{C_1P} = \frac{DP}{D_1P} = r\). Find all possible values of \(r\). | 3 | 4/8 |
With the numbers \(2\), \(3\), \(5\), \(6\), and \(7\), form one-digit or two-digit numbers and arrange them into an equation such that the result is 2010. \(2010 = \) ______ | 2\times3\times5\times67 | 1/8 |
For each integer $n\geq 3$ , find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime. | \lfloor\frac{n}{2}\rfloor+\lfloor\frac{n}{3}\rfloor-\lfloor\frac{n}{6}\rfloor+1 | 2/8 |
Suppose that $a$ and $b$ are nonzero integers such that two of the roots of
\[x^3 + ax^2 + bx + 9a\]coincide, and all three roots are integers. Find $|ab|.$ | 1344 | 6/8 |
A flea jumps along a number line, and the length of each jump cannot be less than $n$. The flea starts its movement from the origin and wants to visit all integer points in the interval $[0, 2013]$ (and only them!) exactly once. For what largest value of $n$ can the flea achieve this? | 1006 | 1/8 |
In the plane coordinate system \( xOy \), consider the point set \( K = \{(x, y) \mid x, y = -1, 0, 1\} \). If three points are randomly selected from \( K \), find the probability that the distance between each pair of these three points does not exceed 2. | \frac{5}{14} | 1/8 |
Arrange the numbers $1, 2, \cdots, n^{2}$ in a clockwise spiral format to form an $n \times n$ table $T_{n}$. The first row consists of $1, 2, \cdots, n$. For example, $T_{3}=\left[\begin{array}{lll}1 & 2 & 3 \\ 8 & 9 & 4 \\ 7 & 6 & 5\end{array}\right]$. Find the position $(i, j)$ in $T_{100}$ where the number 2018 is located. | (34,95) | 1/8 |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.
| 243 | 1/8 |
Consider a rectangle $ABCD$, and inside it are four squares with non-overlapping interiors. Two squares have the same size and an area of 4 square inches each, located at corners $A$ and $C$ respectively. There is another small square with an area of 1 square inch, and a larger square, twice the side length of the smaller one, both adjacent to each other and located centrally from $B$ to $D$. Calculate the area of rectangle $ABCD$. | 12 | 1/8 |
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$ ? | 10 | 7/8 |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | 16 | 7/8 |
If $9:y^3 = y:81$, what is the value of $y$? | 3\sqrt{3} | 4/8 |
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once? | 1/4 | 6/8 |
A square flag features a symmetric red cross with uniform width and two identical blue squares in the center on a white background. The entire cross, including the red arms and the two blue centers, occupies 40% of the flag's area. Determine what percent of the flag's area is occupied by the two blue squares. | 20\% | 1/8 |
Mr. Morgan G. Bloomgarten wants to distribute 1,000,000 dollars among his friends. He has two specific rules for distributing the money:
1. Each gift must be either 1 dollar or a power of 7 (7, 49, 343, 2401, etc.).
2. No more than six people can receive the same amount.
How can he distribute the 1,000,000 dollars under these conditions? | 1,000,000 | 3/8 |
Let \( S \) be a finite set of points in the plane, with no three points collinear. For each convex polygon \( P \) with vertices belonging to set \( S \), let \( a(P) \) be the number of vertices of \( P \), and let \( b(P) \) be the number of points belonging to set \( S \) but outside polygon \( P \).
Prove that for any real number \( x \) (where \( 0 < x < 1 \)), the following holds:
\[
\sum_{P} x^{a(P)} (1-x)^{b(P)} = 1
\]
where \( P \) ranges over all convex polygons with vertices in \( S \) (including triangles, and considering a line segment, a single point, and the empty set as a convex 2-gon, convex 1-gon, and convex 0-gon respectively). | 1 | 2/8 |
A sequence of positive integers \(a_{n}\) begins with \(a_{1}=a\) and \(a_{2}=b\) for positive integers \(a\) and \(b\). Subsequent terms in the sequence satisfy the following two rules for all positive integers \(n\):
\[a_{2 n+1}=a_{2 n} a_{2 n-1}, \quad a_{2 n+2}=a_{2 n+1}+4 .\]
Exactly \(m\) of the numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\) are square numbers. What is the maximum possible value of \(m\)? Note that \(m\) depends on \(a\) and \(b\), so the maximum is over all possible choices of \(a\) and \(b\). | 1012 | 1/8 |
Given \(a > 0\), let \(f: (0, +\infty) \rightarrow \mathbf{R}\) be a function such that \(f(a) = 1\). If for any positive real numbers \(x\) and \(y\), the following condition holds:
\[ f(x)f(y) + f\left(\frac{a}{x}\right)f\left(\frac{a}{y}\right)=2f(xy), \]
prove that \(f(x)\) is a constant function. | 1 | 4/8 |
In triangle \(ABC\), angle bisectors \(AA_{1}\), \(BB_{1}\), and \(CC_{1}\) are drawn. \(L\) is the intersection point of segments \(B_{1}C_{1}\) and \(AA_{1}\), \(K\) is the intersection point of segments \(B_{1}A_{1}\) and \(CC_{1}\). Find the ratio \(LM: MK\) if \(M\) is the intersection point of angle bisector \(BB_{1}\) with segment \(LK\), and \(AB: BC: AC = 2: 3: 4\). (16 points) | 11/12 | 1/8 |
The equation of the line joining the complex numbers $-1 + 2i$ and $2 + 3i$ can be expressed in the form
\[az + b \overline{z} = d\]for some complex numbers $a$, $b$, and real number $d$. Find the product $ab$. | 10 | 7/8 |
Dave's sister baked $3$ dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths contained cayenne, and one-sixths contained salted soy nuts. What is the largest possible number of pies that had none of these ingredients? | 9 | 7/8 |
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{3}{\pi} | 7/8 |
Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime. | 591\times1990^{1989} | 5/8 |
Ivan Semenovich drives to work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich drove at a speed that was 60% greater than usual and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM? | 30 | 4/8 |
The inclination angle of the line given by the parametric equations \[
\begin{cases}
x=1+t \\
y=1-t
\end{cases}
\]
calculate the inclination angle. | \frac {3\pi }{4} | 4/8 |
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace$. Then the area of $S$ has the form $a\pi + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$. | 29 | 1/8 |
Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a sphere that lies within both cones. | \frac{40}{13} | 2/8 |
For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$.
[i] | \frac{3}{32} | 4/8 |
(1) Among the following 4 propositions:
① The converse of "If $a$, $G$, $b$ form a geometric sequence, then $G^2=ab$";
② The negation of "If $x^2+x-6\geqslant 0$, then $x > 2$";
③ In $\triangle ABC$, the contrapositive of "If $A > B$, then $\sin A > \sin B$";
④ When $0\leqslant \alpha \leqslant \pi$, if $8x^2-(8\sin \alpha)x+\cos 2\alpha\geqslant 0$ holds for $\forall x\in \mathbb{R}$, then the range of $\alpha$ is $0\leqslant \alpha \leqslant \frac{\pi}{6}$.
The numbers of the true propositions are ______.
(2) Given an odd function $f(x)$ whose graph is symmetric about the line $x=3$, and when $x\in [0,3]$, $f(x)=-x$, then $f(-16)=$ ______.
(3) The graph of the function $f(x)=a^{x-1}+4$ ($a > 0$ and $a\neq 1$) passes through a fixed point, then the coordinates of this point are ______.
(4) Given a point $P$ on the parabola $y^2=2x$, the minimum value of the sum of the distance from point $P$ to the point $(0,2)$ and the distance from $P$ to the directrix of the parabola is ______. | \frac{\sqrt{17}}{2} | 1/8 |
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is: | 83.33 | 5/8 |
Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum. | 210 | 5/8 |
A club consists of five leaders and some regular members. Each year, all leaders leave the club and each regular member recruits three new people to join as regular members. Subsequently, five new leaders are elected from outside the club to join. Initially, there are eighteen people in total in the club. How many people will be in the club after five years? | 3164 | 1/8 |
4.3535… is a decimal, which can be abbreviated as , and the repeating cycle is . | 35 | 2/8 |
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ${p}_{a,b}$ be the sum of the zeros of $P(x)$. Find the sum of the ${p}_{a,b}$'s for all possible combinations of $a$ and $b$. | 80 | 3/8 |
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be *critical* if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white. | 5000 | 1/8 |
The lengths of the sides of a triangle with positive area are $\log_{2}9$, $\log_{2}50$, and $\log_{2}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | 445 | 1/8 |
Given \(1 < a_i < \sqrt{7}\) for all \(i = 1, 2, \ldots, n\) where \(n\) is a positive integer and \(n \geq 2\).
1. Prove that for any positive integer \(i\), \(\frac{1}{a_i^2 - 1} + \frac{1}{7 - a_i^2} \geq \frac{2}{3}\).
2. Find the minimum value of \(S = \sum_{i=1}^{n} \frac{1}{\sqrt{(a_i^2 - 1)(7 - a_{i+1}^2)}}\), where it is agreed that \(a_{n+1} = a_1\). | \frac{n}{3} | 7/8 |
Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$ ) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$ . | m | 1/8 |
For $ n \in \mathbb{N}$ , let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$ . Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$ | 8/3 | 3/8 |
The cross-section of a wine glass is a section of a parabola given by the equation \( x^2 = 2y \) for \( 0 \leqslant y < 15 \). If a glass sphere with a radius of 3 is placed inside the glass, what is the distance from the highest point of the sphere to the bottom of the glass? | 8 | 6/8 |
Prove the following identity:
$$
\operatorname{tg} x+2 \operatorname{tg} 2 x+4 \operatorname{tg} 4 x+8 \operatorname{ctg} 8 x=\operatorname{ctg} x
$$ | 0 | 1/8 |
There are 10 digit cards from 0 to 9 on the table. Three people, A, B, and C, each take three cards, and calculate the sum of all possible different three-digit numbers that can be formed with their three cards. The results for A, B, and C are $1554, 1688, 4662$. What is the remaining card on the table? (Note: 6 and 9 cannot be flipped to look like 9 or 6.) | 9 | 7/8 |
Let \( a_{1}, a_{2}, \cdots, a_{20} \in \{1, 2, \cdots, 5\} \) and \( b_{1}, b_{2}, \cdots, b_{20} \in \{1, 2, \cdots, 10\} \). Define the set \( X = \{ (i, j) \mid 1 \leq i < j \leq 20, (a_{i} - a_{j})(b_{i} - b_{j}) < 0 \} \). Find the maximum number of elements in \( X \). | 160 | 4/8 |
If $a$ and $b$ are randomly selected real numbers between 0 and 1, find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd. | \frac{1}{3} | 7/8 |
In the nine cells of a $3 \times 3$ square, the numbers from 1 to 9 are placed. Arseni calculated the sum of the numbers on one diagonal and obtained 6. Alice calculated the sum of the numbers on the other diagonal and obtained 20. What number is in the center of the square? | 3 | 6/8 |
The symbol $\odot$ represents a special operation with numbers; some examples are $2 \odot 4 = 10$, $3 \odot 8 = 27$, $4 \odot 27 = 112$, and $5 \odot 1 = 10$. What is the value of $4 \odot (8 \odot 7)$?
(a) 19
(b) 39
(c) 120
(d) 240
(e) 260 | 260 | 7/8 |
On the surface of a spherical planet, there are four continents separated by an ocean. We will call a point in the ocean special if there are at least three closest land points (equidistant from it), each located on different continents. What is the maximum number of special points that can be on this planet? | 4 | 3/8 |
Factorize \( n^{5} - 5n^{3} + 4n \). What can be concluded in terms of divisibility? | 120 | 7/8 |
The median $A D$ and the angle bisector $C E$ of the right triangle $A B C \left(\angle B=90^{\circ}\right)$ intersect at point $M$. Find the area of triangle $A B C$ if $C M =8$, and $M E =5$. | \frac{1352}{15} | 1/8 |
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