Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,300 | In a round glass, the axial cross-section of which is the graph of the function \( y = x^4 \), a cherry (a sphere with radius \( r \)) is placed. For which maximum \( r \) will the sphere touch the bottom point of the glass? (In other words, what is the maximum radius \( r \) of the circle lying in the region \( y \geq... | \frac{3 \cdot 2^{1/3}}{4} | 3.90625 |
14,301 | There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere. | 6/11 | 2.34375 |
14,302 | Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM : MC = 2 \). Line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1. | 11/30 | 10.15625 |
14,303 | Given the function $f(x)=x- \frac {1}{x}+2a\ln x$ $(a\in\mathbb{R})$.
$(1)$ Discuss the monotonicity of $f(x)$;
$(2)$ If $f(x)$ has two extreme values $x_{1}$ and $x_{2}$, where $x_{2}\in[e,+\infty)$, find the minimum value of $f(x_{1})-f(x_{2})$. | \frac {4}{e} | 8.59375 |
14,304 | In triangle \(ABC\), the bisector \(BD\) is drawn, and in triangles \(ABD\) and \(CBD\), the bisectors \(DE\) and \(DF\) are drawn, respectively. It turns out that \(EF \parallel AC\). Find the angle \(\angle DEF\). | 45 | 37.5 |
14,305 | Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$, where $J$ is the center of the octagon. By connecting every second vertex starting from $A$, we obtain a larger equilateral triangle $\triangle ACE$. Compute the ratio of the area of $\triangle ABJ$ to the area of $... | \frac{1}{4} | 28.125 |
14,306 | In the diagram, point \( D \) is on side \( BC \) of \( \triangle ABC \). If \( BD = CD = AD \) and \( \angle ACD = 40^\circ \), what is the measure of \( \angle BAC \)? | 90 | 25 |
14,307 | We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids? | \frac{2}{9} | 0 |
14,308 | The real numbers \(a, b, c\) satisfy the following system of equations:
$$
\left\{\begin{array}{l}
\frac{a b}{a+b}=4 \\
\frac{b c}{b+c}=5 \\
\frac{c a}{c+a}=7
\end{array}\right.
$$
Find the value of the expression \(\frac{a b c}{a b + b c + c a}\). | 280/83 | 55.46875 |
14,309 | Given \((n+1)^{\alpha+1}-n^{\alpha+1} < n^{\alpha}(\alpha+1) < n^{\alpha+1}-(n-1)^{\alpha+1}, -1 < \alpha < 0\). Let \(x = \sum_{k=4}^{10^{6}} \frac{1}{\sqrt[3]{k}}\), find the integer part of \(x\). | 146 | 0 |
14,310 | We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon. | 2:3 | 0 |
14,311 | Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.
At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."
At the end of the first rou... | 70 | 7.8125 |
14,312 | Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, calculate the value of $\angle C$. | \dfrac{2\pi}{3} | 51.5625 |
14,313 | Given $a, b, c, d \in \mathbf{N}$ such that $342(abcd + ab + ad + cd + 1) = 379(bcd + b + d)$, determine the value of $M$ where $M = a \cdot 10^{3} + b \cdot 10^{2} + c \cdot 10 + d$. | 1949 | 0 |
14,314 | A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only... | 40 | 3.90625 |
14,315 | Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$. | \frac{1}{4} | 4.6875 |
14,316 | Compute \( t(0)-t\left(\frac{\pi}{5}\right)+t\left(\frac{2\pi}{5}\right)-t\left(\frac{3\pi}{5}\right)+\ldots+t\left(\frac{8\pi}{5}\right)-t\left(\frac{9\pi}{5}\right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + *^2 \cos x + * \).
The coefficients indicated by * are missing. A math student claime... | 10 | 46.875 |
14,317 | Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30? | 21600 | 39.84375 |
14,318 | Given $|\overrightarrow {a}|=\sqrt {2}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}-\overrightarrow {b})\bot \overrightarrow {a}$, determine the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac{\pi}{4} | 94.53125 |
14,319 | How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)? | 19 | 89.0625 |
14,320 | Three students solved the same problem. The first one said: "The answer is an irrational number. It represents the area of an equilateral triangle with a side length of 2 meters." The second one said: "The answer is divisible by 4 (without remainder). It represents the radius of a circle whose circumference is 2 meters... | \frac{1}{\pi} | 1.5625 |
14,321 | There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3... | 27 | 3.90625 |
14,322 | Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2005}\left|x_{k}-x_{k+1}\right|=2007 \). Define \( y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right) \) for \( k=1, 2, \cdots, 2007 \). Find the maximum possible value of \( \sum_{k=1}^{2006}\left|y_{k}-y_{k-1}\right| \). | 2006 | 33.59375 |
14,323 | A bagel is cut into sectors. Ten cuts were made. How many pieces resulted? | 11 | 57.8125 |
14,324 | Given a square \( ABCD \). Point \( N \) lies on side \( AD \) such that \( AN : ND = 2 : 3 \), point \( F \) lies on side \( CD \) such that \( DF : FC = 1 : 4 \), and point \( K \) lies on side \( AB \) such that \( AK : KB = 1 : 4 \). Find the angle \( \angle KNF \). | 135 | 25 |
14,325 | Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$ , in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$ . What is the largest possible size of $A$ ? | 10 | 0.78125 |
14,326 | Let the function \( f:(0,1) \rightarrow \mathbf{R} \) be defined as
$$
f(x)=\left\{\begin{array}{l}
x, \text{ if } x \text{ is irrational; } \\
\frac{p+1}{q}, \text{ if } x=\frac{p}{q}, \text{ where } (p, q)=1 \text{ and } 0<p<q.
\end{array}\right.
$$
Find the maximum value of \( f(x) \) on the interval \( \left(\frac... | \frac{16}{17} | 58.59375 |
14,327 | Alice and the White Rabbit left the Rabbit's house together at noon to go to the Duchess's reception. Halfway through, the Rabbit remembered that he forgot his gloves and fan, and ran back home at twice the speed he had been walking with Alice. Grabbing the gloves and fan, he then ran towards the Duchess (at the same s... | 12:40 | 31.25 |
14,328 | A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$ . A tangent $t$ to $\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not exceeding $\frac{S}{20}$ | 10 | 16.40625 |
14,329 | On the lateral side \( C D \) of the trapezoid \( A B C D \) (\( A D \parallel B C \)), a point \( M \) is marked. From the vertex \( A \), a perpendicular \( A H \) is dropped onto the segment \( B M \). It turns out that \( A D = H D \). Find the length of the segment \( A D \), given that \( B C = 16 \), \( C M = 8 ... | 18 | 10.9375 |
14,330 | Square $PQRS$ has side length $2$ units. Points $T$ and $U$ are on sides $PQ$ and $SQ$, respectively, with $PT = SU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and lie on diagonal $RQ$. Find the length of segment $PT$ which can be expressed in the form $\sqrt{k}-m$ units. Wha... | 10 | 16.40625 |
14,331 | Sasha records the numbers 1, 2, 3, 4, and 5 in some order, places arithmetic operation signs "+", "-", "x" and parentheses, and looks at the result of the expression obtained. For example, he can get the number 8 using the expression \((4-3) \times (2+5) + 1\). Can he get the number 123?
Forming numbers from multiple ... | 123 | 1.5625 |
14,332 | In the diagram, \( AB \parallel EF \parallel DC \). Given that \( AC + BD = 250 \), \( BC = 100 \), and \( EC + ED = 150 \), find \( CF \). | 60 | 58.59375 |
14,333 | Find and describe the pattern by which the sequence of numbers is formed. Determine the next number in this sequence.
$$
112, 224, 448, 8816, 6612
$$ | 224 | 0 |
14,334 | Given \( f(x) \) is a function defined on \(\mathbf{R}\), for any \( x, y \in \mathbf{R} \), it always holds that
\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\]
Find \( f(x) \) and calculate the value of \( f(\sqrt{2014}) \). | -1006 | 1.5625 |
14,335 | Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in i... | 11 | 1.5625 |
14,336 | 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 | 0.78125 |
14,337 | Given a set $\{4,6,8,12,14,18\}$, select three different numbers, add two of these numbers, multiply their sum by the third number, and finally subtract the smallest number you initially selected. Find the smallest result that can be obtained from this process. | 52 | 0.78125 |
14,338 | A circle is tangent to sides \( AB \) and \( AD \) of rectangle \( ABCD \) and intersects side \( DC \) at a single point \( F \) and side \( BC \) at a single point \( E \).
Find the area of trapezoid \( AFCB \) if \( AB = 32 \), \( AD = 40 \), and \( BE = 1 \). | 1180 | 0 |
14,339 | Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry? | 1956 | 39.84375 |
14,340 | A tram ticket is called "lucky in Leningrad style" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called "lucky in Moscow style" if the sum of its digits in even positions is equal to the sum of its digits in odd positions. How many tickets are there that are both lu... | 6700 | 81.25 |
14,341 | There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible? | 180 | 2.34375 |
14,342 | For all triples \((x, y, z)\) that satisfy the system
$$
\left\{\begin{array}{l}
2 \cos x = \operatorname{ctg} y \\
2 \sin y = \operatorname{tg} z \\
\cos z = \operatorname{ctg} x
\end{array}\right.
$$
find the minimum value of the expression \(\sin x + \cos z\). | -\frac{5 \sqrt{3}}{6} | 0 |
14,343 | A fair six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls is equal to the number rolled on the third roll, what is the probability that at least one of the numbers rolled is 2? | $\frac{8}{15}$ | 0 |
14,344 | In this subtraction problem, \( P, Q, R, S, T \) represent single digits. What is the value of \( P + Q + R + S + T \)?
\[
\begin{array}{rrrrr}
7 & Q & 2 & S & T \\
-P & 3 & R & 9 & 6 \\
\hline
2 & 2 & 2 & 2 & 2
\end{array}
\] | 29 | 10.9375 |
14,345 | Let $x_0$ be a zero of the function $f(x) = \sin \pi x$, and it satisfies $|x_{0}| + f(x_{0} + \frac {1}{2}) < 11$. Determine the number of such zeros. | 21 | 32.8125 |
14,346 | There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determin... | 12 | 9.375 |
14,347 | Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation? | 162 | 91.40625 |
14,348 | In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written? | 19 | 75 |
14,349 | Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. | 972538 | 96.875 |
14,350 | Given the set \( A \) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \((-\infty, \infty)\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \( A \). | 14 | 7.03125 |
14,351 | The first three numbers of a sequence are \(1, 7, 8\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence. | 3028 | 75 |
14,352 | 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 | 3.90625 |
14,353 | In parallelogram \(ABCD\), points \(A_{1}, A_{2}, A_{3}, A_{4}\) and \(C_{1}, C_{2}, C_{3}, C_{4}\) are respectively the quintisection points of \(AB\) and \(CD\). Points \(B_{1}, B_{2}\) and \(D_{1}, D_{2}\) are respectively the trisection points of \(BC\) and \(DA\). Given that the area of quadrilateral \(A_{4} B_{2}... | 15 | 10.9375 |
14,354 | In triangle $ABC$, $\angle C=90^{\circ}, \angle B=30^{\circ}, AC=2$, $M$ is the midpoint of $AB$. Fold triangle $ACM$ along $CM$ such that the distance between points $A$ and $B$ becomes $2\sqrt{2}$. Find the volume of the resulting triangular pyramid $A-BCM$. | \frac{2 \sqrt{2}}{3} | 0 |
14,355 | The region $G$ is bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, and the region $g$ is bounded by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly chosen within the region $G$. What is the probability that it belongs to region $g$ (event $A$)? | \frac{2}{3} | 32.03125 |
14,356 | Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer. | 5600 | 0 |
14,357 | Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]
If for any positive integer \(n\), it holds that
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the m... | 2 + \sqrt{5} | 28.90625 |
14,358 | Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that their union contains all the vertices of the c... | 101 | 31.25 |
14,359 | Find the greatest solution on the interval $[0 ; 10 \pi]$ of the equation $|2 \sin x - 1| + |2 \cos 2x - 1| = 0$. Round the answer to three significant digits according to rounding rules and enter it in the provided field. | 27.7 | 28.90625 |
14,360 | Given a cube \( ABCD A_1 B_1 C_1 D_1 \) with an edge length of 1. A sphere passes through vertices \( A \) and \( C \) and the midpoints \( F \) and \( E \) of edges \( B_1 C_1 \) and \( C_1 D_1 \) respectively. Find the radius \( R \) of this sphere. | \frac{\sqrt{41}}{8} | 4.6875 |
14,361 | Travel along the alley clockwise.
In 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.
In the first case, when moving along the alley, they would need to cover a distance of 6 km, and i... | 0.21 | 0 |
14,362 | The first 14 terms of the sequence $\left\{a_{n}\right\}$ are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, \ldots$. Following this pattern, what is $a_{18}$? | 51 | 6.25 |
14,363 | In the convex quadrilateral \(ABCD\), the length of side \(AD\) is 4, the length of side \(CD\) is 7, the cosine of angle \(ADC\) is \(\frac{1}{2}\), and the sine of angle \(BCA\) is \(\frac{1}{3}\). Find the length of side \(BC\) given that the circumcircle of triangle \(ABC\) also passes through point \(D\). | \frac{\sqrt{37}}{3\sqrt{3}}(\sqrt{24} - 1) | 0 |
14,364 | In the Tenth Kingdom, there are 17 islands, each with 119 inhabitants. The inhabitants are divided into two castes: knights, who always tell the truth, and liars, who always lie. During a population census, each person was first asked, "Not including yourself, are there an equal number of knights and liars on your isla... | 1013 | 14.84375 |
14,365 | Determine the largest positive integer \( n \) with \( n < 500 \) for which \( 6048 \cdot 28^n \) is a perfect cube (that is, it is equal to \( m^3 \) for some positive integer \( m \) ). | 497 | 72.65625 |
14,366 | Find the value(s) of $x$ and $z$ such that $10xz - 15z + 3x - \frac{9}{2} = 0$ is true for all values of $z$. | \frac{3}{2} | 37.5 |
14,367 | Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number. | 520 | 0.78125 |
14,368 | Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the produ... | 99 | 39.0625 |
14,369 | Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:
- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the... | 44 | 0 |
14,370 | In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form? | 23 | 2.34375 |
14,371 | In an old estate, the house is surrounded by tall trees arranged in a circle, including spruces, pines, and birches. There are 96 trees in total. These trees have a peculiar property: for any coniferous tree, among the two trees that are two trees away from it, one is coniferous and the other is deciduous; also, among ... | 32 | 13.28125 |
14,372 |
The side $AB$ of triangle $ABC$ is divided into $n$ equal parts (with division points $B_0 = A, B_1, B_2, \ldots, B_n = B$), and the side $AC$ of this triangle is divided into $n+1$ equal parts (with division points $C_0 = A, C_1, C_2, \ldots, C_{n+1} = C$). The triangles $C_i B_i C_{i+1}$ are shaded. What fraction of... | \frac{1}{2} | 24.21875 |
14,373 | Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2027\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1014 | 9.375 |
14,374 | The side of rhombus \(ABCD\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.
Find the distance between the points where this circle touches the sides \(AB\) and \(BC\), if the diagonal \(AC\) is less than the diagonal \(BD\). | 3.84 | 0.78125 |
14,375 | For the function $f(x)= \sqrt {2}(\sin x+\cos x)$, the following four propositions are given:
$(1)$ There exists $\alpha\in\left(- \frac {\pi}{2},0\right)$, such that $f(\alpha)= \sqrt {2}$;
$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=- \frac {3\pi}{4}$;
$(3)$ There exists $\phi\in\mathb... | (2)(3) | 0.78125 |
14,376 | What is the product of the solutions of the equation $-45 = -2x^2 + 6x?$ | -22.5 | 7.8125 |
14,377 | A natural number \( N \) ends with the digit 5. A ninth-grader Dima found all its divisors and discovered that the sum of the two largest proper divisors is not divisible by the sum of the two smallest proper divisors. Find the smallest possible value of the number \( N \). A divisor of a natural number is called prope... | 725 | 67.96875 |
14,378 | (12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:
(1) If digits can be repeated, how many different five-digit even numbers can be formed?
(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not being 3, can be formed?... | 20 | 4.6875 |
14,379 | Calculate the smallest root \(x_0\) of the equation
$$
x^{2}-\sqrt{\lg x +100}=0
$$
(with a relative error of no more than \(10^{-390} \%\)). | 10^{-100} | 0 |
14,380 | Let the function \( f(x) = 4x^3 + bx + 1 \) with \( b \in \mathbb{R} \). For any \( x \in [-1, 1] \), \( f(x) \geq 0 \). Find the range of the real number \( b \). | -3 | 2.34375 |
14,381 | In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$. | \frac{\sqrt{3}}{3} | 28.125 |
14,382 | Given that a five-digit palindromic number is equal to the product of 45 and a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}} = 45 \times \overline{\text{deed}}$), find the largest possible value of the five-digit palindromic number. | 59895 | 98.4375 |
14,383 | The school table tennis championship was held in an Olympic system format. The winner won six matches. How many participants in the tournament won more games than they lost? (In an Olympic system tournament, participants are paired up. Those who lose a game in the first round are eliminated. Those who win in the first ... | 16 | 1.5625 |
14,384 | Point \( K \) is the midpoint of edge \( A A_{1} \) of cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), and point \( L \) lies on edge \( B C \). Segment \( K L \) touches the sphere inscribed in the cube. In what ratio does the point of tangency divide segment \( K L \)? | 4/5 | 0 |
14,385 | The base of a quadrilateral pyramid is a square \(ABCD\) with each side equal to 2. The lateral edge \(SA\) is perpendicular to the base plane and also equals 2. A plane is passed through the lateral edge \(SC\) and a point on side \(AB\) such that the resulting cross-section of the pyramid has the smallest perimeter. ... | \sqrt{6} | 15.625 |
14,386 | A teacher has prepared three problems for the class to solve. In how many different ways can he present these problems to the students if there are 30 students in the class? | 24360 | 75.78125 |
14,387 | Let the complex number \( z \) satisfy \( |z|=1 \). Given that the equation \( zx^2 + 2\bar{z}x + 2 = 0 \) in terms of \( x \) has a real root, find the sum of all such complex numbers \( z \). | -\frac{3}{2} | 19.53125 |
14,388 | Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches ea... | 2\sqrt{14} | 2.34375 |
14,389 | Given that $17^{-1} \equiv 11 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 42 | 65.625 |
14,390 | For which natural number \( n \) does the quantity \(\frac{n^2}{1.001^n}\) reach its maximum value? | 2001 | 75 |
14,391 | Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest po... | 4.5 | 21.875 |
14,392 | In the acute triangle \( \triangle ABC \),
\[
\sin(A+B) = \frac{3}{5}, \quad \sin(A-B) = \frac{1}{5}, \quad AB = 3.
\]
Find the area of \( \triangle ABC \). | \frac{3(\sqrt{6} + 2)}{2} | 0 |
14,393 | Consider a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n = n^2 - 4n + 2$. Find the sum of the absolute values of the first ten terms: $|a_1| + |a_2| + \cdots + |a_{10}|$. | 68 | 1.5625 |
14,394 | Maria invested $10,000 for 3 years at an annual interest rate of 5 percent compounded annually. Liam invested $10,000 for the same period of time, at the same interest rate, but the interest was compounded semi-annually. Calculate the difference in the amount earned by Liam's investment compared to Maria's, to the near... | 16 | 0 |
14,395 | Ponchik was having a snack at a roadside café when a bus passed by. Three pastries after the bus, a motorcycle passed by Ponchik, and three pastries after that, a car passed by. Syrupchik, who was snacking at another café on the same road, saw them in a different order: first the bus, after three pastries the car, and ... | 40 | 14.0625 |
14,396 | In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the sizes of angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^2 - a^2 = ac \), what is the radian measure of angle \( B \)? | \frac{2 \pi}{7} | 7.03125 |
14,397 | Let \( ABCD \) be a trapezoid such that \( (AB) \) is parallel to \( (CD) \), \( AB = 3 \), \( CD = 3 \), \( DA = 3 \) and \( \widehat{ADC} = 120^\circ \). Determine the angle \( \widehat{CBA} \) in degrees. | 30 | 7.03125 |
14,398 | Inside rectangle \(ABCD\), points \(E\) and \(F\) are located such that segments \(EA, ED, EF, FB, FC\) are all congruent. The side \(AB\) is \(22 \text{ cm}\) long and the circumcircle of triangle \(AFD\) has a radius of \(10 \text{ cm}\).
Determine the length of side \(BC\). | 16 | 0 |
14,399 | On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when later, in a photograph, they saw the bear's t... | 11400 | 0 |
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