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In the tetrahedron \(ABCD\), \(AB \perp BC\), \(CD \perp BC\), \(BC = 2\), and the angle between the skew lines \(AB\) and \(CD\) is \(60^\circ\). If the circumradius of the tetrahedron \(ABCD\) is \(\sqrt{5}\), what is the maximum possible volume of the tetrahedron \(ABCD\)? | 2\sqrt{3} | 1/8 |
Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$ , such that $AB||CD$ . Let $P$ be a point on the circle $\omega$ . Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle. | H_1H_2 | 2/8 |
Given positive numbers \(x, y, z\) such that \(x^2 + y^2 + z^2 = 1\), find the minimum value of the following expression:
\[ S = \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y}. \] | \sqrt{3} | 5/8 |
In a regular triangular pyramid with a base side length of 4 and height of 2, first place a sphere \( B_1 \) that is tangent to the base and all three lateral faces. Then, for \( n = 2, 3, \cdots \), place sphere \( B_n \) in the pyramid such that it is externally tangent to sphere \( B_{n-1} \) and tangent to all three lateral faces of the pyramid. Find the total volume of all these spheres. | \frac{16\pi}{39} | 1/8 |
Given a binary string \( S \) of length \( 10^4 \) containing only '0' and '1'. A \( k \)-block of \( S \) is defined as a contiguous substring of \( S \) of length \( k \). Two \( k \)-blocks \( a_1a_2 \cdots a_k \) and \( b_1b_2 \cdots b_k \) are considered identical if and only if \( a_i = b_i \) for \( i = 1, 2, \cdots, k \).
Consider all binary strings of length \( 10^4 \) that contain at most seven different 3-blocks. Determine the maximum number of distinct 10-blocks that such a string can contain. | 504 | 1/8 |
Compute the value of $k$ such that the equation
\[\frac{x + 2}{kx - 1} = x\]has exactly one solution. | 0 | 6/8 |
How many natural numbers from 1 to 700, inclusive, contain the digit 7 at least once? | 133 | 1/8 |
Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
*Proposed by Michael Tang* | 20 | 2/8 |
There are 9 digits: 0, 1, 2, …, 8. Using five cards, with the two sides respectively marked as 0/8, 1/7, 2/5, 3/4, 6/9; and 6 can be used as 9. How many different four-digit numbers can be formed with these five cards? | 1728 | 4/8 |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | 7/8 |
Find the maximum real number \( k \) such that for any positive real numbers \( a, b, c \), the following inequality holds:
$$
\frac{(b-c)^{2}(b+c)}{a}+\frac{(c-a)^{2}(c+a)}{b}+\frac{(a-b)^{2}(a+b)}{c}
\geqslant k\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)
$$ | 2 | 1/8 |
From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group with this property? | 40 | 5/8 |
The rational numbers \( x \) and \( y \), when written in lowest terms, have denominators 60 and 70, respectively. What is the smallest possible denominator of \( x + y \)? | 84 | 7/8 |
Given that \(a_{1}, a_{2}, \cdots, a_{n}\) are \(n\) people corresponding to \(A_{1}, A_{2}, \cdots, A_{n}\) cards (\(n \geq 2\), \(a_{i}\) corresponds to \(A_{i}\)). Now \(a_{1}\) picks a card from the deck randomly, and then each person in sequence picks a card. If their corresponding card is still in the deck, they take it, otherwise they pick a card randomly. What is the probability that \(a_{n}\) gets \(A_{n}\)? | \frac{1}{2} | 3/8 |
Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$ , then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is at most $C$ .
*Merlijn Staps* | \frac{50}{51} | 1/8 |
The teacher plans to give children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately he will provide the value of the expression \( P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right) \). Based on this information, the children must find the polynomial that the teacher might have in mind. What is the smallest possible \( k \) for which the teacher can compose a problem of this type such that the polynomial found by the children will necessarily match the intended one? | 2017 | 1/8 |
Consider the points $A(0,12), B(10,9), C(8,0),$ and $D(-4,7).$ There is a unique square $S$ such that each of the four points is on a different side of $S.$ Let $K$ be the area of $S.$ Find the remainder when $10K$ is divided by $1000$. | 936 | 1/8 |
Find all real values of $a$ for which the polynomial
\[x^4 + ax^3 - x^2 + ax + 1 = 0\]has at least one real root. | (-\infty,-\frac{1}{2}]\cup[\frac{1}{2},\infty) | 5/8 |
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$ ? | 7 | 6/8 |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots? | 100 | 7/8 |
Given that the asymptotic line of the hyperbola $\frac{x^2}{a}+y^2=1$ has a slope of $\frac{5π}{6}$, determine the value of $a$. | -3 | 3/8 |
A sphere with radius 1 is drawn through vertex \( D \) of a tetrahedron \( ABCD \). This sphere is tangent to the circumscribed sphere of the tetrahedron \( ABCD \) at point \( D \) and is also tangent to the plane \( ABC \). Given that \( AD = 2\sqrt{3} \), \( \angle BAC = 60^\circ \), and \( \angle BAD = \angle CAD = 45^\circ \), find the radius of the circumscribed sphere of tetrahedron \( ABCD \). | 3 | 3/8 |
Sunshine High School is planning to order a batch of basketballs and jump ropes from an online store. After checking on Tmall, they found that each basketball is priced at $120, and each jump rope is priced at $25. There are two online stores, Store A and Store B, both offering free shipping and their own discount schemes:<br/>Store A: Buy one basketball and get one jump rope for free;<br/>Store B: Pay 90% of the original price for both the basketball and jump rope.<br/>It is known that they want to buy 40 basketballs and $x$ jump ropes $\left(x \gt 40\right)$.<br/>$(1)$ If they purchase from Store A, the payment will be ______ yuan; if they purchase from Store B, the payment will be ______ yuan; (express in algebraic expressions with $x$)<br/>$(2)$ If $x=80$, through calculation, determine which store is more cost-effective to purchase from at this point.<br/>$(3)$ If $x=80$, can you provide a more cost-effective purchasing plan? Write down your purchasing method and calculate the amount to be paid. | 5700 | 5/8 |
A trapezoid with integer sides has an inscribed circle. The median line of the trapezoid divides it into two parts, the areas of which are 15 and 30. Find the radius of the inscribed circle. | \frac{5}{2} | 2/8 |
How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?
A sequence of three numbers \( a, b, c \) is said to form an arithmetic progression if \( a + c = 2b \).
A correct numerical answer without justification will earn 4 points. For full points, a detailed reasoning is expected. | 45 | 6/8 |
Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits. | 209 | 3/8 |
Let \( x, y, z \) be real numbers such that
\[ x + y + z = 1 \quad \text{and} \quad x^2 + y^2 + z^2 = 1. \]
Let \( m \) denote the minimum value of \( x^3 + y^3 + z^3 \). Find \( 9m \). | 5 | 6/8 |
A car must pass through 4 intersections during its journey. At each intersection, the probability of encountering a green light (allowing passage) is $\frac{3}{4}$, and the probability of encountering a red light (stopping) is $\frac{1}{4}$. Assuming that the car only stops when it encounters a red light or reaches its destination, and letting $\xi$ represent the number of intersections passed when the car stops, find: (I) the probability distribution and expectation $E(\xi)$ of $\xi$; (II) the probability that at most 3 intersections have been passed when the car stops. | \frac{175}{256} | 7/8 |
Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$ . | 505 | 2/8 |
Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.
$(1)$ Find the standard equation of parabola $C$.
$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$. | \frac{3\sqrt{2}}{2} | 7/8 |
Consider a square on the complex plane whose four vertices correspond exactly to the four roots of a monic quartic polynomial with integer coefficients \( x^{4} + p x^{3} + q x^{2} + r x + s = 0 \). Find the minimum possible area of such a square. | 2 | 2/8 |
The sides of triangle $CAB$ are in the ratio of $2:3:4$. Segment $BD$ is the angle bisector drawn to the shortest side, dividing it into segments $AD$ and $DC$. What is the length, in inches, of the longer subsegment of side $AC$ if the length of side $AC$ is $10$ inches? Express your answer as a common fraction. | \frac {40}7 | 7/8 |
A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$. | 36 | 1/8 |
Arrange numbers $ 1,\ 2,\ 3,\ 4,\ 5$ in a line. Any arrangements are equiprobable. Find the probability such that the sum of the numbers for the first, second and third equal to the sum of that of the third, fourth and fifth. Note that in each arrangement each number are used one time without overlapping. | 1/15 | 1/8 |
In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ to the area of triangle $ABC$ can be written in the form $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$. | 295 | 6/8 |
There are 50 musketeers serving at the court. Every day, they split into pairs and conduct training duels. Is it true that after 24 days, there will be three musketeers who have not participated in training duels with each other? | 3 | 1/8 |
In the expression \((x+y+z)^{2034}+(x-y-z)^{2034}\), the brackets were expanded, and like terms were combined. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient? | 1036324 | 3/8 |
At Easter-Egg Academy, each student has two eyes, each of which can be eggshell, cream, or cornsilk. It is known that 30% of the students have at least one eggshell eye, 40% of the students have at least one cream eye, and 50% of the students have at least one cornsilk eye. What percentage of the students at Easter-Egg Academy have two eyes of the same color? | 80 | 5/8 |
Points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \) divide the circumference of a circle with unit radius into five equal parts; prove that the following equality holds between the chords \( A_{1} A_{2} \) and \( A_{1} A_{3} \):
\[
\left( A_{1} A_{2} \cdot A_{1} A_{3} \right)^{2}=5
\] | 5 | 6/8 |
Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers? | 32 | 4/8 |
In a convex hexagon, two diagonals are chosen at random independently of each other. Find the probability that these diagonals intersect inside the hexagon (inside means not at a vertex). | \frac{5}{12} | 6/8 |
What is the product of the solutions of the equation $45 = -x^2 - 4x?$ | -45 | 1/8 |
Find the smallest positive integer $M$ such that the three numbers $M$, $M+1$, and $M+2$, one of them is divisible by $3^2$, one of them is divisible by $5^2$, and one is divisible by $7^2$. | 98 | 4/8 |
One of the factors of $x^4+4$ is:
$\textbf{(A)}\ x^2+2 \qquad \textbf{(B)}\ x+1 \qquad \textbf{(C)}\ x^2-2x+2 \qquad \textbf{(D)}\ x^2-4\\ \textbf{(E)}\ \text{none of these}$ | \textbf{(C)}\x^2-2x+2 | 1/8 |
Given that the arithmetic square root of $m$ is $3$, and the square roots of $n$ are $a+4$ and $2a-16$.
$(1)$ Find the values of $m$ and $n$.
$(2)$ Find $\sqrt[3]{{7m-n}}$. | -1 | 6/8 |
Given a multi-digit number 201312210840, which has a total of 12 digits. By removing 8 of these digits, a four-digit number can be formed. What is the difference between the maximum value and the minimum value of this four-digit number? | 2800 | 2/8 |
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \geqslant \frac{b+c}{3}$, then the following inequality holds:
$$
ac + bc - c^2 \leqslant \lambda \left( a^2 + b^2 + 3c^2 + 2ab - 4bc \right).
$$ | \frac{2\sqrt{2} + 1}{7} | 1/8 |
Nine hundred forty-three minus eighty-seven equals | 856 | 7/8 |
Linda is tasked with writing a report on extracurricular activities at her school. The school offers two clubs: Robotics and Science. Linda has a list of 30 students who are members of at least one club. She knows that 22 students are in the Robotics club and 24 students are in the Science club. If Linda picks two students randomly from her list to interview, what is the probability that she will be able to gather information about both clubs? Express your answer as a fraction in simplest form. | \frac{392}{435} | 7/8 |
Detached calculation.
327 + 46 - 135
1000 - 582 - 128
(124 - 62) × 6
500 - 400 ÷ 5 | 420 | 7/8 |
Circles \(K_{1}\) and \(K_{2}\) share a common point \(A\). Through point \(A\), three lines are drawn: two pass through the centers of the circles and intersect them at points \(B\) and \(C\), and the third is parallel to \(BC\) and intersects the circles at points \(M\) and \(N\). Find the length of segment \(MN\) if the length of segment \(BC\) is \(a\). | a | 2/8 |
What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits? | 1/203 | 3/8 |
Find the area of the region \(D\) bounded by the curves
\[ x^{2} + y^{2} = 12, \quad x \sqrt{6} = y^{2} \quad (x \geq 0) \] | 3\pi + 2 | 1/8 |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333 | 1/8 |
Let \(a\) and \(b\) be positive integers, and let \(c\) be a positive real number, for which
$$
\frac{a+1}{b+c} = \frac{b}{a}
$$
is satisfied.
Prove that \(c \geq 1\). | \ge1 | 5/8 |
Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation
$$
x \sqrt{y} + y \sqrt{x} + \sqrt{2009 x y} - \sqrt{2009 x} - \sqrt{2009 y} - 2009 = 0.
$$ | 6 | 3/8 |
There are 35 students in a class. Their total age is 280 years. Will there be 25 students in this class whose total age is at least 225 years? | No | 2/8 |
Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$ , with $x_1+x_2+\ldots +x_n=n$ , for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value. | n^2 | 2/8 |
Find the number of $4$ -digit numbers (in base $10$ ) having non-zero digits and which are divisible by $4$ but not by $8$ . | 729 | 4/8 |
The polynomial \( x^3 - 3x^2 + 1 \) has three real roots \( r_1, r_2, \) and \( r_3 \). Compute:
\[
\sqrt[3]{3r_1 - 2} + \sqrt[3]{3r_2 - 2} + \sqrt[3]{3r_3 - 2}.
\] | 0 | 1/8 |
Taking a box of matches, I found that I could use them to form any pair of the regular polygons shown in the illustration, using all the matches each time. For example, if I had 11 matches, I could form either a triangle and a pentagon, a pentagon and a hexagon, or a square and a triangle (using only 3 matches for the triangle). However, 11 matches cannot form a triangle with a hexagon, a square with a pentagon, or a square with a hexagon. Of course, each side of the shape must use an equal number of matches.
What is the smallest number of matches that could be in my box? | 36 | 1/8 |
Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$.
(1) Determine the range of the real number $a$ and the expression for $S_n$;
(2) Does there exist a real number $a$ such that $S_1$, $S_3$, $S_2$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain why. | \frac{1}{2} | 6/8 |
If the maximum and minimum values of the exponential function $f(x) = a^x$ on the interval $[1, 2]$ differ by $\frac{a}{2}$, then find the value of $a$. | \frac{3}{2} | 7/8 |
The side length of the base of a regular quadrilateral pyramid is \( a \), and the slant edge is \( b \). Find the radius of the sphere that touches all the edges of the pyramid. | \frac{(2b-)}{2\sqrt{2b^2-^2}} | 1/8 |
A reconnaissance plane flies in a circle with center at point $A$. The radius of the circle is $10 \mathrm{kм}$, and the plane's speed is $1000 \mathrm{км/ч}$. At a certain moment, a missile is launched from point $A$, which has the same speed as the plane and is guided such that it always lies on the line connecting the plane with point A. How long after launch will the missile catch up with the plane? | \frac{\pi}{200}\, | 1/8 |
A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuckoo strike during this period? | 43 | 5/8 |
Given a point P(3, 2) outside the circle $x^2+y^2-2x-2y+1=0$, find the cosine of the angle between the two tangents drawn from this point to the circle. | \frac{3}{5} | 5/8 |
When numbers are represented in "base fourteen", the digit before it becomes full is fourteen. If in "base fourteen", the fourteen digits are sequentially noted as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ten, J, Q, K, then convert the three-digit number JQK in "base fourteen" into a "binary" number and determine the number of digits. | 11 | 1/8 |
For which pairs of positive integers \( a \) and \( b \) is the expression \( a^2 + b^2 - a - b + 1 \) divisible by \( ab \)? | 1,1 | 1/8 |
A boy and a girl. Assume that the birth of a girl and a boy are equally likely. It is known that a certain family has two children.
a) What is the probability that they have one boy and one girl?
b) Additionally, it is known that one of the children is a boy. What is the probability now that the family has one boy and one girl?
c) Additionally, it is known that the boy was born on a Monday. What is the probability now that the family has one boy and one girl? | \frac{14}{27} | 6/8 |
In a heavy rocketry club, there are 40 students. Each of them has bolts, screws, and nails. It is known that there are exactly 15 students whose number of nails is not equal to the number of bolts. The number of students whose number of screws is equal to the number of nails is 10. Prove that there are at least 15 students whose number of screws is not equal to the number of bolts. | 15 | 7/8 |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | 20 | 7/8 |
A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven? | 96,\!569 | 6/8 |
Let $L$ be the intersection point of the lines $AP$ and $CM$, and $S$ be the intersection point of the lines $AN$ and $CQ$. Prove that $LS \| PQ$. | LS\parallelPQ | 1/8 |
Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). | \frac{3}{4} | 7/8 |
The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term?
$\textbf{(A) }1\qquad\textbf{(B) }\sqrt[7]3\qquad\textbf{(C) }\sqrt[8]3\qquad\textbf{(D) }\sqrt[9]3\qquad\textbf{(E) }\sqrt[10]3\qquad$ | \textbf{(A)}1 | 1/8 |
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
| 25 | 2/8 |
In how many ways can a million be factored into three factors? Factorizations that differ only in the order of the factors are considered identical. | 139 | 3/8 |
A two-digit number \( A \) is called a supernumber if it is possible to find two two-digit numbers \( B \) and \( C \) such that:
- \( A = B + C \);
- sum of the digits of \( A \) = (sum of the digits of \( B \)) + (sum of the digits of \( C \)).
For example, 35 is a supernumber. Two different ways to show this are \( 35 = 11 + 24 \) and \( 35 = 21 + 14 \), because \( 3 + 5 = (1 + 1) + (2 + 4) \) and \( 3 + 5 = (2 + 1) + (1 + 4) \). The only way to show that 21 is a supernumber is \( 21 = 10 + 11 \).
a) Show in two different ways that 22 is a supernumber and in three different ways that 25 is a supernumber.
b) How many different ways is it possible to show that 49 is a supernumber?
c) How many supernumbers exist? | 80 | 2/8 |
The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \in (0, +\infty)$. The value of the real number $m$ is | -1 | 1/8 |
Prove that the sum of the squares of the lengths of the projections of the edges of a regular tetrahedron onto any plane is equal to $4 a^2$, where $a$ is the edge length of the tetrahedron. | 4a^2 | 4/8 |
The area of an isosceles trapezoid is 32. The cotangent of the angle between the diagonal and the base is 2. Find the height of the trapezoid. | 4 | 5/8 |
For a positive integer $n$, let $1 \times 2 \times \cdots \times n = n!$. If $\frac{2017!}{2^{n}}$ is an integer, find the maximum value of $n$. | 2010 | 6/8 |
Anya calls a date beautiful if all 6 digits in its recording are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not.
a) How many beautiful dates will there be in April 2023?
b) How many beautiful dates will there be in the entire year of 2023? | 30 | 1/8 |
A workshop has 11 workers, of which 5 are fitters, 4 are turners, and the remaining 2 master workers can act as both fitters and turners. If we need to select 4 fitters and 4 turners to repair a lathe from these 11 workers, there are __ different methods for selection. | 185 | 2/8 |
Find all integers \( n > 3 \) for which there exist \( n \) points \( A_1, \ldots, A_n \) in the plane and real numbers \( r_1, \ldots, r_n \) such that:
1. Any three points are never collinear.
2. For any \( i, j, k \), the area of the triangle \( A_i A_j A_k \) is equal to \( r_i + r_j + r_k \). | 4 | 1/8 |
The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets. | 1250 | 1/8 |
A deck of playing cards without jokers has 4 suits with a total of 52 cards, each suit has 13 cards numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of these 4 drawn cards is exactly 34, how many of them are 2s? | 4 | 2/8 |
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$ respectively, draw a line $l$ through the right focus that intersects the ellipse at points $P$ and $Q$. Find the maximum area of the inscribed circle of triangle $F_{1} PQ$. | \frac{9 \pi}{16} | 1/8 |
On the coordinate plane, an isosceles right triangle with vertices at points with integer coordinates is depicted. It is known that there are exactly 2019 points with integer coordinates on the sides of the triangle (including the vertices). What is the smallest possible length of the hypotenuse of the triangle under these conditions? In the answer, indicate the length of the hypotenuse rounded to the nearest whole number. | 952 | 2/8 |
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? | 22.5 | 7/8 |
Given an arithmetic sequence $\{a_n\}$, where $a_n \in \mathbb{N}^*$, and $S_n = \frac{1}{8}(a_n+2)^2$. If $b_n = \frac{1}{2}a_n - 30$, find the minimum value of the sum of the first \_\_\_\_\_\_ terms of the sequence $\{b_n\}$. | 15 | 5/8 |
Consider the equation $F O R T Y+T E N+T E N=S I X T Y$, where each of the ten letters represents a distinct digit from 0 to 9. Find all possible values of $S I X T Y$. | 31486 | 1/8 |
The cross-section of a regular hexagonal pyramid SABCDEF is formed by a plane passing through the center of the base ABCDEF and parallel to the median CM of the lateral face SCD and the apothem SN of the lateral face SAF. The side of the base of the pyramid is 8, and the distance from the vertex S to the cutting plane is \(3 \sqrt{\frac{13}{7}}\). Find the cosine of the angle between the cutting plane and the base plane. (20 points) | \frac{\sqrt{3}}{4} | 7/8 |
Olya drew $N$ different lines on a plane, any two of which intersect. It turned out that among any 15 lines, there are always two that form an angle of $60^{\circ}$ between them. What is the largest possible value of $N$ for which this is possible? | 42 | 1/8 |
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need? | n | 3/8 |
From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected. | 14 | 1/8 |
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.
In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24$ | \textbf{(A)}\6 | 1/8 |
Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many interesting ordered quadruples are there?
| 80 | 3/8 |
Find all primes $p$ such that $p^2-p+1$ is a perfect cube. | 19 | 2/8 |
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