Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
15,200 | Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that from any subset of \( S \) containing \( n \) elements, we can always select 10 numbers. No matter how we divide these 10 numbers into two groups, there will always be one group containing a number that is coprime with the other 4 nu... | 50 | 32.8125 |
15,201 | Given triangle \( \triangle ABC \) with \( AB < AC \), the altitude \( AD \), angle bisector \( AE \), and median \( AF \) are drawn from \( A \), with \( D, E, F \) all lying on \(\overline{BC}\). If \( \angle BAD = 2 \angle DAE = 2 \angle EAF = \angle FAC \), what are all possible values of \( \angle ACB \)? | 30 | 32.8125 |
15,202 | Two motorcyclists departed simultaneously from points \( A \) and \( B \) towards each other and met 50 km from point \( B \). After arriving at points \( A \) and \( B \), the motorcyclists immediately turned back and met for the second time 25 km from point \( A \). How many kilometers are there from \( A \) to \( B ... | 125 | 30.46875 |
15,203 | Expanding the expression \((1+\sqrt{11})^{212}\) using the binomial theorem, we obtain terms of the form \(C_{212}^{k}(\sqrt{11})^{k}\). Find the value of \(k\) for which this term takes on the greatest value. | 163 | 6.25 |
15,204 | In the base of the pyramid \( S A B C D \), there is a trapezoid \( A B C D \) with bases \( B C \) and \( A D \), where \( B C = 2 A D \). Points \( K \) and \( L \) are taken on the edges \( S A \) and \( S B \) such that \( 2 S K = K A \) and \( 3 S L = L B \). In what ratio does the plane \( K L C \) divide the edg... | 2:1 | 9.375 |
15,205 |
Calculate the arc lengths of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$ | \sqrt{2} - 1 | 51.5625 |
15,206 | Andrey found the product of all the numbers from 1 to 11 inclusive and wrote the result on the board. During a break, someone accidentally erased three digits, leaving the number $399 * 68 * *$. Help restore the missing digits without recalculating the product. | 39916800 | 26.5625 |
15,207 | Let \( p(x) \) be a polynomial with integer coefficients such that \( p(m) - p(n) \) divides \( m^2 - n^2 \) for all integers \( m \) and \( n \). If \( p(0) = 1 \) and \( p(1) = 2 \), find the largest possible value of \( p(100) \). | 10001 | 29.6875 |
15,208 | A store purchased a batch of New Year cards at a price of 21 cents each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how many cents did the store earn in total? | 470 | 27.34375 |
15,209 | For any real number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \), and let \( \{ x \} \) denote the fractional part of \( x \). Then
$$
\left\{\frac{2014}{2015}\right\}+\left\{\frac{2014^{2}}{2015}\right\}+\cdots+\left\{\frac{2014^{2014}}{2015}\right\}
$$
equals: | 1007 | 72.65625 |
15,210 | Two individuals undertake a certain task and work for an equal amount of time. $A$ misses 2 days and earns 80 forints in total, while $B$ misses 5 days and earns 63 forints. If $A$ had missed 5 days and $B$ had missed 2 days, then $A$ would earn 2 forints more than $B$. How many days did the work last? | 32 | 19.53125 |
15,211 | The minimum value of the function \( y = \sin^4{x} + \cos^4{x} + \sec^4{x} + \csc^4{x} \). | \frac{17}{2} | 57.8125 |
15,212 | How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute values of the numbers in the sequence are nonincreasing? | 128 | 42.1875 |
15,213 | There is a four-digit number \( A \). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \( A \), and the smallest possible number is 594 less than \( A \). What is \( A \)? | 1963 | 78.90625 |
15,214 | Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). A chord \( OB \) of the circle \( \Omega \) intersects the second circ... | \sqrt{5} | 59.375 |
15,215 | The numbers \( a, b, c, d \) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 240 | 0 |
15,216 | A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM? | 5/16 | 0 |
15,217 | Given a positive integer \( N \) that has exactly nine positive divisors, with three of these divisors \( a, b, \) and \( c \) satisfying
\[ a + b + c = 2017 \]
and
\[ ac = b^2. \]
Find the value of \( N \). | 82369 | 17.1875 |
15,218 | A courier set out on a moped from city $A$ to city $B$, which are 120 km apart. One hour later, a second courier on a motorcycle left from $A$. The second courier caught up to the first, delivered a message, and immediately returned to $A$ at the same speed, arriving back in $A$ at the same time the first courier reach... | 30 | 28.90625 |
15,219 | Three circles are drawn around vertices \( A, B, \) and \( C \) of a regular hexagon \( ABCDEF \) with side length 2 units, such that the circles touch each other externally. What is the radius of the smallest circle? | 2 - \sqrt{3} | 0 |
15,220 | Xiao Zhang departs from point A to point B at 8:00 AM, traveling at a speed of 60 km/h. At 9:00 AM, Xiao Wang departs from point B to point A. After arriving at point B, Xiao Zhang immediately returns along the same route and arrives at point A at 12:00 PM, at the same time as Xiao Wang. How many kilometers from point ... | 96 | 4.6875 |
15,221 | A regular tetrahedron is inscribed in a cylinder such that two opposite edges of the tetrahedron are the diameters of the cylinder's bases. Find the ratio of the volume of the cylinder to the volume of the tetrahedron. | \frac{3 \pi}{2} | 0 |
15,222 | Given a right quadrilateral pyramid \( V-ABCD \) with the height equal to half the length of \( AB \), \( M \) is the midpoint of the lateral edge \( VB \), and \( N \) is a point on the lateral edge \( VD \) such that \( DN=2VN \). Find the cosine of the angle between the skew lines \( AM \) and \( BN \). | \frac{\sqrt{11}}{11} | 0.78125 |
15,223 | Find the smallest natural number that begins with the digit five, which is reduced by four times if this five is removed from the beginning of its decimal representation and appended to its end. | 512820 | 89.0625 |
15,224 | It is known that the numbers \(x, y, z \) form an arithmetic progression in that order with a common difference \(\alpha = \arcsin \frac{\sqrt{7}}{4}\), and the numbers \(\frac{1}{\sin x}, \frac{4}{\sin y}, \frac{1}{\sin z}\) also form an arithmetic progression in that order. Find \(\sin ^{2} y\). | \frac{7}{13} | 0 |
15,225 | Five identical balls roll on a smooth horizontal surface towards each other. The velocities of the first and second are $v_{1}=v_{2}=0.5$ m/s, and the velocities of the others are $v_{3}=v_{4}=v_{5}=0.1$ m/s. The initial distances between the balls are the same, $l=2$ m. All collisions are perfectly elastic. How much t... | 10 | 7.03125 |
15,226 | Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy's treasure and halving the number of pirat... | 30 | 12.5 |
15,227 | The base of a pyramid is an equilateral triangle with a side length of 1. Out of the three vertex angles at the apex of the pyramid, two are right angles.
Find the maximum volume of the pyramid. | \frac{1}{16} | 5.46875 |
15,228 | How many solutions in integers $x$ and $y$ does the inequality
$$
|x| + |y| < 10
$$
have? | 181 | 96.09375 |
15,229 | Given that the function $f(x+1)$ is an odd function, $f(x-1)$ is an even function, and $f(0)=2$, find $f(4)=$ | -2 | 11.71875 |
15,230 | Xiao Liang went to the science museum. The science museum is located 400 meters northeast of Xiao Liang's home. When Xiao Liang left home, he walked 600 meters in the northwest direction by mistake. At this time, he did not see the science museum. He asked a lady, and she didn't know where the science museum was, so sh... | 600 | 8.59375 |
15,231 | Given a moving point \( P \) on the \( x \)-axis, \( M \) and \( N \) lie on the circles \((x-1)^{2}+(y-2)^{2}=1\) and \((x-3)^{2}+(y-4)^{2}=3\) respectively. Find the minimum value of \( |PM| + |PN| \). | 2\sqrt{10} - \sqrt{3} - 1 | 3.125 |
15,232 | Given \( 1991 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}} \), where \( \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \) are distinct non-negative integers, find the sum \( \alpha_{1} + \alpha_{2} + \cdots + \alpha_{n} \). | 43 | 43.75 |
15,233 | Find the area of a parallelogram if one of its sides is 51 and the diagonals are 40 and 74. | 1224 | 4.6875 |
15,234 |
The different ways to obtain the number of combinations of dice, as discussed in Example 4-15 of Section 4.6, can also be understood using the generating function form of Pólya’s enumeration theorem as follows:
$$
\begin{aligned}
P= & \frac{1}{24} \times\left[\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{6}\right.... | 30 | 40.625 |
15,235 |
The analysis of bank accounts revealed that the balances on each of them are more than 10 rubles. Additionally, there was a group of clients, each having the same amount of money on their account. This amount is a number consisting solely of ones. If the total amount of money on the accounts of this group of clients i... | 101 | 18.75 |
15,236 | Given a positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \).... | 4\sqrt{2} | 21.875 |
15,237 | On a cubic planet, cubic mice live only on the faces of the cube and not on the edges or vertices. The number of mice on different faces is different and on any two neighboring faces, this number differs by at least 2. What is the minimum number of cubic mice that can live on this planet if there is at least one mouse ... | 27 | 1.5625 |
15,238 | The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second attendee counts one less plate than the f... | 1006 | 64.84375 |
15,239 | On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$.
Find the ratio of the sides of rectangle $ABCD$. | \sqrt{3}/2 | 0 |
15,240 |
The medians \( A M \) and \( B E \) of triangle \( A B C \) intersect at point \( O \). Points \( O, M, E, C \) lie on the same circle. Find \( A B \) if \( B E = A M = 3 \). | 2\sqrt{3} | 39.84375 |
15,241 | Calculate the arc length of the curve defined by the equation in the rectangular coordinate system.
\[ y = \ln 7 - \ln x, \sqrt{3} \leq x \leq \sqrt{8} \] | 1 + \frac{1}{2} \ln \frac{3}{2} | 0 |
15,242 | This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours I could have exercised this month? | 22 | 3.90625 |
15,243 | From a container filled to the brim with $100\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water... | 1.75 | 7.03125 |
15,244 |
If \( x_{1} \) satisfies \( 2x + 2^{x} = 5 \) and \( x_{2} \) satisfies \( 2x + 2 \log_{2}(x - 1) = 5 \), then \( x_{1} + x_{2} = \) ? | \frac{7}{2} | 0 |
15,245 | There is a round table with 9 chairs, and 4 people are seated randomly. What is the probability that no two people are sitting next to each other? | 1/14 | 3.90625 |
15,246 | In the repeating decimal 0.2017, if the sum of all digits from the $m$-th digit to the $n$-th digit after the decimal point is 2017, find the value of $n$ when $m$ takes the minimum value. | 808 | 15.625 |
15,247 | Determine the minimum of the expression
$$
\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}
$$
under the conditions that \(ab + bc + ca > 0\), \(a + b + c = 1\), and \(a, b, c\) are distinct. | 10\sqrt{6} | 0.78125 |
15,248 | On the coordinate plane, the graph of \( y = \frac{2020}{x} \) is plotted. How many points on the graph have a tangent line that intersects both coordinate axes at points with integer coordinates? | 40 | 0 |
15,249 | Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finishe... | 120 | 14.0625 |
15,250 | Person A departs from location A to location B, while persons B and C depart from location B to location A. After A has traveled 50 kilometers, B and C start simultaneously from location B. A and B meet at location C, and A and C meet at location D. It is known that A's speed is three times that of C and 1.5 times that... | 130 | 2.34375 |
15,251 | A vessel with a capacity of 6 liters contains 4 liters of a 70% sulfuric acid solution. Another vessel of the same capacity contains 3 liters of a 90% sulfuric acid solution. Some amount of the solution is transferred from the second vessel to the first one so that the first vessel ends up with an \( r\% \) sulfuric ac... | 76 | 8.59375 |
15,252 | In triangle \( \triangle ABC \), \(E\) and \(F\) are the midpoints of \(AC\) and \(AB\) respectively, and \( AB = \frac{2}{3} AC \). If \( \frac{BE}{CF} < t \) always holds, then the minimum value of \( t \) is ______. | \frac{7}{8} | 0 |
15,253 | All the edges of the regular tetrahedron \(P-ABC\) have length \(1\). Let \( L, M, N \) be the midpoints of the edges \( PA, PB, PC \) respectively. Find the area of the cross-section of the circumsphere of the tetrahedron when cut by the plane \( LMN \). | \frac{\pi}{3} | 5.46875 |
15,254 | The base of the pyramid \( SABC \) is a triangle \( ABC \) such that \( AB = AC = 10 \) cm and \( BC = 12 \) cm. The face \( SBC \) is perpendicular to the base and \( SB = SC \). Calculate the radius of the sphere inscribed in the pyramid if the height of the pyramid is 1.4 cm. | 12/19 | 0 |
15,255 | When triangle $EFG$ is rotated by an angle $\arccos(_{1/3})$ around point $O$, which lies on side $EG$, vertex $F$ moves to vertex $E$, and vertex $G$ moves to point $H$, which lies on side $FG$. Find the ratio in which point $O$ divides side $EG$. | 3:1 | 0 |
15,256 | Piercarlo chooses \( n \) integers from 1 to 1000 inclusive. None of his integers is prime, and no two of them share a factor greater than 1. What is the greatest possible value of \( n \)? | 12 | 2.34375 |
15,257 | Come up with at least one three-digit number PAU (all digits are different), such that \((P + A + U) \times P \times A \times U = 300\). (Providing one example is sufficient) | 235 | 0 |
15,258 | How many divisors of \(88^{10}\) leave a remainder of 4 when divided by 6? | 165 | 10.15625 |
15,259 | Let \( a_{1}, a_{2}, \ldots, a_{2000} \) be real numbers in the interval \([0,1]\). Find the maximum possible value of
\[
\sum_{1 \leq i < j \leq 2000}(j - i) \left| a_{j} - a_{i} \right|
\] | 1000000000 | 28.90625 |
15,260 | A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone. | 94.5 | 0 |
15,261 | In the center of a circular field, there is a geologists' hut. From it, 6 straight roads radiate out, dividing the field into 6 equal sectors. Two geologists leave their hut at a speed of 4 km/h, each choosing a road at random. Determine the probability that the distance between them after one hour will be at least 6 k... | 0.5 | 34.375 |
15,262 | Given the triangular pyramid \( P-ABC \) with edge lengths \( PA=1 \), \( PB=2 \), and \( PC=3 \), and \( PA \perp PB \), \( PB \perp PC \), \( PC \perp PA \), determine the maximum distance from a point \( Q \) on the circumsphere of this pyramid to the face \( ABC \). | \frac{3}{7} + \frac{\sqrt{14}}{2} | 6.25 |
15,263 | Let \( O \) be a point inside \( \triangle ABC \), such that \(\overrightarrow{AO} = \frac{1}{3}\overrightarrow{AB} + \frac{1}{4}\overrightarrow{AC}\). Find the ratio of the area of \( \triangle OAB \) to the area of \( \triangle OBC \). | \frac{3}{5} | 19.53125 |
15,264 | In triangle \(ABC\), the angle bisector \(BL\) is drawn. Find the area of the triangle, given that \(AL = 2\), \(BL = \sqrt{30}\), and \(CL = 5\). | \frac{7\sqrt{39}}{4} | 1.5625 |
15,265 | Given \( P \) is the product of \( 3,659,893,456,789,325,678 \) and \( 342,973,489,379,256 \), find the number of digits of \( P \). | 34 | 100 |
15,266 | Among all victims of zombie bites, 10% are prescribed the experimental drug Undetenin to treat them. Overall, 4% of the human population suffer an adverse reaction to Undetenin. Out of all the patients being treated with Undetenin, 2% suffer an adverse reaction to the drug. What is the probability that a patient allerg... | 0.05 | 16.40625 |
15,267 | Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$. | \frac{4}{3} | 4.6875 |
15,268 | There are 1991 participants at a sporting event. Each participant knows at least $n$ other participants (the acquaintance is mutual). What is the minimum value of $n$ for which there necessarily exists a group of 6 participants who all know each other? | 1593 | 10.9375 |
15,269 | A train leaves station K for station L at 09:30, while another train leaves station L for station K at 10:00. The first train arrives at station L 40 minutes after the trains pass each other. The second train arrives at station K 1 hour and 40 minutes after the trains pass each other. Each train travels at a constant s... | 10:50 | 23.4375 |
15,270 | A 100-digit number has the form \(a = 1777 \ldots 76\) (with 98 digits of 7 in the middle). The number \(\frac{1}{a}\) is represented as an infinite repeating decimal. Find its period and justify your answer. | 99 | 33.59375 |
15,271 | A right triangle \(ABC\) is inscribed in a circle. A chord \(CM\) is drawn from the vertex \(C\) of the right angle, intersecting the hypotenuse at point \(K\). Find the area of triangle \(ABM\) if \(AK : AB = 1 : 4\), \(BC = \sqrt{2}\), and \(AC = 2\). | \frac{9}{19} \sqrt{2} | 0 |
15,272 | There are 10 boys, each with different weights and heights. For any two boys $\mathbf{A}$ and $\mathbf{B}$, if $\mathbf{A}$ is heavier than $\mathbf{B}$, or if $\mathbf{A}$ is taller than $\mathbf{B}$, we say that " $\mathrm{A}$ is not inferior to B". If a boy is not inferior to the other 9 boys, he is called a "strong... | 10 | 60.9375 |
15,273 | The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_1 = 1\), and for each \(n \in \mathbf{N}^{*}\), \(a_n\) and \(a_{n+1}\) are the roots of the equation \(x^2 + 3nx + b_n = 0\). Find \(\sum_{k=1}^{20} b_k\). | 6385 | 72.65625 |
15,274 | How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.). | 13 | 2.34375 |
15,275 | In trapezoid \(ABCD\), \(AB\) is parallel to \(DC\) and \(\angle DAF = 90^\circ\). Point \(E\) is on \(DC\) such that \(EB = BC = CE\). Point \(F\) is on \(AB\) such that \(DF\) is parallel to \(EB\). In degrees, what is the measure of \(\angle FDA\)? | 30 | 78.125 |
15,276 | The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the lengths of the lateral edges? | \sqrt{2} | 10.15625 |
15,277 | In an isosceles trapezoid, the lengths of the bases are 9 cm and 21 cm, and the height is 8 cm. Find the radius of the circumscribed circle around the trapezoid. | 10.625 | 1.5625 |
15,278 | On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either "You are a knight!" or "You are a liar!". The phrase "You are a liar!" was said exactly 230 time... | 526 | 26.5625 |
15,279 | Once upon a time, a team of Knights and a team of Liars met in the park and decided to ride a circular carousel that can hold 40 people (the "Chain" carousel, where everyone sits one behind the other). When they took their seats, each person saw two others: one in front and one behind. Each person then said, "At least ... | 26 | 0 |
15,280 | In trapezoid \(ABCD\), the sides \(AB\) and \(CD\) are parallel and \(CD = 2AB\). Points \(P\) and \(Q\) are chosen on sides \(AD\) and \(BC\), respectively, such that \(DP : PA = 2\) and \(BQ : QC = 3 : 4\). Find the ratio of the areas of quadrilaterals \(ABQP\) and \(CDPQ\). | 19/44 | 0 |
15,281 | Losyash is walking to Sovunya's house along the river at a speed of 4 km/h. Every half hour, he launches paper boats that travel towards Sovunya at a speed of 10 km/h. At what time intervals do the boats arrive at Sovunya's house? (Provide a complete solution, not just the answer.) | 18 | 17.96875 |
15,282 | Given that the area of the parallelogram $ABCD$ is $240$, $E$ and $H$ are the midpoints of sides $AD$ and $AB$ respectively. Point $G$ is on side $BC$ such that $BG = 2GC$, and point $F$ is on side $CD$ such that $DF = 3FC$. Point $K$ is on side $AC$ such that the area of $\triangle EKF$ is 33. Find the area of $\trian... | 32 | 6.25 |
15,283 | A cone is inscribed in a sphere such that the slant height of the cone is equal to the diameter of the base. Find the ratio of the total surface area of the cone to the surface area of the sphere. | 9/16 | 4.6875 |
15,284 | Points \( P \) and \( Q \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) such that \( AP:PB = 1:4 \) and \( AQ:QC = 3:1 \). Point \( M \) is chosen randomly on side \( BC \). Find the probability that the area of triangle \( ABC \) exceeds the area of triangle \( PQM \) by no more than two times.... | 13/40 | 3.90625 |
15,285 | Divide all coins into two parts of 20 coins each and weigh them. Since the number of fake coins is odd, one of the piles will be heavier. Thus, there is at most one fake coin in that pile. Divide it into two piles of 10 coins and weigh them. If the balance is even, then all 20 coins weighed are genuine. If one of the p... | 16 | 12.5 |
15,286 | Given that real numbers \( x \) and \( y \) satisfy the equation \( 1 + \cos^{2}(x + y - 1) = \frac{x^{2} + y^{2} + 2(x + 1)(1 - y)}{x - y + 1} \), find the minimum value of \( xy \). | \frac{1}{4} | 67.1875 |
15,287 | Given \( n = 7^{3} \times 11^{2} \times 13^{4} \), find the number of integers that are divisors of \( n \). | 60 | 100 |
15,288 | On the side \( BC \) of an equilateral triangle \( ABC \), points \( K \) and \( L \) are marked such that \( BK = KL = LC \). On the side \( AC \), point \( M \) is marked such that \( AM = \frac{1}{3} AC \). Find the sum of the angles \( \angle AKM \) and \( \angle ALM \). | 30 | 0 |
15,289 | Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3}+y^{2}=1 \). A line \( l \) with a positive slope passes through point \( F \) and intersects \( E \) at points \( A \) and \( B \). Through points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn such that \( AM \perp l \) and \( BN \perp l \)... | \sqrt{6} | 5.46875 |
15,290 | Let \( n = 1990 \), then evaluate the expression \(\frac{1}{2^{n}}\left(1 - 3 C_{n}^{2} + 3^{2} C_{n}^{4} - 3^{3} C_{n}^{6} + \cdots + 3^{994} C_{n}^{1988} - 3^{995} C_{n}^{1990}\right)\). | -\frac{1}{2} | 89.0625 |
15,291 | In quadrilateral \(ABCD\), \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), \(\angle CBD = 40^\circ\). Find \(\angle ACD\). | 22 | 9.375 |
15,292 | An engraver makes plates with letters. He engraves the same letters in the same amount of time, but different letters may take different times. On two plates, "ДОМ МОДЫ" (DOM MODY) and "ВХОД" (VKHOD), together he spent 50 minutes, and one plate "В ДЫМОХОД" (V DYMOHOD) took him 35 minutes. How much time will it take him... | 20 | 10.9375 |
15,293 | Let the function \( f(x) \) be defined on \([0,1]\), satisfying \( f(0) = f(1) \) and for any \( x, y \in [0,1] \) having \( |f(x) - f(y)| < |x - y| \). Find the smallest real number \( m \) such that for any function \( f(x) \) meeting the above conditions and for any \( x, y \in [0,1] \), we have \( |f(x) - f(y)| < m... | \frac{1}{2} | 63.28125 |
15,294 | The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face? | 16 | 88.28125 |
15,295 |
How many solutions in natural numbers \(x, y\) does the system of equations have?
$$
\left\{\begin{array}{l}
\text{GCD}(x, y) = 20! \\
\text{LCM}(x, y) = 30!
\end{array}\right.
$$
(where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\)) | 1024 | 22.65625 |
15,296 | The points \( M \) and \( N \) are chosen on the angle bisector \( AL \) of a triangle \( ABC \) such that \( \angle ABM = \angle ACN = 23^\circ \). \( X \) is a point inside the triangle such that \( BX = CX \) and \( \angle BXC = 2\angle BML \). Find \( \angle MXN \). | 46 | 9.375 |
15,297 | Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces
$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$ | 4\pi | 14.0625 |
15,298 | Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}-14 x-16 y+61}{\left(4-16 x^{2}-8 x y-y^{2}\right)^{7 / 2}}\). If necessary, round your answer to the nearest hundredth. | 0.44 | 1.5625 |
15,299 | Does there exist a six-digit number that increases by 6 times if its first three digits, without changing their order, are moved to the end of the number? | 142857 | 67.1875 |
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