Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,100 | In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $a^{2}=b^{2}+4bc\sin A$ and $\tan A \cdot \tan B=2$, then $\tan B-\tan A=$ ______. | -8 | 3.90625 |
14,101 | Let \( d_{1}, d_{2}, \cdots, d_{k} \) be all the divisors of the positive integer \( n \), denoted as:
\[ 1 = d_{1} < d_{2} < d_{3} < \cdots < d_{k} = n. \]
Find all \( n \) such that \( k \geq 4 \) and \( d_{1}^{2} + d_{2}^{2} + d_{3}^{2} + d_{4}^{2} = n \). | 130 | 100 |
14,102 | Let $\mathrm{O}$ be the intersection point of the diagonals of a convex quadrilateral $A B C D$, and let $P, Q, R$, and $S$ be the centroids of triangles $A O B$, $B O C$, $C O D$, and $D O A$, respectively. Find the ratio of the areas of the quadrilateral $P Q R S$ to that of $A B C D$. | \frac{2}{9} | 0 |
14,103 | Triangles $\triangle DEF$ and $\triangle D'E'F'$ are in the coordinate plane with vertices $D(2,2)$, $E(2,14)$, $F(18,2)$, $D'(32,26)$, $E'(44,26)$, $F'(32,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\triangle DEF$ to $\triangle D'E'F'$. Find $n+u+v$. | 124 | 0 |
14,104 | Kolya, after walking one-fourth of the way from home to school, realized that he forgot his problem book. If he does not go back for it, he will arrive at school 5 minutes before the bell rings, but if he goes back, he will be 1 minute late. How long (in minutes) does it take to get to school? | 12 | 1.5625 |
14,105 | A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \(a\) and \(b\) such that \(|2^a - 3^b| = n\). Find the smallest value of \(n\). | 35 | 57.8125 |
14,106 | How many lattice points (points with integer coordinates) are inside (but not on the boundary) the region formed by the right branch of the hyperbola $x^{2} - y^{2} = 1$ and the line $x = 100$? | 9800 | 81.25 |
14,107 | Primes like $2, 3, 5, 7$ are natural numbers greater than 1 that can only be divided by 1 and themselves. We split 2015 into the sum of 100 prime numbers, requiring that the largest of these prime numbers be as small as possible. What is this largest prime number? | 23 | 2.34375 |
14,108 | Grandma told her grandchildren: "Today I am 60 years and 50 months and 40 weeks and 30 days old."
How old was Grandma on her last birthday? | 65 | 47.65625 |
14,109 | Xiaopang, Xiaodingding, Xiaoya, and Xiaoqiao have a total of 8 parents and 4 children in their four families. They are going to an amusement park together. The ticket pricing is as follows: Adult tickets are 100 yuan per person, children's tickets are 50 yuan per person. If there are 10 or more people, they can buy gro... | 800 | 0 |
14,110 | A certain store sells a product, and because the purchase price decreased by 6.4% compared to the original purchase price, the profit margin increased by 8 percentage points. What was the original profit margin for selling this product? | 17\% | 63.28125 |
14,111 | Given $\sin (α- \frac {π}{6})= \frac {2}{3}$, $α∈(π, \frac {3π}{2})$, $\cos ( \frac {π}{3}+β)= \frac {5}{13}$, $β∈(0,π)$, find the value of $\cos (β-α)$. | - \frac{10+12 \sqrt{5}}{39} | 2.34375 |
14,112 | Two Wei Qi teams, $A$ and $B$, each comprising 7 members, compete against each other. Players from each team face off in sequence. The first game is between the first player of each team. The loser is eliminated, and the winner moves on to face the next player of the opposing team. This process continues until one team... | 3432 | 46.875 |
14,113 | There are 8 blue, 7 red, and 12 white light bulbs. In how many ways can they all be arranged to form a garland such that no two white light bulbs are next to each other? | 11711700 | 32.03125 |
14,114 | There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible sum? | 2005 | 0 |
14,115 | Given that $\triangle ABC$ is an isosceles right triangle with one leg length of $1$, determine the volume of the resulting geometric solid when $\triangle ABC$ is rotated around one of its sides | \frac{\sqrt{2}\pi}{6} | 3.90625 |
14,116 |
Olga Ivanovna, the class teacher of Grade 5B, is organizing a "Mathematical Ballet." She wants to arrange boys and girls so that at a distance of 5 meters from each girl there are exactly 2 boys. What is the maximum number of girls that can participate in the ballet, given that 5 boys are participating? | 20 | 0 |
14,117 | The legs \( AC \) and \( CB \) of the right triangle \( ABC \) are 15 and 8, respectively. A circular arc with radius \( CB \) is drawn from center \( C \), cutting off a part \( BD \) from the hypotenuse. Find \( BD \). | \frac{128}{17} | 7.8125 |
14,118 | How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit? | 715 | 13.28125 |
14,119 | For the largest natural \( m \), when will the product \( m! \cdot 2022! \) be a factorial of a natural number? | 2022! - 1 | 0 |
14,120 | Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two parts such that with the two pieces of this stick and the two whole sticks, he could form the contour of a right triangle. How many square centimeters is the area of this triangle? | 216 | 7.03125 |
14,121 | Let $T$ denote the value of the sum\[\sum_{n=0}^{432} (-1)^{n} {1500 \choose 3n}\]Determine the remainder obtained when $T$ is divided by $100$. | 66 | 3.90625 |
14,122 | Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression. | -\frac{82}{9} | 1.5625 |
14,123 | A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sphere. Determine the probability that $Q$ li... | \frac{1}{3} | 3.125 |
14,124 | Find the smallest prime number that can be represented as the sum of two, three, four, five, and six distinct prime numbers. | 61 | 0 |
14,125 | 3 red marbles, 4 blue marbles, and 5 green marbles are distributed to 12 students. Each student gets one and only one marble. In how many ways can the marbles be distributed so that Jamy and Jaren get the same color and Jason gets a green marble? | 3150 | 0 |
14,126 | In the trapezium \(ABCD\), the lines \(AB\) and \(DC\) are parallel, \(BC = AD\), \(DC = 2 \times AD\), and \(AB = 3 \times AD\). The angle bisectors of \(\angle DAB\) and \(\angle CBA\) intersect at the point \(E\). What fraction of the area of the trapezium \(ABCD\) is the area of the triangle \(ABE\)? | 3/5 | 25 |
14,127 | The sequence of real numbers \( a_1, a_2, \cdots, a_n, \cdots \) is defined by the following equation: \( a_{n+1} = 2^n - 3a_n \) for \( n = 0, 1, 2, \cdots \).
1. Find an expression for \( a_n \) in terms of \( a_0 \) and \( n \).
2. Find \( a_0 \) such that \( a_{n+1} > a_n \) for any positive integer \( n \). | \frac{1}{5} | 28.90625 |
14,128 | Each face of a regular tetrahedron is labeled with one of the numbers 1, 2, 3, 4. Four identical regular tetrahedrons are simultaneously rolled onto a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. | \frac{13}{16} | 21.09375 |
14,129 | Given the coordinates of the vertices of triangle $\triangle O A B$ are $O(0,0), A(4,4 \sqrt{3}), B(8,0)$, with its incircle center being $I$. Let the circle $C$ pass through points $A$ and $B$, and intersect the circle $I$ at points $P$ and $Q$. If the tangents drawn to the two circles at points $P$ and $Q$ are perpen... | 2\sqrt{7} | 0.78125 |
14,130 | The center of sphere $\alpha$ lies on the surface of sphere $\beta$. The ratio of the surface area of sphere $\beta$ that is inside sphere $\alpha$ to the entire surface area of sphere $\alpha$ is $1 / 5$. Find the ratio of the radii of spheres $\alpha$ and $\beta$. | \sqrt{5} | 0 |
14,131 | In a certain business district parking lot, temporary parking is charged by time period. The charging standard is: a charge of 6 yuan for parking not exceeding 1 hour per car, and for the part exceeding 1 hour, a charge of 8 yuan per hour (parts of an hour are rounded up to the next hour). Now, two people, A and B, par... | \frac{1}{4} | 55.46875 |
14,132 | Calculate the following expression (accurate to 8 decimal places):
$$
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{... | 3.14159265 | 95.3125 |
14,133 | In the regular quadrangular pyramid \(P-ABCD\), \(M\) and \(N\) are the midpoints of \(PA\) and \(PB\) respectively. If the tangent of the dihedral angle between a side face and the base is \(\sqrt{2}\), find the cosine of the angle between skew lines \(DM\) and \(AN\). | 1/6 | 28.125 |
14,134 | A palindrome is a number, word, or text that reads the same backward as forward. How much time in a 24-hour day display palindromes on a clock, showing time from 00:00:00 to 23:59:59? | 144 | 0 |
14,135 | On the board, there are \( n \) different integers, each pair of which differs by at least 10. The sum of the squares of the three largest among them is less than three million. The sum of the squares of the three smallest among them is also less than three million. What is the greatest possible \( n \)? | 202 | 0.78125 |
14,136 | Two spheres touch the plane of triangle \(ABC\) at points \(B\) and \(C\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \(5 \sqrt{17}\). The center of a third sphere with a radius of 8 is at point \(A\), and it is externally tange... | 2\sqrt{19} | 1.5625 |
14,137 | In the quadrilateral pyramid \(P-ABCD\), given that \(AB\) is parallel to \(CD\), \(AB\) is perpendicular to \(AD\), \(AB=4\), \(AD=2\sqrt{2}\), \(CD=2\), and \(PA\) is perpendicular to the plane \(ABCD\), with \(PA=4\). Let \(Q\) be a point on line segment \(PB\) such that the sine of the angle between line \(QC\) and... | 7/12 | 5.46875 |
14,138 | In $\triangle ABC$, $AB=10$, $AC=8$, and $BC=6$. Circle $P$ passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection of circle $P$ with sides $AC$ and $BC$ (excluding $C$). The length of segment $QR$ is | 4.8 | 9.375 |
14,139 | Given the set $H$ defined by the points $(x,y)$ with integer coordinates, $2\le|x|\le8$, $2\le|y|\le8$, calculate the number of squares of side at least $5$ that have their four vertices in $H$. | 14 | 0.78125 |
14,140 | In triangle \( A B C \) with side \( A C = 8 \), a bisector \( B L \) is drawn. It is known that the areas of triangles \( A B L \) and \( B L C \) are in the ratio \( 3: 1 \). Find the bisector \( B L \), for which the height dropped from vertex \( B \) to the base \( A C \) will be the greatest. | 3\sqrt{2} | 2.34375 |
14,141 | Given numbers \(a, b, c\) satisfy \(a b c+a+c-b\). Then the maximum value of the algebraic expression \(\frac{1}{1+a^{2}}-\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\) is | \frac{5}{4} | 1.5625 |
14,142 | Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$ . Find $w_1+w_2$ . | 27 | 3.125 |
14,143 | There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long? | 12 | 61.71875 |
14,144 | Given that $| \overrightarrow{a}|=12$, $| \overrightarrow{b}|=9$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-54 \sqrt {2}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 63.28125 |
14,145 | If
\[(1 + \tan 0^\circ)(1 + \tan 1^\circ)(1 + \tan 2^\circ) \dotsm (1 + \tan 30^\circ) = 2^m,\]
find the value of $m$. | 16 | 32.03125 |
14,146 | We say that two natural numbers form a perfect pair when the sum and the product of these two numbers are perfect squares. For example, 5 and 20 form a perfect pair because $5+20=25=5^{2}$ and $5 \times 20=100=10^{2}$. Does 122 form a perfect pair with any other natural number? | 122 \times 121 | 0 |
14,147 | Given the circle $C$: $x^{2}+y^{2}-2x=0$, find the coordinates of the circle center $C$ and the length of the chord intercepted by the line $y=x$ on the circle $C$. | \sqrt{2} | 56.25 |
14,148 | There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers ... | 10 | 16.40625 |
14,149 | In Mr. Smith's class, the ratio of boys to girls is 3 boys for every 4 girls and there are 42 students in his class, calculate the percentage of students that are boys. | 42.857\% | 0 |
14,150 | Let \( a_{n} = 1 + 2 + \cdots + n \), where \( n \in \mathbf{N}_{+} \), and \( S_{m} = a_{1} + a_{2} + \cdots + a_{m} \), \( m = 1, 2, \cdots, m \). Find the number of values among \( S_{1}, S_{2}, \cdots, S_{2017} \) that are divisible by 2 but not by 4. | 252 | 0 |
14,151 | A bus traveling a 100 km route is equipped with a computer that forecasts the remaining time to arrival at the final destination. This time is calculated based on the assumption that the average speed of the bus on the remaining part of the route will be the same as it was on the part already traveled. Forty minutes af... | 85 | 4.6875 |
14,152 | Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it also holds that \( x y z \mid (x+y+z)^{n} \). | 13 | 16.40625 |
14,153 | In the center of a circular field stands the geologists' house. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set off on a journey from their house at a speed of 5 km/h along randomly chosen roads. Determine the probability that the distance between them after one hour will b... | 0.5 | 37.5 |
14,154 | Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{5 \operatorname{tg} x+2}{2 \sin 2 x+5} d x
$$ | \frac{1}{2} \ln \left(\frac{14}{5}\right) | 3.125 |
14,155 | The secant \( ABC \) intercepts an arc \( BC \), which contains \( 112^\circ \); the tangent \( AD \) at point \( D \) divides this arc in the ratio \( 7:9 \). Find \(\angle BAD\). | 31.5 | 18.75 |
14,156 | A certain operation is performed on a positive integer: if it is even, divide it by 2; if it is odd, add 1. This process continues until the number becomes 1. How many integers become 1 after exactly 10 operations? | 55 | 57.03125 |
14,157 | In the Cartesian coordinate plane, the number of integer points (points where both the x-coordinate and y-coordinate are integers) that satisfy the system of inequalities
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100
\end{cases}
\]
is ___. | 2551 | 87.5 |
14,158 | Three lathes \( A, B, C \) each process the same type of standard parts at a certain work efficiency. Lathe \( A \) starts 10 minutes earlier than lathe \( C \), and lathe \( C \) starts 5 minutes earlier than lathe \( B \). After lathe \( B \) has been working for 10 minutes, the number of standard parts processed by ... | 15 | 42.96875 |
14,159 | In the quadrilateral pyramid \( S A B C D \):
- The lateral faces \( S A B \), \( S B C \), \( S C D \), and \( S D A \) have areas 9, 9, 27, 27 respectively;
- The dihedral angles at the edges \( A B \), \( B C \), \( C D \), \( D A \) are equal;
- The quadrilateral \( A B C D \) is inscribed in a circle, and its are... | 54 | 2.34375 |
14,160 | Solve the equation
$$
\sin ^{4} x + 5(x - 2 \pi)^{2} \cos x + 5 x^{2} + 20 \pi^{2} = 20 \pi x
$$
Find the sum of its roots that belong to the interval $[-\pi ; 6 \pi]$, and provide the answer, rounding to two decimal places if necessary. | 31.42 | 35.15625 |
14,161 | A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposite direction, also equally spaced. The shi... | 4.6 | 2.34375 |
14,162 | Perpendiculars \( B E \) and \( D F \), dropped from the vertices \( B \) and \( D \) of parallelogram \( A B C D \) onto sides \( A D \) and \( B C \) respectively, divide the parallelogram into three parts of equal area. On the extension of diagonal \( B D \) past vertex \( D \), a segment \( D G \) is laid off equal... | 1:1 | 14.84375 |
14,163 |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane containing the given edge. (12 points) | \frac{\sqrt{3}}{4} | 10.9375 |
14,164 | Let the set \( M = \{1, 2, 3, \cdots, 50\} \). For any subset \( S \subseteq M \) such that for any \( x, y \in S \) with \( x \neq y \), it holds that \( x + y \neq 7k \) for any \( k \in \mathbf{N} \). If \( S_0 \) is the subset with the maximum number of elements that satisfies this condition, how many elements are ... | 23 | 75.78125 |
14,165 | Triangle $PQR$ has $PQ = 28.$ The incircle of the triangle evenly trisects the median $PS.$ If the area of the triangle is $p \sqrt{q}$ where $p$ and $q$ are integers, and $q$ is prime, find $p+q.$ | 199 | 6.25 |
14,166 | The incircle of triangle \( ABC \) with center \( O \) touches the sides \( AB \), \( BC \), and \( AC \) at points \( M \), \( N \), and \( K \) respectively. It is given that angle \( AOC \) is four times larger than angle \( MKN \). Find angle \( B \). | 108 | 15.625 |
14,167 | In a right triangle \(ABC\), the legs \(AB\) and \(AC\) measure 4 and 3 respectively. Point \(D\) bisects the hypotenuse \(BC\). Find the distance between the centers of the incircles of triangles \(ADC\) and \(ABD\). | \frac{5 \sqrt{13}}{12} | 0.78125 |
14,168 | I bought a lottery ticket with a five-digit number such that the sum of its digits equals the age of my neighbor. Determine the number of this ticket, given that my neighbor easily solved this problem. | 99999 | 3.90625 |
14,169 | Let \( S = \{1, 2, \cdots, 98\} \). Find the smallest positive integer \( n \) such that, in any subset of \( S \) with \( n \) elements, it is always possible to select 10 numbers, and no matter how these 10 numbers are evenly divided into two groups, there will always be one number in one group that is relatively pri... | 50 | 24.21875 |
14,170 | A hotel has 5 distinct rooms, each with single beds for up to 2 people. The hotel has no other guests, and 5 friends want to stay there for the night. In how many ways can the 5 friends choose their rooms? | 2220 | 0 |
14,171 | In triangle $PQR$, let the side lengths be $PQ = 7,$ $PR = 8,$ and $QR = 5$. Calculate:
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{16}{7} | 31.25 |
14,172 | Consider a $2\times 3$ grid where each entry is either $0$ , $1$ , or $2$ . For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$ ? One valid grid is shown below: $$ \begin{bmatrix} 1 & 2 & 0 2 & 1 & 0 \end{bmatrix} $$ | 10 | 0 |
14,173 | Given the sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x| + |y| = a, a > 0\}, \\
B=\{(x, y) \mid |xy| + 1 = |x| + |y|\}
\end{array}
$$
If $A \cap B$ forms the vertices of a regular octagon in the plane, find the value of $a$. | \sqrt{2} | 50.78125 |
14,174 | There are 13 students in a class (one of them being the monitor) and 13 seats in the classroom. Every day, the 13 students line up in random order and then enter the classroom one by one. Except for the monitor, each student will randomly choose an unoccupied seat and sit down. The monitor, however, prefers the seat ne... | 7/13 | 0 |
14,175 | What is the sum and the product of the values of $x$ that satisfy the equation $x^2 - 7x + 12 = 0$? | 12 | 35.15625 |
14,176 | Given the complex numbers \( z_{1} \) and \( z_{2} \) such that \( \left| z_{2} \right| = 4 \) and \( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \), find the maximum value of \( \left| \left( z_{1} + 1 \right)^{2} \left( z_{1} - 2 \right) \right| \). | 6\sqrt{6} | 0.78125 |
14,177 | Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the point of intersection of its diagonals, $O_1$ and $O_2$ are the centers of the inscribed circles of triangles $ABM$ and $CMD$ respectively, $K$ is the midpoint of the arc $AD$ that does not contain points $B$ and $C$, $\angle O_1 K O_2 = 60^{\circ}$, $K O_1 = 10... | 10 | 91.40625 |
14,178 | In triangle \(ABC\), angle \(A\) is \(60^\circ\) and \(AB:AC = 3:2\). Points \(M\) and \(N\) are located on sides \(AB\) and \(AC\) respectively, such that \(BM = MN = NC\). Find the ratio of the area of triangle \(AMN\) to the area of triangle \(ABC\). | 4/25 | 1.5625 |
14,179 | Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
\[ y = \ln \frac{5}{2 x}, \quad \sqrt{3} \leq x \leq \sqrt{8} \] | 1 + \frac{1}{2} \ln \frac{3}{2} | 0 |
14,180 | A certain school holds a men's table tennis team competition. The final match adopts a points system. The two teams in the final play three matches in sequence, with the first two matches being men's singles matches and the third match being a men's doubles match. Each participating player can only play in one match in... | 36 | 8.59375 |
14,181 | Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \). | \frac{1}{3} | 6.25 |
14,182 | Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves
$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$ | \pi + \frac{3\sqrt{3}}{8} | 0.78125 |
14,183 | Find the sum of the areas of all distinct rectangles that can be formed from 9 squares (not necessarily all), if the side of each square is $1 \text{ cm}$. | 72 | 0.78125 |
14,184 | For all real numbers $x$ and $y$, define the mathematical operation $\star$ such that the following conditions apply: $x\ \star\ 0 = x+1, x\ \star\ y = y\ \star\ x$, and $(x + 2)\ \star\ y = (x\ \star\ y) + y + 2$. What is the value of $7\ \star\ 3$? | 24 | 3.90625 |
14,185 | There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each? | 10 | 7.03125 |
14,186 | How many positive integers \( n \) exist such that both \(\frac{n+1}{3}\) and \(3n+1\) are three-digit integers? | 12 | 10.9375 |
14,187 | Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds. | 2 + 2\sqrt{2} | 0 |
14,188 | The bases of a trapezoid are 2 cm and 3 cm long. A line passing through the intersection point of the diagonals and parallel to the bases intersects the legs at points X and Y. What is the distance between points X and Y? | 2.6 | 0 |
14,189 | Chords \(AB\) and \(CD\) of a circle with center \(O\) both have a length of 5. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) intersect at point \(P\), where \(DP=13\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL:LC\). | 13/18 | 7.03125 |
14,190 | A block of iron solidifies from molten iron, and its volume reduces by $\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase? | \frac{1}{33} | 23.4375 |
14,191 | The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\sqrt{85}$, $\sqrt{58}$, and $\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere. | 14/9 | 0 |
14,192 | A positive integer \( n \) with \( n \) digits is called an "auspicious number" if, when appended to the end of any two positive integers, the product of these two new numbers ends in \( x \). For example, 6 is an "auspicious number," but 16 is not, because \( 116 \times 216 = 25056 \), which does not end in 16. What i... | 1114 | 4.6875 |
14,193 | When Bendegúz boarded the 78-seat train car with his valid seat reservation, he was shocked to find that all seats were already taken. What had happened was that Dömötör boarded without a seat reservation. The other 77 passengers, including Elek, had purchased a seat reservation, but did not necessarily sit in their as... | 1/2 | 35.9375 |
14,194 | Two adjacent faces of a tetrahedron, each being equilateral triangles with side length 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | \frac{\sqrt{3}}{4} | 5.46875 |
14,195 | Let \( E(n) \) denote the largest integer \( k \) such that \( 5^{k} \) divides the product \( 1^{1} \cdot 2^{2} \cdot 3^{3} \cdot 4^{4} \cdots \cdots n^{n} \). What is the value of \( E(150) \)? | 2975 | 39.0625 |
14,196 | Given $S$, $P$ (not the origin) are two different points on the parabola $y=x^{2}$, the tangent line at point $P$ intersects the $x$ and $y$ axes at $Q$ and $R$, respectively.
(Ⅰ) If $\overrightarrow{PQ}=\lambda \overrightarrow{PR}$, find the value of $\lambda$;
(Ⅱ) If $\overrightarrow{SP} \perp \overrightarrow{PR}$,... | \frac{4\sqrt{3}}{9} | 0 |
14,197 | Four two-inch squares are placed with their bases on a line. The second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on both sides. Determine the distance, in inches, of point P, the top vertex of the rotated square, from the line on which ... | 1 + \sqrt{2} | 25.78125 |
14,198 | It is known that \( b^{16} - 1 \) has four distinct prime factors. Determine the largest one, denoted by \( c \). | 257 | 85.15625 |
14,199 | An employee receives an average of two requests per hour. Assuming a simple flow of requests, what is the probability of receiving four requests in four hours? | 0.0572 | 0 |
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