Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
13,400 | An ant situated at point \( A \) decides to walk 1 foot east, then \( \frac{1}{2} \) foot northeast, then \( \frac{1}{4} \) foot east, then \( \frac{1}{8} \) foot northeast, then \( \frac{1}{16} \) foot east, and so on (that is, the ant travels alternately between east and northeast, and the distance travelled is decre... | \frac{2}{3} \sqrt{2 \sqrt{2}+5} | 0 |
13,401 | Car A and Car B start simultaneously from points $A$ and $B$ respectively, traveling towards each other. The initial speed ratio of car A to car B is 5:4. Shortly after departure, car A has a tire blowout, stops to replace the tire, and then resumes the journey, increasing its speed by $20\%$. They meet at the midpoint... | 52 | 0 |
13,402 | Find all values of \(a\) such that the roots \(x_1, x_2, x_3\) of the polynomial \(x^3 - 6x^2 + ax + a\) satisfy \((x_1 - 3)^2 + (x_2 - 3)^3 + (x_3 - 3)^3 = 0\). | -9 | 27.34375 |
13,403 | In triangle \(ABC\), the angle bisector of \(\angle ABC\) intersects side \(AC\) at point \(K\). It is given that \(BC = 2\), \(KC = 1\), and \(BK = \frac{3\sqrt{2}}{2}\). Find the area of triangle \(ABC\). | \frac{15 \sqrt{7}}{16} | 53.125 |
13,404 | We write the following equation: \((x-1) \ldots (x-2020) = (x-1) \ldots (x-2020)\). What is the minimal number of factors that need to be erased so that there are no real solutions? | 1010 | 6.25 |
13,405 | A large rectangular garden contains two flower beds in the shape of congruent isosceles right triangles and a trapezoidal playground. The parallel sides of the trapezoid measure $30$ meters and $46$ meters. Determine the fraction of the garden occupied by the flower beds.
A) $\frac{1}{10}$
B) $\frac{1}{11}$
C) $\frac{4... | \frac{4}{23} | 32.8125 |
13,406 | The clock runs 3 minutes fast every 24 hours. The clock was set accurately. After what minimum time will the clock show the correct time again? | 240 | 36.71875 |
13,407 | At McDonald's restaurants, we can order Chicken McNuggets in packages of 6, 9, or 20 pieces. (For example, we can order 21 pieces because $21=6+6+9$, but there is no way to get 19 pieces.) What is the largest number of pieces that we cannot order? | 43 | 85.9375 |
13,408 | In a $7 \times 7$ grid, choose $k$ cells such that the centers of any 4 chosen cells do not form the vertices of a rectangle. Find the maximum value of $k$ that satisfies this condition. | 21 | 2.34375 |
13,409 | The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.
$$
AB = 18, EO = 7, AE = 2 BE, R = ?
$$ | 11 | 2.34375 |
13,410 | 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 two roots of the equation \(x^{2}+3nx+b_{n}=0\). Find \(\sum_{k=1}^{20} b_{k}\). | 6385 | 66.40625 |
13,411 | Three vertices of a rectangle are at points $(2, 7)$, $(13, 7)$, and $(13, -6)$. What is the area of the intersection between this rectangle and the circular region described by equation $(x - 2)^2 + (y + 6)^2 = 25$? | \frac{25}{4}\pi | 0 |
13,412 | If three different lines $x+y=1$, $x-y=1$, and $ax+y=1$ cannot form a triangle, then the value of the real number $a$ is. | -1 | 9.375 |
13,413 | For a $5 \times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this? | 1440 | 0.78125 |
13,414 | Given \(0 < a < \sqrt{3} \cos \theta\), \(\theta \in \left[-\frac{\pi}{4}, \frac{\pi}{3}\right]\), find the minimum value of \(f(a, \theta) = \cos^3 \theta + \frac{4}{3a \cos^2 \theta - a^3}\). | \frac{17 \sqrt{2}}{4} | 3.125 |
13,415 | Calculate: $5 \times 13 \times 31 \times 73 \times 137$ | 20152015 | 56.25 |
13,416 | Angles $C$ and $D$ are supplementary. If the measure of angle $C$ is $12$ times angle $D$, what is the measure of angle $C$? | 166.15 | 0.78125 |
13,417 | Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of the original square larger than the area o... | 81 | 14.84375 |
13,418 | For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary. | 0.79 | 29.6875 |
13,419 | In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Furthermore, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "To my left is a g... | 176 | 71.875 |
13,420 | Let \(\left\{a_{n}\right\}\) be a sequence of positive integers such that \(a_{1}=1\), \(a_{2}=2009\) and for \(n \geq 1\), \(a_{n+2} a_{n} - a_{n+1}^{2} - a_{n+1} a_{n} = 0\). Determine the value of \(\frac{a_{993}}{100 a_{991}}\). | 89970 | 18.75 |
13,421 | In how many ways can 10 fillér and 50 fillér coins be placed side by side (with all centers on a straight line) to cover a $1 \mathrm{~m}$ long segment (not more), using at least 50 coins, and considering the order of the two types of coins? (Coins of the same value are not distinguished. The diameter of the 10 fillér ... | 270725 | 32.03125 |
13,422 | A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road, which received the mail and turned back i... | 40 | 10.9375 |
13,423 | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $c=4$. Point $D$ is on $CD\bot AB$, and $c\cos C\cos \left(A-B\right)+4=c\sin ^{2}C+b\sin A\sin C$. Find the maximum value of the length of segment $CD$. | 2\sqrt{3} | 14.84375 |
13,424 |
The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.
How many fruits in total grew on the miracle tree? | 30 | 3.125 |
13,425 | In the quadrilateral \(ABCD\), it is known that \(\angle ABD = \angle ACD = 45^\circ\), \(\angle BAC = 30^\circ\), and \(BC = 1\). Find \(AD\). | \sqrt{2} | 13.28125 |
13,426 | Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer. | 1159 | 67.96875 |
13,427 | Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$. | 61 | 79.6875 |
13,428 | Calculate the definite integral:
$$
\int_{0}^{\sqrt{2} / 2} \frac{x^{4} \cdot d x}{\sqrt{\left(1-x^{2}\right)^{3}}}
$$ | \frac{5}{4} - \frac{3\pi}{8} | 38.28125 |
13,429 | There are 3 screw-in light bulbs and 5 bayonet light bulbs in the box, and light bulbs are randomly drawn without replacement until the 5th light bulb is drawn to have all the screw-in light bulbs. Calculate the probability of this event. | \frac{3}{28} | 0.78125 |
13,430 | When \( N \) takes all the values from 1, 2, 3, \ldots, 2015, how many numbers of the form \( 3^{n} + n^{3} \) are divisible by 7? | 288 | 85.9375 |
13,431 | Given two integers $a$ and $b$, if they are not coprime and neither is a multiple of the other, they are called a "league" pair. Let $A$ be an $n$-element subset of the set $M = \{1, 2, \cdots, 2014\}$ such that every pair of numbers in $A$ is a league pair. Determine the maximum value of $n$. | 504 | 10.15625 |
13,432 | Find the value of \(\cos ^{5} \frac{\pi}{9}+\cos ^{5} \frac{5 \pi}{9}+\cos ^{5} \frac{7 \pi}{9}\). | \frac{15}{32} | 15.625 |
13,433 | In an arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$, and $a_{11} > |a_{10}|$. The maximum negative value of the partial sum $S_n$ of the first $n$ terms of the sequence $\{a_n\}$ is the sum of the first ______ terms. | 19 | 7.03125 |
13,434 | Given a sequence of length 15 consisting of zeros and ones, find the number of sequences that have either all zeros consecutive or all ones consecutive, but not both. | 238 | 5.46875 |
13,435 | Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games, meaning into unordered pairs of unordered pairs of people? | 315 | 4.6875 |
13,436 | Given a square \(ABCD\) with side length 2, \(E\) is the midpoint of \(AB\). The square is folded along lines \(EC\) and \(ED\) so that \(AE\) coincides with \(BE\), and point \(A\) coincides with point \(B\), named point \(O\). Calculate the volume of the tetrahedron \(O-CDE\). | \frac{\sqrt{3}}{3} | 5.46875 |
13,437 | Given the relationship $P={P}_{0}{e}^{-kt}$ between the concentration of toxic and harmful substances $P$ (unit: $mg/L$) in the exhaust gas and time $t$ (unit: $h$) during the filtration process, determine the percentage of the original toxic and harmful substances that will remain after $5$ hours, given that $20\%$ of... | 57\% | 0 |
13,438 |
In an olympiad, 2006 students participated. It was found that a student, Vasia, solved only one out of the six problems. Additionally, the number of participants who solved at least 1 problem is 4 times greater than those who solved at least 2 problems;
the number who solved at least 2 problems is 4 times greater than... | 982 | 41.40625 |
13,439 | A basketball championship has been played in a round-robin format, with each pair of teams playing twice and no ties (overtime is played until one team wins). The winner of a match receives 2 points, and the loser receives 1 point. At the end of the championship, the sum of the points obtained by all the teams except t... | 39 | 0.78125 |
13,440 | Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM:MC = 1:2 \). The 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. | \frac{11}{24} | 6.25 |
13,441 | There are 5 girls sitting in a row on five chairs, and opposite them, on five chairs, there are 5 boys sitting. It was decided that the boys would switch places with the girls. In how many ways can this be done? | 14400 | 92.1875 |
13,442 | Given a line $l$ passing through point $A(1,1)$ with a slope of $-m$ ($m>0$) intersects the x-axis and y-axis at points $P$ and $Q$, respectively. Perpendicular lines are drawn from $P$ and $Q$ to the line $2x+y=0$, and the feet of the perpendiculars are $R$ and $S$. Find the minimum value of the area of quadrilateral ... | 3.6 | 0 |
13,443 | In rectangle $ABCD$, $AB=8$ and $BC=6$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF=3$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$. | 24 | 1.5625 |
13,444 | There are 55 points marked on a plane: the vertices of a regular 54-gon and its center. Petya wants to color a set of three marked points in red so that the colored points form the vertices of a regular triangle. In how many ways can Petya do this? | 72 | 7.8125 |
13,445 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$ respectively. Given that angle $A = \frac{\pi}{4}$, $\sin A + \sin (B - C) = 2\sqrt{2}\sin 2C$, and the area of triangle $ABC$ is $1$. Find the length of side $BC$. | \sqrt{5} | 5.46875 |
13,446 | Find the smallest value of $n$ for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds $3^{2007}$ . | 2000 | 3.125 |
13,447 | The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration. | 15 | 4.6875 |
13,448 | Given two groups of numerical sequences, each containing 15 arithmetic progressions with 10 terms each. The first terms of the progressions in the first group are $1, 2, 3, \ldots, 15$, and their differences are respectively $2, 4, 6, \ldots, 30$. The second group of progressions has the same first terms $1, 2, 3, \ldo... | 160/151 | 94.53125 |
13,449 | Points \( M \) and \( N \) are the midpoints of the sides \( AC \) and \( CB \) of the isosceles triangle \( ACB \). Point \( L \) lies on the median \( BM \) such that \( BL : BM = 4 : 9 \). A circle with center at point \( L \) is tangent to the line \( MN \) and intersects the line \( AB \) at points \( Q \) and \( ... | 2(2 + \sqrt{13}) | 0 |
13,450 | Triangle \( ABC \) has a right angle at \( B \). Point \( D \) lies on side \( BC \) such that \( 3 \angle BAD = \angle BAC \). Given \( AC = 2 \) and \( CD = 1 \), compute \( BD \). | \frac{3}{8} | 0.78125 |
13,451 | In triangle ABC, point D is on line segment AB such that AD bisects $\angle CAB$. Given that $BD = 36$, $BC = 45$, and $AC = 27$, find the length of segment $AD$. | 24 | 0.78125 |
13,452 | A square piece of paper has a side length of 1. It is folded such that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Given $C'D = \frac{1}{4}$, find the perimeter of triangle $\bigtriangleup AEC'$.
**A)** $\frac{25}{12}$
**B)** $\frac{33}{12... | \frac{10}{3} | 27.34375 |
13,453 | Let \(a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of all positive integers that are relatively prime to 75, where \(a_{1}<a_{2}<a_{3}<\cdots\). (The first five terms of the sequence are: \(a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8\).) Find the value of \(a_{2008}\). | 3764 | 69.53125 |
13,454 | In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those who did not solve problem A, the number of p... | 23 | 21.09375 |
13,455 | Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously? | 35 | 22.65625 |
13,456 | Find all values of \( a \) for which the equation \( x^{2} + 2ax = 8a \) has two distinct integer roots. In your answer, record the product of all such values of \( a \), rounding to two decimal places if necessary. | 506.25 | 20.3125 |
13,457 | Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \). | 1987 | 92.1875 |
13,458 | Positive real numbers \( x, y, z \) satisfy: \( x^{4} + y^{4} + z^{4} = 1 \). Find the minimum value of the algebraic expression \( \frac{x^{3}}{1-x^{8}} + \frac{y^{3}}{1-y^{8}} + \frac{z^{3}}{1-z^{8}} \). | \frac{9 \sqrt[4]{3}}{8} | 0.78125 |
13,459 | Given a set \( A = \{0, 1, 2, \cdots, 9\} \), and a family of non-empty subsets \( B_1, B_2, \cdots, B_j \) of \( A \), where for \( i \neq j \), \(\left|B_i \cap B_j\right| \leqslant 2\), determine the maximum value of \( k \). | 175 | 63.28125 |
13,460 | Radovan read an interesting book. Yesterday, he read 15 pages, and today he read 12 pages. He realized with surprise that the sum of the page numbers he read yesterday is the same as the sum of the page numbers he read today. On which page will he start reading tomorrow? (Radovan does not skip any pages or read any pag... | 74 | 0.78125 |
13,461 | Given that $\frac{a}{36-a}+\frac{b}{48-b}+\frac{c}{72-c}=9$, evaluate $\frac{4}{36-a}+\frac{6}{48-b}+\frac{9}{72-c}$. | \frac{13}{3} | 0.78125 |
13,462 | Let \( r_{1}, r_{2}, \cdots, r_{20} \) be the roots of the polynomial \( x^{20}-7x^{3}+1 \). If \(\frac{1}{r_{1}^{2}+1}+\frac{1}{r_{2}^{2}+1}+\cdots+\frac{1}{r_{20}^{2}+1} \) can be expressed in the form \( \frac{m}{n} \) (with \( m \) and \( n \) coprime), find the value of \( m+n \). | 240 | 0 |
13,463 | Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. | \frac{9\sqrt{3}}{4} | 28.90625 |
13,464 | In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\alpha$ through $AM$ parallel to $BC$, and denote $\angle PAM$ as $\theta_1$. The acute dihedral angle between $\alpha$ and the base pl... | \frac{\sqrt{2}}{2} | 0 |
13,465 | From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board? | 23 | 4.6875 |
13,466 | Let \( x, y, z, w \) be four consecutive vertices of a regular \( A \)-gon. If the length of the line segment \( xy \) is 2 and the area of the quadrilateral \( xyzw \) is \( a + \sqrt{b} \), find the value of \( B = 2^a \cdot 3^b \). | 108 | 1.5625 |
13,467 | A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square? | 4 - 2\sqrt{3} | 11.71875 |
13,468 | In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maint... | 1896 | 0 |
13,469 | Every day from Monday to Friday, an old man went to the blue sea and cast his net. Each day, the number of fish caught in the net was not greater than the number caught the previous day. Over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three day... | 50 | 64.84375 |
13,470 | Given an equilateral triangle of side 10, divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part. Repeat this step for each side of the resulting polygon. Find \( S^2 \), where \( S \) is the area of the region obtained by repeating this procedure ... | 4800 | 21.875 |
13,471 | Let non-negative real numbers \(a_1, a_2, \ldots, a_{100}\) satisfy: \( a_i + a_{i+1} + a_{i+2} \leq 1 \) for \( 1 \leq i \leq 100 \), where \(a_{101} = a_1\) and \(a_{102} = a_2\). Find the maximum value of \(\sum_{i=1}^{100} a_i a_{i+2}\). | 25/2 | 17.96875 |
13,472 | Two people, A and B, start from the same point on a 300-meter circular track and run in opposite directions. A runs at 2 meters per second, and B runs at 4 meters per second. When they first meet, A turns around and runs back. When A and B meet again, B turns around and runs back. Following this pattern, after how many... | 250 | 4.6875 |
13,473 | In triangle \(ABC\), sides \(AB\) and \(BC\) are equal, \(AC = 2\), and \(\angle ACB = 30^\circ\). From vertex \(A\), the angle bisector \(AE\) and the median \(AD\) are drawn to the side \(BC\). Find the area of triangle \(ADE\). | \frac{2 \sqrt{3} - 3}{6} | 0 |
13,474 | If point \( P \) is on the curve \( y=\frac{1}{2} e^{x} \) and point \( Q \) is on the curve \( y=\ln (2 x) \), then the minimum value of \( |PQ| \) is \( \qquad \). | \sqrt{2}(1 - \ln 2) | 0.78125 |
13,475 | A solid cube of side length $4$ has $12$ edges, and a solid cube of side length $2$ has $12$ edges. The remaining solid is created by removing $8$ solid cubes of side length $2$ from the corners of the solid cube of side length $4$. Calculate the total number of edges of the remaining solid. | 24 | 2.34375 |
13,476 | I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of the ticket, given that my neighbor easily solved this problem. | 99999 | 3.125 |
13,477 | Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake ca... | \frac{\pi}{12} | 7.03125 |
13,478 | Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BP... | 24 | 0.78125 |
13,479 | Given that \( n \) is a positive integer, \( P \) is a prime number, and \( pn \) has exactly 8 positive divisors, arrange them in ascending order as \( 1=d_{1}<d_{2}< \cdots <d_{8}=pn \). Additionally, let \( d_{17p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13p\right) \). Find \( n \). | 2021 | 0 |
13,480 | Out of the 200 natural numbers between 1 and 200, how many numbers must be selected to ensure that there are at least 2 numbers among them whose product equals 238? | 198 | 3.90625 |
13,481 | A circle has 2017 distinct points $A_{1}, \ldots, A_{2017}$ marked on it, and all possible chords connecting pairs of these points are drawn. A line is drawn through the point $A_{1}$, which does not pass through any of the points $A_{2}, \ldots A_{2017}$. Find the maximum possible number of chords that can intersect t... | 1018080 | 0 |
13,482 | Let $f(x) = \frac{x^6 - 2}{4}$. Find $f^{-1}\left(-\frac{1}{8}\right)$. | \left(\frac{3}{2}\right)^{\frac{1}{6}} | 5.46875 |
13,483 | In the quadrilateral \(ABCD\), it is given that \(\cos \angle BAD = \frac{3}{4}\), \(\angle BAC = \angle DAC\), \(AD < AB\), and \(AB = 5\), \(AC = BD = \sqrt{14}\). If \(\overrightarrow{AC} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AD}\) (\(\lambda, \mu \in \mathbf{R}\)), find \(\lambda + \mu\). | \frac{7}{5} | 3.125 |
13,484 | Let \( m \) and \( n \) (with \( m > n \)) be positive integers such that \( 70^2 \) divides \( 2023^m - 2023^n \). What is the smallest value of \( m+n \)? | 24 | 5.46875 |
13,485 | Form natural numbers without repeating digits using the digits 0, 1, and 2, and calculate the total number of such natural numbers. | 11 | 0.78125 |
13,486 | Vovochka adds three-digit numbers in a column in the following way: he does not carry over tens, but writes the sum of pairs of digits in the same place value position under each pair of digits, even if the result is a two-digit number. For example, for the sum \(248 + 208\), he would get the result 4416. Find the smal... | 1800 | 2.34375 |
13,487 | The center of a circle with a radius of 5, circumscribed around an isosceles trapezoid, lies on the longer base, and the shorter base is equal to 6. Find the area of the trapezoid. | 32 | 1.5625 |
13,488 | 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 is always true that \( x y z \mid (x+y+z)^{n} \). | 13 | 9.375 |
13,489 | An electrician was called to repair a garland of four light bulbs connected in series, one of which has burned out. It takes 10 seconds to unscrew any bulb from the garland and 10 seconds to screw it back in. The time spent on other actions is negligible. What is the minimum time in which the electrician can definitely... | 60 | 14.84375 |
13,490 | What is the maximum number of parts into which the coordinate plane \(xOy\) can be divided by the graphs of 100 quadratic polynomials of the form
\[ y = a_{n} x^{2} + b_{n} x + c_{n} \quad (n=1, 2, \ldots, 100) ? \] | 10001 | 11.71875 |
13,491 | A triangle with vertices \(A = (4, 3)\), \(B = (6, -2)\), and \(C = (7, 1)\) is reflected about the line \(x = 6\) to create a second triangle. Determine the area of the union of the two triangles. | 10 | 3.90625 |
13,492 | Prince Gvidon had three sons. Among his descendants, 93 each had two sons and no daughters, while the rest died childless. How many total descendants did Prince Gvidon have? | 189 | 43.75 |
13,493 | An isosceles triangle has sides with lengths that are composite numbers, and the square of the sum of the lengths is a perfect square. What is the smallest possible value for the square of its perimeter? | 256 | 3.125 |
13,494 | There are 6 blue, 7 red, and 9 white light bulbs. In how many ways can you arrange them (using all the light bulbs) in a garland so that no two white light bulbs are consecutive? | 3435432 | 71.875 |
13,495 | For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression. | \frac{3 + \sqrt{5}}{2} | 32.03125 |
13,496 | The distance from the point where a diameter of a circle intersects a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.
Given:
\[ AB = 18 \, \text{cm}, \, EO = 7 \, \text{cm}, \, AE = 2 \, BE \]
Find the radius \( R \). | 11 | 8.59375 |
13,497 | Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \mathrm{~km} / \mathrm{h}$.
What is the average speed at which Vendelín ... | 15 | 1.5625 |
13,498 | One of the three cards had the number 18, another had the number 75, and the third had some two-digit number. If you sum all the distinct six-digit numbers that can be obtained by arranging these cards in a row, you get the number 2606058. What number is written on the third card? | 36 | 7.8125 |
13,499 | If for any positive integer \( m \), the set
$$
\{m, m+1, m+2, \cdots, m+99\}
$$
in any \( n \)-element subset with \( n \geq 3 \), there are always three elements that are pairwise coprime, find the smallest value of \( n \). | 68 | 7.03125 |
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