Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,200 | In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on sides $YZ$, $XZ$, and $XY$ respectively. Given that $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\frac{XP}{PX'} + \frac{YP}{PY'} + \frac{ZP}{PZ'} = 100$, find $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$. | 98 | 0.78125 |
14,201 | When the number of repeated experiments is large enough, probability can be estimated using frequency. The mathematician Pearson once tossed a fair coin 24,000 times in an experiment. The number of times the coin landed heads up was 12,012 times, with a frequency of 0.5005. Therefore, the probability of a fair coin lan... | 0.5005 | 15.625 |
14,202 | Find all integers \( n \) such that \( n^{4} + 6 n^{3} + 11 n^{2} + 3 n + 31 \) is a perfect square. | 10 | 85.9375 |
14,203 | How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \] | 12 | 24.21875 |
14,204 | Let $a > 3$. Determine the value of $a$ given that $f(g(a)) = 16$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$. | \sqrt{\sqrt{6} + 6} | 0 |
14,205 | Find the number of distinct arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3. | 1728 | 53.90625 |
14,206 | Given two four-digit numbers \( M \) and \( N \) which are reverses of each other, and have \( q^{p}-1 \) identical positive divisors, \( M \) and \( N \) can be factorized into prime factors as \( p q^{q} r \) and \( q^{p+q} r \) respectively, where \( p \), \( q \), and \( r \) are prime numbers. Find the value of \(... | 1998 | 0.78125 |
14,207 | In a computer game, a player can choose to play as one of three factions: \( T \), \( Z \), or \( P \). There is an online mode where 8 players are divided into two teams of 4 players each. How many total different matches are possible, considering the sets of factions? The matches are considered different if there is ... | 120 | 0 |
14,208 | Given the equation \(x^2 + y^2 = 2(|x| + |y|)\), calculate the area of the region enclosed by its graph. | 2\pi | 28.125 |
14,209 | What is the smallest positive angle \( x \) for which
\[
2^{\sin^2 x} \cdot 4^{\cos^2 x} \cdot 2^{\tan x} = 8
\] | 60 | 13.28125 |
14,210 | Given the real numbers $a, x, y$ that satisfy the equation:
$$
x \sqrt{a(x-a)}+y \sqrt{a(y-a)}=\sqrt{|\lg (x-a)-\lg (a-y)|},
$$
find the value of the algebraic expression $\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$. | \frac{1}{3} | 7.8125 |
14,211 | Given the function $f(x)=ax^{3}-4x+4$, where $a\in\mathbb{R}$, $f′(x)$ is the derivative of $f(x)$, and $f′(1)=-3$.
(1) Find the value of $a$;
(2) Find the extreme values of the function $f(x)$. | -\frac{4}{3} | 25 |
14,212 | In different historical periods, the conversion between "jin" and "liang" was different. The idiom "ban jin ba liang" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in today's terms. Currently, 10 liang equals 1 jin, ... | 2800 | 25 |
14,213 | Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$. | 502 | 13.28125 |
14,214 | Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence $\{a_n\}$ is $S_n$, and $\frac{a_{n+1}}{a_n} < 1$, if $a_3 + a_5 = 20$ and $a_2 \cdot a_6 = 64$, calculate $S_6$. | 126 | 43.75 |
14,215 | In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$. | 147 | 20.3125 |
14,216 | At the first site, high-class equipment was used, while at the second site, first-class equipment was used, with the amount of high-class equipment being less than that of the first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment that ended up at... | 17 | 0 |
14,217 | 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 to Sovunya at a speed of 10 km/h. What is the time interval at which the boats arrive at Sovunya's house? | 18 | 14.84375 |
14,218 | Natural numbers \( x, y, z \) are such that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \cdot \operatorname{LCM}(\operatorname{GCD}(x, y), z) = 1400 \).
What is the maximum value that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \) can take? | 10 | 3.125 |
14,219 | The coach of the math training team needs to photocopy a set of materials for 23 team members. The on-campus copy shop charges 1.5 yuan per page for the first 300 pages and 1 yuan per page for any additional pages. The cost of photocopying these 23 sets of materials together is exactly 20 times the cost of photocopying... | 950 | 10.15625 |
14,220 | Given triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ lying on the line $x + 2y = 8$, find the area of triangle $ABC$. | 4.5 | 3.90625 |
14,221 | Given that \( O \) is the circumcenter of \(\triangle ABC\), and \( 3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \overrightarrow{0} \), find the value of \( \cos \angle BAC \). | \frac{\sqrt{10}}{10} | 10.9375 |
14,222 | Right triangles \(ABC\) and \(ABD\) share a common hypotenuse \(AB = 5\). Points \(C\) and \(D\) are located on opposite sides of the line passing through points \(A\) and \(B\), with \(BC = BD = 3\). Point \(E\) lies on \(AC\), and \(EC = 1\). Point \(F\) lies on \(AD\), and \(FD = 2\). Find the area of the pentagon \... | 9.12 | 0 |
14,223 | Months of the year are usually labeled numerically by '01' for January, '02' for February, and so on, through to '12' for December. Lydia notices that during January, the number of letters in the name of the month is greater than the month's numerical label (i.e., $7>1$). For how many days during 2024 will the date hav... | 121 | 25.78125 |
14,224 | Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([1, 3]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{3+\sqrt{7}}{2}\right)) \ldots) \). If necessary, round your answer to two decimal places. | 0.18 | 3.125 |
14,225 | What is the sum and the average of the prime numbers between 20 and 40? | 30 | 0.78125 |
14,226 | A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. | 64 | 0 |
14,227 | In $\triangle ABC$, where $A > B > C$, if $2 \cos 2B - 8 \cos B + 5 = 0$, $\tan A + \tan C = 3 + \sqrt{3}$, and the height $CD$ from $C$ to $AB$ is $2\sqrt{3}$, then find the area of $\triangle ABC$. | 12 - 4\sqrt{3} | 2.34375 |
14,228 | The number of books issued from the library to readers constitutes $\frac{1}{16}$ of the number of books on the shelves. After transferring 2000 books from the library to the reading room, the number of books absent from the shelves became $\frac{1}{15}$ of the number of books remaining on the shelves. How many books d... | 544000 | 0.78125 |
14,229 | A $2018 \times 2018$ square was cut into rectangles with integer side lengths. Some of these rectangles were used to form a $2000 \times 2000$ square, and the remaining rectangles were used to form a rectangle whose length differs from its width by less than 40. Find the perimeter of this rectangle. | 1076 | 12.5 |
14,230 | Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the train 30 seconds to completely pass Car A and... | 1.25 | 18.75 |
14,231 | Let the width and length of the pan be $w$ and $l$ respectively. If the number of interior pieces is twice the number of perimeter pieces, then find the greatest possible value of $w \cdot l$. | 294 | 0 |
14,232 | Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots. | \frac{4\sqrt{6}}{9} | 24.21875 |
14,233 | Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that
\[
a^2 + b^2 + c^2 + d^2 = 9,
\]
\[
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.
\] | 15 | 0.78125 |
14,234 | Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After meeting, Person B also doubles their speed ... | 420 | 0.78125 |
14,235 | In triangle \(ABC\), side \(BC\) is equal to 5. A circle passes through vertices \(B\) and \(C\) and intersects side \(AC\) at point \(K\), where \(CK = 3\) and \(KA = 1\). It is known that the cosine of angle \(ACB\) is \(\frac{4}{5}\). Find the ratio of the radius of this circle to the radius of the circle inscribed ... | \frac{10\sqrt{10} + 25}{9} | 0 |
14,236 | How many four-digit numbers, formed using the digits 0, 1, 2, 3, 4, 5 without repetition, are greater than 3410? | 132 | 2.34375 |
14,237 | In parallelogram \(ABCD\), \(OE = EF = FD\). The area of the parallelogram is 240 square centimeters. The area of the shaded region is _______ square centimeters. | 20 | 4.6875 |
14,238 | Given $x \gt 0$, $y \gt 0$, $x+2y=1$, calculate the minimum value of $\frac{{(x+1)(y+1)}}{{xy}}$. | 8+4\sqrt{3} | 1.5625 |
14,239 | Find the number of eight-digit numbers whose product of digits equals 1400. The answer must be presented as an integer. | 5880 | 0 |
14,240 | How many factors are there in the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ if we know that it ends with 1981 zeros? | 7935 | 77.34375 |
14,241 | What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other? | 16 | 17.1875 |
14,242 | In the equation, $\overline{\mathrm{ABCD}}+\overline{\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\qquad$? | 31 | 2.34375 |
14,243 | How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3? | 10 | 75.78125 |
14,244 | A square field is enclosed by a wooden fence, which is made of 10-meter-long boards placed horizontally. The height of the fence is four boards. It is known that the number of boards in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions of the field. | 16000 | 3.90625 |
14,245 | Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get? | 28077 | 27.34375 |
14,246 | Let $f(x) = |3\{x\} - 1.5|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation \[nf(xf(x)) = 2x\] has at least $1000$ real solutions. | 250 | 0.78125 |
14,247 | Point \( D \) lies on side \( BC \) of triangle \( ABC \), and point \( O \) is located on segment \( AD \) with \( AO : OD = 9 : 4 \). A line passing through vertex \( B \) and point \( O \) intersects side \( AC \) at point \( E \) with \( BO : OE = 5 : 6 \). Determine the ratio in which point \( E \) divides side \(... | 21 : 44 | 12.5 |
14,248 | In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.
After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$? | 34 | 0.78125 |
14,249 | Given real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, m]$ for some positive integer $m$. The probability that no two of $x$, $y$, and $z$ are within 2 units of each other is greater than $\frac{1}{2}$. Determine the smallest possible value of $m$. | 16 | 3.90625 |
14,250 | Given that \( p \) is a prime number, the decimal part of \( \sqrt{p} \) is \( x \). The decimal part of \( \frac{1}{x} \) is \( \frac{\sqrt{p} - 31}{75} \). Find all prime numbers \( p \) that satisfy these conditions. | 2011 | 1.5625 |
14,251 | In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\). | \sqrt{3} | 21.875 |
14,252 | Please write an irrational number that is smaller than $3$. | \sqrt{2} | 77.34375 |
14,253 | The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure? | 100 | 16.40625 |
14,254 | Given \( x \in [0, 2\pi] \), determine the maximum value of the function
\[
f(x) = \sqrt{4 \cos^2 x + 4 \sqrt{6} \cos x + 6} + \sqrt{4 \cos^2 x - 8 \sqrt{6} \cos x + 4 \sqrt{2} \sin x + 22}.
\] | 2(\sqrt{6} + \sqrt{2}) | 0 |
14,255 | Given that Steve's empty swimming pool holds 30,000 gallons of water when full and will be filled by 5 hoses, each supplying 2.5 gallons of water per minute, calculate the time required to fill the pool. | 40 | 20.3125 |
14,256 | A right triangle has integer side lengths. One of its legs is 1575 units shorter than its hypotenuse, and the other leg is less than 1991 units. Find the length of the hypotenuse of this right triangle. | 1799 | 46.875 |
14,257 | Let \( f(x) = x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \), and \( g(x) = 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \). Find \( \int_{0}^{\infty} f(x) g(x) \, dx \). | \sqrt{e} | 0 |
14,258 | A rod with a length of four meters has weights attached as follows: $20 \mathrm{~kg}$ at one end, and at distances of one, two, and three meters from that end, weights of $30, 40, 50$ $\mathrm{kg}$ respectively. Additionally, a weight of $60 \mathrm{~kg}$ is attached at the other end. Where should the rod be supported ... | 2.5 | 74.21875 |
14,259 | Laura conducted a survey in her neighborhood about pest awareness. She found that $75.4\%$ of the people surveyed believed that mice caused electrical fires. Of these, $52.3\%$ incorrectly thought that mice commonly carried the Hantavirus. Given that these 31 people were misinformed, how many total people did Laura sur... | 78 | 0 |
14,260 | As shown in the diagram, in the square \(ABCD\), \(AB = 2\). Draw an arc with center \(C\) and radius equal to \(CD\), and another arc with center \(B\) and radius equal to \(BA\). The two arcs intersect at \(E\). What is the area of the sector \(BAE\)? | \frac{\pi}{3} | 10.9375 |
14,261 | Let \( a_{1}, a_{2}, \cdots, a_{n} \) be distinct positive integers such that \( a_{1} + a_{2} + \cdots + a_{n} = 2014 \), where \( n \) is some integer greater than 1. Let \( d \) be the greatest common divisor of \( a_{1}, a_{2}, \cdots, a_{n} \). For all values of \( n \) and \( a_{1}, a_{2}, \cdots, a_{n} \) that s... | 530 | 0.78125 |
14,262 | Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score of the four students is 0, calculate the nu... | 44 | 0 |
14,263 | Vitya Perestukin always incorrectly calculates percentages during surveys: he divides the number of respondents who answered a certain way by the number of all remaining respondents. For instance, in the survey "What is your name?" conducted among 7 Annas, 9 Olgas, 8 Julias, Vitya calculated 50% Julias.
Vitya conducte... | 110 | 0.78125 |
14,264 | Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), \( M \) is the center of the face \( A B B_{1} A_{1} \), \( N \) is a point on the edge \( B_{1} C_{1} \), \( L \) is the midpoint of \( A_{1} B_{1} \); \( K \) is the foot of the perpendicular dropped from \( N \) to \( BC_{1} \). In what ratio does point \( N \) div... | \sqrt{2} + 1 | 0 |
14,265 | Given a rectangle \(ABCD\), a circle intersects the side \(AB\) at points \(K\) and \(L\), and the side \(CD\) at points \(M\) and \(N\). Find the length of segment \(MN\) if \(AK = 10\), \(KL = 17\), and \(DN = 7\). | 23 | 9.375 |
14,266 | Given a point P $(x, y)$ on the circle $x^2 - 4x - 4 + y^2 = 0$, find the maximum value of $x^2 + y^2$. | 12 + 8\sqrt{2} | 96.875 |
14,267 | Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$
Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$. | 3 - 2\sqrt{2} | 0 |
14,268 |
There are 256 players in a tennis tournament who are ranked from 1 to 256, with 1 corresponding to the highest rank and 256 corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability \(\frac{3}{5}\).
In each round of the tournamen... | 103 | 3.90625 |
14,269 | On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour? | 23 | 63.28125 |
14,270 |
In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\). | 10 | 23.4375 |
14,271 | Given the function \( f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\} \), and for each \( k=1,2, \cdots, 9 \), it is true that \( |f(k+1)-f(k)| \geq 3 \). Find the number of functions \( f \) that satisfy these conditions. | 288 | 46.875 |
14,272 | There are \( n \) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \( n \). | 52 | 24.21875 |
14,273 | Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___. | -1 | 22.65625 |
14,274 | In a trapezoid, the smaller base is 1 decimeter, and the angles adjacent to it are $135^{\circ}$. The angle between the diagonals, opposite to the base, is $150^{\circ}$. Find the area of the trapezoid. | 0.5 | 2.34375 |
14,275 | Given that in square ABCD, AE = 3EC and BF = 2FB, and G is the midpoint of CD, find the ratio of the area of triangle EFG to the area of square ABCD. | \frac{1}{24} | 0 |
14,276 | 132009 students are taking a test which comprises ten true or false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers. | 513 | 2.34375 |
14,277 | The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables. | 79 | 53.125 |
14,278 | On the side \( AD \) of the rhombus \( ABCD \), a point \( M \) is taken such that \( MD = 0.3 \, AD \) and \( BM = MC = 11 \). Find the area of triangle \( BCM \). | 20\sqrt{6} | 0.78125 |
14,279 | Solve for $X$ if $\sqrt[4]{X^5} = 32\sqrt[16]{32}$. | 16\sqrt[4]{2} | 18.75 |
14,280 | Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is not known which coin is not symmetric, but it is given that the non-symmetric coin lands heads with a probability of $p = 0.6$.
Billy Bones flipped the gold coin, and it landed heads immediately. Then Billy B... | 0.6 | 0.78125 |
14,281 | Factorize the number \( 989 \cdot 1001 \cdot 1007 + 320 \) into prime factors. | 991 * 997 * 1009 | 0 |
14,282 | Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)? | 47/32 | 0 |
14,283 | Mr. Pipkins said, "I was walking along the road at a speed of $3 \frac{1}{2}$ km/h when suddenly a car sped past me, almost knocking me off my feet."
"What was its speed?" his friend asked.
"I can tell you now. From the moment it sped past me until it disappeared around the bend, I took 27 steps. Then I continued wal... | 21 | 5.46875 |
14,284 | On the sides \( AB \) and \( AD \) of a square \( ABCD \) with side length 108, semicircles are constructed inward. Find the radius of a circle that touches one side of the square and the semicircles: one externally and the other internally. | 27 | 14.0625 |
14,285 | Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull? | \frac{\sqrt{3}}{2} | 0 |
14,286 | Each of $b_1, b_2, \dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of
\[\sum_{1 \le i < j \le 150} b_i b_j.\] | 38 | 53.90625 |
14,287 | A sequence of numbers \(a_1, a_2, \cdots, a_n, \cdots\) is defined. Let \(S(a_i)\) be the sum of all the digits of \(a_i\). For example, \(S(22) = 2 + 2 = 4\). If \(a_1 = 2017\), \(a_2 = 22\), and \(a_n = S(a_{n-1}) + S(a_{n-2})\), what is the value of \(a_{2017}\)? | 10 | 63.28125 |
14,288 | Triangle \(A B C\) has side lengths \(A B = 65\), \(B C = 33\), and \(A C = 56\). Find the radius of the circle tangent to sides \(A C\) and \(B C\) and to the circumcircle of triangle \(A B C\). | 24 | 7.03125 |
14,289 | If $e^{i \theta} = \frac{3 + i \sqrt{8}}{4},$ then find $\sin 6 \theta.$ | -\frac{855 \sqrt{2}}{1024} | 6.25 |
14,290 | Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\overline{c d a b}$ and $\overline{a b c d}$ is a four-digit number of the form $\overline{p q e f}$. If $\overline{e f}$ is a perfect square and $\overline{p q}$ is not divisible by 5, determine ... | 1983 | 23.4375 |
14,291 | Households A, B, and C plan to subscribe to newspapers. There are 5 different types of newspapers available. Each household subscribes to two different newspapers. It is known that each pair of households shares exactly one common newspaper. How many different subscription ways are there for the three households? | 180 | 1.5625 |
14,292 | Determine the maximum number of different sets consisting of three terms that form arithmetic progressions and can be chosen from a sequence of real numbers \( a_1, a_2, \ldots, a_{101} \), where
\[
a_1 < a_2 < a_3 < \cdots < a_{101} .
\] | 2500 | 15.625 |
14,293 | The bases of a trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\circ}$. Find the perimeter of the trapezoid. | 22 | 7.03125 |
14,294 | Let $s$ be a table composed of positive integers (the table may contain the same number). Among the numbers in $s$ there is the number 68. The arithmetic mean of all numbers in $s$ is 56. However, if 68 is removed, the arithmetic mean of the remaining numbers drops to 55. What is the largest possible number that can ap... | 649 | 82.03125 |
14,295 | Find the sum of all roots of the equation:
$$
\begin{gathered}
\sqrt{2 x^{2}-2024 x+1023131} + \sqrt{3 x^{2}-2025 x+1023132} + \sqrt{4 x^{2}-2026 x+1023133} = \\
= \sqrt{x^{2}-x+1} + \sqrt{2 x^{2}-2 x+2} + \sqrt{3 x^{2}-3 x+3}
\end{gathered}
$$ | 2023 | 17.96875 |
14,296 | Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. L... | 13213 | 0 |
14,297 | Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11. | 98765 | 28.90625 |
14,298 | What is the sum and product of the distinct prime factors of 420? | 210 | 39.0625 |
14,299 | Let $\sigma(n)$ be the number of positive divisors of $n$ , and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$ . By convention, $\operatorname{rad} 1 = 1$ . Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\... | 164 | 0.78125 |
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