problem stringlengths 10 5.15k | answer stringlengths 0 1.22k |
|---|---|
In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances ... | 160 |
Suppose \( a \) is an integer. A sequence \( x_1, x_2, x_3, x_4, \ldots \) is constructed with:
- \( x_1 = a \),
- \( x_{2k} = 2x_{2k-1} \) for every integer \( k \geq 1 \),
- \( x_{2k+1} = x_{2k} - 1 \) for every integer \( k \geq 1 \).
For example, if \( a = 2 \), then:
\[ x_1 = 2, \quad x_2 = 2x_1 = 4, \quad x_3 =... | 1409 |
The numbers \(a, b, c, d\) belong to the interval \([-6.5 ; 6.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 182 |
Workshop A and Workshop B together have 360 workers. The number of workers in Workshop A is three times that of Workshop B. How many workers are there in each workshop? | 270 |
Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points... | 36 |
Given that the quadratic equation \( (5a + 2b)x^2 + ax + b = 0 \) has a unique solution for \( x \), find the value of \( x \). | \frac{5}{2} |
Using three rectangular pieces of paper (A, C, D) and one square piece of paper (B), an area of 480 square centimeters can be assembled into a large rectangle. It is known that the areas of B, C, and D are all 3 times the area of A. Find the total perimeter of the four pieces of paper A, B, C, and D in centimeters. | 184 |
Let \(ABC\) be a triangle with circumradius \(R = 17\) and inradius \(r = 7\). Find the maximum possible value of \(\sin \frac{A}{2}\). | \frac{17 + \sqrt{51}}{34} |
Given that the function \( f(x) \) satisfies the equation \( 2 f(x) + x^{2} f\left(\frac{1}{x}\right) = \frac{3 x^{3} - x^{2} + 4 x + 3}{x + 1} \) and \( g(x) = \frac{5}{x + 1} \), determine the minimum value of \( f(x) + g(x) \). | \frac{15}{4} |
A person bequeathed an amount of money, slightly less than 1500 dollars, to be distributed as follows. His five children and the notary received amounts such that the square root of the eldest son's share, half of the second son's share, the third son's share minus 2 dollars, the fourth son's share plus 2 dollars, the ... | 1464 |
On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90? | 15180 |
Four princesses thought of two-digit numbers, and Ivan thought of a four-digit number. After they wrote their numbers in a row in some order, the result was 132040530321. Find Ivan's number. | 5303 |
In a regular 2017-gon, all diagonals are drawn. Petya randomly selects some number $\mathrm{N}$ of diagonals. What is the smallest $N$ such that among the selected diagonals there are guaranteed to be two diagonals of the same length? | 1008 |
Snow White has a row of 101 plaster dwarfs in her garden, arranged by weight from heaviest to lightest, with the weight difference between each pair of adjacent dwarfs being the same. Once, Snow White weighed the dwarfs and discovered that the first, heaviest dwarf weighs exactly $5 \mathrm{~kg}$. Snow White was most s... | 2.5 |
Farmer Yang has a \(2015 \times 2015\) square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased? | 2014 |
What is the maximum number of numbers that can be selected from the set \( 1, 2, \ldots, 1963 \) such that the sum of no two numbers is divisible by their difference? | 655 |
A regular hexagon \( K L M N O P \) is inscribed in an equilateral triangle \( A B C \) such that the points \( K, M, O \) lie at the midpoints of the sides \( A B, B C, \) and \( A C \), respectively. Calculate the area of the hexagon \( K L M N O P \) given that the area of triangle \( A B C \) is \( 60 \text{ cm}^2 ... | 30 |
Let \(ABCD\) be a quadrilateral inscribed in a circle with center \(O\). Let \(P\) denote the intersection of \(AC\) and \(BD\). Let \(M\) and \(N\) denote the midpoints of \(AD\) and \(BC\). If \(AP=1\), \(BP=3\), \(DP=\sqrt{3}\), and \(AC\) is perpendicular to \(BD\), find the area of triangle \(MON\). | 3/4 |
For any real number \( x \), let \([x]\) denote the greatest integer less than or equal to \( x \). When \( 0 \leqslant x \leqslant 100 \), how many different integers are in the range of the function \( f(x) = [2x] + [3x] + [4x] + [5x] \)? | 101 |
A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the separated parts was then alloyed with the rem... | 2.4 |
At a bus stop near Absent-Minded Scientist's house, two bus routes stop: #152 and #251. Both go to the subway station. The interval between bus #152 is exactly 5 minutes, and the interval between bus #251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other... | 5/14 |
From point \( A \), two rays are drawn intersecting a given circle: one at points \( B \) and \( C \), and the other at points \( D \) and \( E \). It is known that \( AB = 7 \), \( BC = 7 \), and \( AD = 10 \). Determine \( DE \). | 0.2 |
In the Cartesian coordinate system, given the set of points $I=\{(x, y) \mid x$ and $y$ are integers, and $0 \leq x \leq 5,0 \leq y \leq 5\}$, find the number of distinct squares that can be formed with vertices from the set $I$. | 105 |
Since December 2022, various regions in the country have been issuing multiple rounds of consumption vouchers in different forms to boost consumption recovery. Let the amount of issued consumption vouchers be denoted as $x$ (in hundreds of million yuan) and the consumption driven be denoted as $y$ (in hundreds of milli... | 35.25 |
An electronic clock always displays the date as an eight-digit number. For example, January 1, 2011, is displayed as 20110101. What is the last day of 2011 that can be evenly divided by 101? The date is displayed as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$? | 1221 |
A company has calculated that investing x million yuan in project A will yield an economic benefit y that satisfies the relationship: $y=f(x)=-\frac{1}{4}x^{2}+2x+12$. Similarly, the economic benefit y from investing in project B satisfies the relationship: $y=h(x)=-\frac{1}{3}x^{2}+4x+1$.
(1) If the company has 10 mil... | 6.5 |
Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers? | 85 |
How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. | 248 |
Let $ABCDEFGH$ be a regular octagon, and let $I, J, K$ be the midpoints of sides $AB, DE, GH$ respectively. If the area of $\triangle IJK$ is $144$, what is the area of octagon $ABCDEFGH$? | 1152 |
For which natural numbers \( n \) is the sum \( 5^n + n^5 \) divisible by 13? What is the smallest \( n \) that satisfies this condition? | 12 |
Given triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) in... | 3/2 |
There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues u... | 10 |
A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.
How many additional rods will the farmer need? | 21 |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $3, n$, and $n+1$ cents, $115$ cents is the greatest postage that cannot be formed. | 59 |
A sphere with a radius of \(\sqrt{3}\) has a cylindrical hole drilled through it; the axis of the cylinder passes through the center of the sphere, and the diameter of the base of the cylinder is equal to the radius of the sphere. Find the volume of the remaining part of the sphere. | \frac{9 \pi}{2} |
The segments \( AP \) and \( AQ \) are tangent to circle \( O \) at points \( P \) and \( Q \), respectively. Moreover, \( QE \) is perpendicular to diameter \( PD \) of length 4. If \( PE = 3.6 \) and \( AP = 6 \), what is the length of \( QE \)? | 1.2 |
Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)? | 674 |
Attach a single digit to the left and right of the eight-digit number 20222023 so that the resulting 10-digit number is divisible by 72. (Specify all possible solutions.) | 3202220232 |
The area of triangle $ABC$ is $2 \sqrt{3}$, side $BC$ is equal to $1$, and $\angle BCA = 60^{\circ}$. Point $D$ on side $AB$ is $3$ units away from point $B$, and $M$ is the intersection point of $CD$ with the median $BE$. Find the ratio $BM: ME$. | 3 : 5 |
The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are equal to 65 and 31 respectively, and its lateral sides are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AC} \) and \( \overrightarrow{BD} \). | -2015 |
Four spheres, each with a radius of 1, are placed on a horizontal table with each sphere tangential to its neighboring spheres (the centers of the spheres form a square). There is a cube whose bottom face is in contact with the table, and each vertex of the top face of the cube just touches one of the four spheres. Det... | \frac{2}{3} |
At the end of $1997$, the desert area in a certain region was $9\times 10^{5}hm^{2}$ (note: $hm^{2}$ is the unit of area, representing hectares). Geologists started continuous observations from $1998$ to understand the changes in the desert area of this region. The observation results at the end of each year are record... | 2021 |
On each of the one hundred cards, a different non-zero number is written such that each number equals the square of the sum of all the others.
What are these numbers? | \frac{1}{99^2} |
The *cross* of a convex $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$ .
Suppose $S$ is a dodecagon ( $12$ -gon) inscribed in a unit circle.... | \frac{2\sqrt{66}}{11} |
Ms. Linda teaches mathematics to 22 students. Before she graded Eric's test, the average score for the class was 84. After grading Eric's test, the class average rose to 85. Determine Eric's score on the test. | 106 |
For real numbers \(x, y, z\), the matrix
\[
\begin{pmatrix}
x & y & z \\
y & z & x \\
z & x & y
\end{pmatrix}
\]
is not invertible. Find all possible values of
\[
\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.
\] | \frac{3}{2} |
For a natural number $n \ge 3$ , we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$ -gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are eq... | 41 |
In an isosceles triangle \(ABC\) with \(\angle B\) equal to \(30^{\circ}\) and \(AB = BC = 6\), the altitude \(CD\) of triangle \(ABC\) and the altitude \(DE\) of triangle \(BDC\) are drawn.
Find \(BE\). | 4.5 |
How many positive integers \(N\) possess the property that exactly one of the numbers \(N\) and \((N+20)\) is a 4-digit number? | 40 |
A square has sides of length 3 units. A second square is formed having sides that are $120\%$ longer than the sides of the first square. This process is continued sequentially to create a total of five squares. What will be the percent increase in the perimeter from the first square to the fifth square? Express your an... | 107.4\% |
Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immediately return and meet again at point C. Gi... | 1680 |
The height \( PO \) of the regular quadrilateral pyramid \( PABC D \) is 4, and the side of the base \( ABCD \) is 6. Points \( M \) and \( K \) are the midpoints of segments \( BC \) and \( CD \). Find the radius of the sphere inscribed in the pyramid \( PMKC \). | \frac{12}{13+\sqrt{41}} |
[asy]size(8cm);
real w = 2.718; // width of block
real W = 13.37; // width of the floor
real h = 1.414; // height of block
real H = 7; // height of block + string
real t = 60; // measure of theta
pair apex = (w/2, H); // point where the strings meet
path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the bl... | 13 |
A rectangular tank with a horizontal cross-sectional area of \(S = 6 \ \text{m}^2\) is filled with water up to a height of \(H = 5 \ \text{m}\). Determine the time it takes for all the water to flow out of the tank through a small hole at the bottom with an area of \(s = 0.01 \ \text{m}^2\), assuming that the outflow s... | 1010 |
Calculate the value of $\frac12\cdot\frac41\cdot\frac18\cdot\frac{16}{1} \dotsm \frac{1}{2048}\cdot\frac{4096}{1}$, and multiply the result by $\frac34$. | 1536 |
Given the arithmetic sequence $\{a_n\}$ that satisfies the recursive relation $a_{n+1} = -a_n + n$, calculate the value of $a_5$. | \frac{9}{4} |
The base of a pyramid is an isosceles triangle with a base of 6 and a height of 9. Each lateral edge is 13. Find the volume of the pyramid. | 108 |
In a right triangle $PQR$, medians are drawn from $P$ and $Q$ to divide segments $\overline{QR}$ and $\overline{PR}$ in half, respectively. If the length of the median from $P$ to the midpoint of $QR$ is $8$ units, and the median from $Q$ to the midpoint of $PR$ is $4\sqrt{5}$ units, find the length of segment $\overli... | 16 |
If the fractional equation in terms of $x$, $\frac{x-2}{x-3}=\frac{n+1}{3-x}$ has a positive root, then $n=\_\_\_\_\_\_.$ | -2 |
There is a \(4 \times 4\) square. Its cells are called neighboring if they share a common side. All cells are painted in two colors: red and blue. It turns out that each red cell has more red neighbors than blue ones, and each blue cell has an equal number of red and blue neighbors. It is known that cells of both colo... | 12 |
Find the maximum value of the expression for \( a, b > 0 \):
$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$ | 2 \sqrt{87} |
The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are 155 and 13 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \). | 2015 |
Three fair six-sided dice are rolled (meaning all outcomes are equally likely). What is the probability that the numbers on the top faces form an arithmetic sequence with a common difference of 1? | $\frac{1}{9}$ |
The function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) has how many zeros in the interval \([0, \pi]\)? | 2017 |
Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$. | 117 |
Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to select, Congcong cannot even determine whe... | 420 |
In the sequence $5, 8, 15, 18, 25, 28, \cdots, 2008, 2015$, how many numbers have a digit sum that is an even number? (For example, the digit sum of 138 is $1+3+8=12$) | 202 |
In a kingdom of animals, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 animals of each kind, divided into 100 groups, with each group containing exactly 2 animals of one kind and 1 animal of another kind. After grouping, Kung Fu Panda asked each an... | 76 |
In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board? | 480 |
A box contains $3$ pennies, $5$ nickels, $7$ dimes, and $4$ quarters. Eight coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $1.00$ (100 cents)?
A) $\frac{325}{75582}$
B) $0$
C) $\frac{5000}{75582}$
D) $\fr... | \frac{2345}{75582} |
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have partic... | 15 |
Let $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ be the two foci of the ellipse, and $e$ be the eccentricity of the ellipse. Given $\angle P F_{1} F_{2}=\alpha$ and $\angle P F_{2} F_{1}=\beta$, express $\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}$ in terms of $e$. | \frac{1 - e}{1 + e} |
Dima took the fractional-linear function \(\frac{a x + 2b}{c x + 2d}\), where \(a, b, c, d\) are positive numbers, and summed it with the remaining 23 functions obtained from it by permuting the numbers \(a, b, c, d\). Find the root of the sum of all these functions, independent of the numbers \(a, b, c, d\). | -1 |
Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, he caught no more fish than the previous day. In total, he caught exactly 100 fish over the five days. What is the minimum total number of fish he could have caught on Monday, Wednesday, and Friday? | 50 |
The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \). | 75 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $B= \frac {\pi}{3}$ and $(a-b+c)(a+b-c)= \frac {3}{7}bc$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) If $a=5$, find the area of $\triangle ABC$. | 10 \sqrt {3} |
Let \( M = \{1, 2, 3, \ldots, 1995\} \). Subset \( A \) of \( M \) satisfies the condition: If \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)? | 1870 |
We shuffle a deck of French playing cards and then draw the cards one by one. In which position is it most likely to draw the second ace? | 18 |
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$? | 2/5 |
Given the real numbers \( a, b, c \) satisfy \( a + b + c = 6 \), \( ab + bc + ca = 5 \), and \( abc = 1 \), determine the value of \( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \). | 38 |
100 participants came to the international StarCraft championship. The game is played in a knockout format, meaning two players participate in each match, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who won exactly two matches. | 49 |
In the parallelogram \(ABCD\), the longer side \(AD\) is 5. The angle bisectors of angles \(A\) and \(B\) intersect at point \(M\).
Find the area of the parallelogram, given that \(BM = 2\) and \(\cos \angle BAM = \frac{4}{5}\). | 16 |
Let $S$ be the sum of all positive integers $n$ such that $\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3$ . Compute $\frac{S}{36}$ . | 2345 |
Vasya wrote consecutive natural numbers \(N\), \(N+1\), \(N+2\), and \(N+3\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.
The sum of the numbers in the first and second circles equals 200, and the sum of the numbers in the third and fourth circles ... | 103 |
A \( 5 \mathrm{~cm} \) by \( 5 \mathrm{~cm} \) pegboard and a \( 10 \mathrm{~cm} \) by \( 10 \mathrm{~cm} \) pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in \( 1 \mathrm{~cm} \) intervals from each edge. Pegs are placed into the holes on the two main diagonals of b... | 100 |
Fill the numbers $1, 2, \cdots, 36$ into a $6 \times 6$ grid with each cell containing one number, such that each row is in ascending order from left to right. What is the minimum possible sum of the six numbers in the third column? | 63 |
For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $215216_n$ a multiple of $5$? | 20 |
Given a regular tetrahedron $S-ABC$ with a base that is an equilateral triangle of side length 1 and side edges of length 2. If a plane passing through line $AB$ divides the tetrahedron's volume into two equal parts, the cosine of the dihedral angle between the plane and the base is: | $\frac{2 \sqrt{15}}{15}$ |
Find the smallest prime number that can be expressed as the sum of five different prime numbers. | 43 |
In a regular hexagon \(ABCDEF\), points \(M\) and \(K\) are taken on the diagonals \(AC\) and \(CE\) respectively, such that \(AM : AC = CK : CE = n\). Points \(B, M,\) and \(K\) are collinear. Find \(n\). | \frac{\sqrt{3}}{3} |
In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = ... | 8041 |
Let \( M = \{1, 2, \cdots, 2005\} \), and \( A \) be a subset of \( M \). If for any \( a_i, a_j \in A \) with \( a_i \neq a_j \), an isosceles triangle can be uniquely determined with \( a_i \) and \( a_j \) as side lengths, find the maximum value of \( |A| \). | 11 |
How many integers between $3250$ and $3500$ have four distinct digits arranged in increasing order? | 20 |
On each side of a right-angled triangle, a semicircle is drawn with that side as a diameter. The areas of the three semicircles are \( x^{2} \), \( 3x \), and 180, where \( x^{2} \) and \( 3x \) are both less than 180. What is the area of the smallest semicircle? | 144 |
Oleg drew an empty 50×50 table and wrote a number above each column and next to each row.
It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its col... | 1250 |
Lyla and Isabelle run on a circular track both starting at point \( P \). Lyla runs at a constant speed in the clockwise direction. Isabelle also runs in the clockwise direction at a constant speed 25% faster than Lyla. Lyla starts running first and Isabelle starts running when Lyla has completed one third of one lap. ... | 17 |
A dark room contains 120 red socks, 100 green socks, 70 blue socks, 50 yellow socks, and 30 black socks. A person randomly selects socks from the room without the ability to see their colors. What is the smallest number of socks that must be selected to guarantee that the selection contains at least 15 pairs? | 146 |
Let \( S = \{1, 2, 3, 4, \ldots, 16\} \). Each of the following subsets of \( S \):
\[ \{6\},\{1, 2, 3\}, \{5, 7, 9, 10, 11, 12\}, \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \]
has the property that the sum of all its elements is a multiple of 3. Find the total number of non-empty subsets \( A \) of \( S \) such that the sum of all... | 21855 |
A snail crawls from one tree to another. In half a day, it covered \( l_{1}=5 \) meters. Then, it got tired of this and turned back, crawling \( l_{2}=4 \) meters. It got tired and fell asleep. The next day, the same process repeats. The distance between the trees is \( s=30 \) meters. On which day of its journey will ... | 26 |
Given that \( a \) and \( b \) are real numbers, and the following system of inequalities in terms of \( x \):
\[
\left\{\begin{array}{l}
20x + a > 0, \\
15x - b \leq 0
\end{array}\right.
\]
has integer solutions of only 2, 3, and 4, find the maximum value of \( ab \). | -1200 |
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