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28,800 | An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$ , $1986$ , $1996$ , $2008$ and it didn't occur in $1976$ , $1993$ , $2006$ , $2013$ . What is the first year in that the event will occur again? | 2018 | 12.5 |
28,801 | From the six digits 0, 1, 2, 3, 4, 5, select two odd numbers and two even numbers to form a four-digit number without repeating digits. The total number of such four-digit numbers is ______. | 180 | 38.28125 |
28,802 | In the plane rectangular coordinate system $O-xy$, if $A(\cos\alpha, \sin\alpha)$, $B(\cos\beta, \sin\beta)$, $C\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, then one possible value of $\beta$ that satisfies $\overrightarrow{OC}=\overrightarrow{OB}-\overrightarrow{OA}$ is ______. | \frac{2\pi}{3} | 16.40625 |
28,803 | What is the smallest positive integer with exactly 20 positive divisors? | 432 | 0 |
28,804 | Evaluate the expression $\frac{16^{24}}{64^{8}}$.
A) $16^2$
B) $16^4$
C) $16^8$
D) $16^{16}$
E) $16^{24}$ | 16^8 | 42.96875 |
28,805 | Isabella and Evan are cousins. The 10 letters from their names are placed on identical cards so that each of 10 cards contains one letter. Without replacement, two cards are selected at random from the 10 cards. What is the probability that one letter is from each cousin's name? Express your answer as a common fraction. | \frac{16}{45} | 0 |
28,806 | I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction. | \frac{3}{20} | 11.71875 |
28,807 | How many two-digit numbers have digits whose sum is either a perfect square or a prime number up to 25? | 41 | 0 |
28,808 | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 41 | 0.78125 |
28,809 | What is the smallest positive integer with exactly 20 positive divisors? | 240 | 92.1875 |
28,810 | Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 72 | 0 |
28,811 | A pyramid has a square base with sides of length 2 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and the center of its opposite face exactly meets the apex of the pyramid. Calculate the volume of the cube.
A) $\frac{3\sqrt{6}}{8}$
B) $\frac{6\sqrt{6}}{8}$
C) $\sqrt{6}$
D) $\frac{3\sqrt{6}}{4}$
E) $\frac{6\sqrt{2}}{4}$ | \frac{3\sqrt{6}}{4} | 10.15625 |
28,812 | What is the probability that all 4 blue marbles are drawn before all 3 yellow marbles are drawn? | \frac{4}{7} | 14.84375 |
28,813 | For an arithmetic sequence $b_1,$ $b_2,$ $b_3,$ $\dots,$ let
\[P_n = b_1 + b_2 + b_3 + \dots + b_n,\]and let
\[Q_n = P_1 + P_2 + P_3 + \dots + P_n.\]If you are told the value of $P_{2023},$ then you can uniquely determine the value of $Q_n$ for some integer $n.$ What is this integer $n$? | 3034 | 10.15625 |
28,814 | In a right triangle PQR with right angle at P, suppose $\sin Q = 0.6$. If the length of QP is 15, what is the length of QR? | 25 | 52.34375 |
28,815 | Given that Liliane has $30\%$ more cookies than Jasmine and Oliver has $10\%$ less cookies than Jasmine, and the total number of cookies in the group is $120$, calculate the percentage by which Liliane has more cookies than Oliver. | 44.44\% | 95.3125 |
28,816 | Rodney is now guessing a secret number based on these clues:
- It is a two-digit integer.
- The tens digit is even.
- The units digit is odd.
- The number is greater than 50. | \frac{1}{10} | 0 |
28,817 | Find the area of quadrilateral \(ABCD\) if \(AB = BC = 3\sqrt{3}\), \(AD = DC = \sqrt{13}\), and vertex \(D\) lies on a circle of radius 2 inscribed in the angle \(ABC\), where \(\angle ABC = 60^\circ\). | 3\sqrt{3} | 7.03125 |
28,818 | Find the sum of the distinct prime factors of $7^7 - 7^4$. | 31 | 92.96875 |
28,819 | Four friends initially plan a road trip and decide to split the fuel cost equally. However, 3 more friends decide to join at the last minute. Due to the increase in the number of people sharing the cost, the amount each of the original four has to pay decreases by $\$$8. What was the total cost of the fuel? | 74.67 | 0 |
28,820 | Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 2520 | 10.9375 |
28,821 | Sara conducted a survey among a group to find out how many people were aware that bats can transmit diseases. She found out that $75.3\%$ believed bats could transmit diseases. Among those who believed this, $60.2\%$ incorrectly thought that all bats transmit Zika virus, which amounted to 37 people. Determine how many people were surveyed in total by Sara. | 81 | 14.0625 |
28,822 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given that $b=3a$ and $c=2$, find the area of $\triangle ABC$ when angle $A$ is at its maximum value. | \frac { \sqrt {2}}{2} | 0 |
28,823 | An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\). | 10 | 0 |
28,824 | Alice's favorite number has the following properties:
- It has 8 distinct digits.
- The digits are decreasing when read from left to right.
- It is divisible by 180.
What is Alice's favorite number?
*Author: Anderson Wang* | 97654320 | 74.21875 |
28,825 | Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | 15.5 | 0 |
28,826 | Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called *stable* if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?
*Proposed by Evan Chen* | 20 | 0 |
28,827 | What is the sum of the number of faces, edges, and vertices of a square pyramid that has a square base? Additionally, find the number of diagonals in the square base. | 18 | 1.5625 |
28,828 | Given \\(a < 0\\), \\((3x^{2}+a)(2x+b) \geqslant 0\\) holds true over the interval \\((a,b)\\), then the maximum value of \\(b-a\\) is \_\_\_\_\_\_. | \dfrac{1}{3} | 7.8125 |
28,829 | Vasya has:
a) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;
b) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;
c) a book of lyrical poems with a height of 40 cm, published by Vasya himself.
Vasya wants to arrange these books on a shelf so that his own work is in the center, and the books located at the same distance from it on both the left and the right have equal heights. In how many ways can this be done?
a) $3 \cdot 2! \cdot 4!$;
b) $2! \cdot 3!$;
c) $\frac{51}{3! \cdot 2!}$;
d) none of the above answers are correct. | 144 | 2.34375 |
28,830 | The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \ldots\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). What is the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when the 1994th term of the sequence is divided by 1000? | 063 | 0.78125 |
28,831 | Two circles are centered at \( (5,5) \) and \( (25,15) \) respectively, each tangent to the \( y \)-axis. Find the distance between the closest points of these two circles. | 10\sqrt{5} - 20 | 0 |
28,832 | In square $ABCD$ , $\overline{AC}$ and $\overline{BD}$ meet at point $E$ .
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$ .
If $\overline{AF}$ meets $\overline{ED}$ at point $G$ , and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$ . | 48 | 2.34375 |
28,833 | After the appearance of purple sand flowerpots in the Ming and Qing dynasties, their development momentum was soaring, gradually becoming the collection target of collectors. With the development of pot-making technology, purple sand flowerpots have been integrated into the daily life of ordinary people. A certain purple sand product factory is ready to mass-produce a batch of purple sand flowerpots. The initial cost of purchasing equipment for the factory is $10,000. In addition, $27 is needed to produce one purple sand flowerpot. When producing and selling a thousand purple sand flowerpots, the total sales of the factory is given by $P(x)=\left\{\begin{array}{l}5.7x+19,0＜x⩽10,\\ 108-\frac{1000}{3x},x＞10.\end{array}\right.($unit: ten thousand dollars).<br/>$(1)$ Find the function relationship of total profit $r\left(x\right)$ (unit: ten thousand dollars) with respect to the output $x$ (unit: thousand pieces). (Total profit $=$ total sales $-$ cost)<br/>$(2)$ At what output $x$ is the total profit maximized? And find the maximum value of the total profit. | 39 | 3.90625 |
28,834 | Alice and Bob live on the same road. At time $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$ , and Bob arrived at Alice's house at $3:29\text{pm}$ . Charlie, who was driving by, noted that Alice and Bob passed each other at $3:11\text{pm}$ . Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day.
*Proposed by Kevin You* | 179 | 0 |
28,835 | On the surface of a sphere with a radius of $2$, there is a triangular prism with an equilateral triangle base and lateral edges perpendicular to the base. All vertices of the prism are on the sphere's surface. Determine the maximum lateral area of this triangular prism. | 12\sqrt{3} | 32.8125 |
28,836 | In the quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(K\). Points \(L\) and \(M\) are the midpoints of sides \(BC\) and \(AD\), respectively. Segment \(LM\) contains point \(K\). The quadrilateral \(ABCD\) is such that a circle can be inscribed in it. Find the radius of this circle, given that \(AB = 3\), \(AC = \sqrt{13}\), and \(LK: KM = 1: 3\). | 3/2 | 3.90625 |
28,837 | Given that \(2 \cdot 50N\) is an integer and its representation in base \(b\) is 777, find the smallest positive integer \(b\) such that \(N\) is a fourth power of an integer. | 18 | 52.34375 |
28,838 | To stabilize housing prices, a local government decided to build a batch of affordable housing for the community. The plan is to purchase a piece of land for 16 million yuan and build a residential community with 10 buildings on it. Each building has the same number of floors, and each floor has a construction area of $1,000 m^{2}$. The construction cost per square meter is related to the floor level, with the cost for the $x^{th}$ floor being $(kx+800)$ yuan (where $k$ is a constant). It has been calculated that if each building has 5 floors, the average comprehensive cost per square meter for the community is 1,270 yuan.
Note: The average comprehensive cost per square meter $= \dfrac{\text{land purchase cost} + \text{all construction costs}}{\text{total construction area}}$.
$(1)$ Find the value of $k$;
$(2)$ To minimize the average comprehensive cost per square meter for the community, how many floors should the 10 buildings have? What is the average comprehensive cost per square meter at this time? | 1225 | 17.96875 |
28,839 | A zoo has six pairs of different animals (one male and one female for each species). The zookeeper wishes to begin feeding the male lion and must alternate between genders each time. Additionally, after feeding any lion, the next animal cannot be a tiger. How many ways can he feed all the animals? | 14400 | 0 |
28,840 | Given a tetrahedron $P-ABC$ with its circumscribed sphere's center $O$ on $AB$, and $PO \perp$ plane $ABC$, $2AC = \sqrt{3}AB$. If the volume of the tetrahedron $P-ABC$ is $\frac{3}{2}$, find the volume of the sphere. | 4\sqrt{3}\pi | 3.90625 |
28,841 | Trapezoid $EFGH$ has base $EF = 15$ units and base $GH = 25$ units. Diagonals $EG$ and $FH$ intersect at $Y$. If the area of trapezoid $EFGH$ is $200$ square units, what is the area of triangle $FYG$? | 46.875 | 7.8125 |
28,842 | Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the maximum whole number of minutes Petya's sleep could have lasted? | 361 | 0.78125 |
28,843 | A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle? | 338350 | 0 |
28,844 | I have a bag with $8$ marbles numbered from $1$ to $8.$ Mathew has a bag with $16$ marbles numbered from $1$ to $16.$ Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such that the sum of the numbers on my marbles is a prime number and equals the number on his marble? | 22 | 0 |
28,845 | Given that there are $m$ distinct positive even numbers and $n$ distinct positive odd numbers such that their sum is 2015. Find the maximum value of $20m + 15n$. | 1105 | 0 |
28,846 | Let $g(x) = dx^3 + ex^2 + fx + g$, where $d$, $e$, $f$, and $g$ are integers. Suppose that $g(1) = 0$, $70 < g(5) < 80$, $120 < g(6) < 130$, $10000m < g(50) < 10000(m+1)$ for some integer $m$. What is $m$? | 12 | 24.21875 |
28,847 | Find the minimum value of the discriminant of a quadratic trinomial whose graph does not intersect the regions below the x-axis and above the graph of the function \( y = \frac{1}{\sqrt{1-x^2}} \). | -4 | 21.09375 |
28,848 | Given that in triangle $ABC$, $\overrightarrow{A B} \cdot \overrightarrow{B C} = 3 \overrightarrow{C A} \cdot \overrightarrow{A B}$, find the maximum value of $\frac{|\overrightarrow{A C}| + |\overrightarrow{A B}|}{|\overrightarrow{B C}|}$: | $\sqrt{3}$ | 0 |
28,849 | Find all \( t \) such that \( x-t \) is a factor of \( 10x^2 + 21x - 10 \). | -\frac{5}{2} | 0 |
28,850 | How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 | 0.78125 |
28,851 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $4\sin A\sin B-4\cos ^{2} \frac {A-B}{2}= \sqrt {2}-2$.
$(1)$ Find the magnitude of angle $C$;
$(2)$ Given $\frac {a\sin B}{\sin A}=4$ and the area of $\triangle ABC$ is $8$, find the value of side length $c$. | c=4 | 83.59375 |
28,852 | Calculate the product $(\frac{4}{8})(\frac{8}{12})(\frac{12}{16})\cdots(\frac{2016}{2020})$. Express your answer as a common fraction. | \frac{2}{505} | 0 |
28,853 | Given the function $f(x) = x^2 + (a+8)x + a^2 + a - 12$ ($a < 0$), and $f(a^2 - 4) = f(2a - 8)$, determine the minimum value of $\frac {f(n)-4a}{n+1}$ for $n \in \mathbb{N}^{+}$. | \frac {37}{4} | 0.78125 |
28,854 | The smallest possible value of $m$ for which Casper can buy exactly $10$ pieces of strawberry candy, $18$ pieces of lemon candy, and $20$ pieces of cherry candy, given that each piece of orange candy costs $15$ cents. | 12 | 0.78125 |
28,855 | The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \( P(-2,0) \) to its asymptote is \( \frac{2 \sqrt{6}}{3} \). If a line with slope \( \frac{\sqrt{2}}{2} \) passes through point \( P \) and intersects the hyperbola at points \( A \) and \( B \), intersects the y-axis at point \( M \), and \( P M \) is the geometric mean of \( P A \) and \( P B \), then the half focal length of the hyperbola is what? | \sqrt{3} | 31.25 |
28,856 | Given points A, B, and C on the surface of a sphere O, the height of the tetrahedron O-ABC is 2√2 and ∠ABC=60°, with AB=2 and BC=4. Find the surface area of the sphere O. | 48\pi | 67.96875 |
28,857 | Let $a$, $b$, and $c$ be positive real numbers. Find the minimum value of
\[
\frac{(a^2 + 4a + 4)(b^2 + 4b + 4)(c^2 + 4c + 4)}{abc}.
\] | 64 | 20.3125 |
28,858 | On a mathematics quiz, there were $6x$ problems. Lucky Lacy missed $2x$ of them. What percent of the problems did she get correct? | 66.67\% | 44.53125 |
28,859 | A bagel is cut into sectors. A total of 10 cuts were made. How many pieces were formed? | 11 | 49.21875 |
28,860 | A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? | \frac{5}{8} | 10.15625 |
28,861 | A cube with edge length 2 cm has a dot marked at the center of the top face. The cube is on a flat table and rolls without slipping, making a full rotation back to its initial orientation, with the dot back on top. Calculate the length of the path followed by the dot in terms of $\pi. $ | 2\sqrt{2}\pi | 9.375 |
28,862 | Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$ , $AB=16$ , $CD=12$ , and $BC<AD$ . A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$ . | 13 | 3.90625 |
28,863 | Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face. | 10 | 15.625 |
28,864 | What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \( d \) contained in this segment, there are no more than \( 1 + 1000 d^2 \) points? | 32 | 10.15625 |
28,865 | A school club designed an unconventional rectangular checkerboard that measures 10 by 6. Calculate the probability that a randomly chosen unit square does not touch any edge of the board or is not adjacent to any edge. | \frac{1}{5} | 10.15625 |
28,866 | How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 | 0.78125 |
28,867 | Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0), (2010,0), (2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? | \frac{335}{2011} | 35.9375 |
28,868 | Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements. | 216 | 32.8125 |
28,869 | Evaluate the expression $\dfrac{\sqrt[6]{5}}{\sqrt[4]{5}}$. What power of 5 does this expression represent? | -\frac{1}{12} | 42.1875 |
28,870 | A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both these circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction. | 18 | 0.78125 |
28,871 | Given the numbers 1, 2, 3, 4, 5, there are $5!$ permutations $a_1, a_2, a_3, a_4, a_5$. Find the number of distinct permutations where $a_k \geq k - 2$ for all $k = 1, 2, 3, 4, 5$. | 54 | 45.3125 |
28,872 | In triangle $XYZ,$ angle bisectors $\overline{XU}$ and $\overline{YV}$ intersect at $Q.$ If $XY = 8,$ $XZ = 6,$ and $YZ = 4,$ find $\frac{YQ}{QV}.$ | 1.5 | 0 |
28,873 | Determine the value of:
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).\] | 9006.5 | 0 |
28,874 | There is a hemispherical raw material. If this material is processed into a cube through cutting, the maximum value of the ratio of the volume of the obtained workpiece to the volume of the raw material is ______. | \frac { \sqrt {6}}{3\pi } | 0 |
28,875 | For how many four-digit whole numbers does the sum of the digits equal $30$? | 20 | 0 |
28,876 | A and B plays a game on a pyramid whose base is a $2016$ -gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarantee that all sides are colored. | 2016 | 13.28125 |
28,877 | Let $\phi$ be the smallest acute angle for which $\cos \phi,$ $\cos 2\phi,$ $\cos 4\phi$ form an arithmetic progression, in some order. Find $\sin \phi.$ | \frac{1}{\sqrt{2}} | 0 |
28,878 | The numbers $1, 2, \dots, 16$ are randomly placed into the squares of a $4 \times 4$ grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is even. | \frac{36}{20922789888000} | 0 |
28,879 | If 1983 were expressed as a sum of distinct powers of 2, with the requirement that at least five distinct powers of 2 are used, what would be the least possible sum of the exponents of these powers? | 55 | 21.875 |
28,880 | Let $\{b_k\}$ be a sequence of integers where $b_1 = 2$ and $b_{m+n} = b_m + b_n + m^2 + n^2$ for all positive integers $m$ and $n$. Find $b_{12}$. | 160 | 16.40625 |
28,881 | Define a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are spacy? | 60 | 0 |
28,882 | Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \le i \le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists. | 64 | 56.25 |
28,883 | Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing at his original speed. By the time the document is completed, A and B have typed an equal number of characters. What is the total number of characters in the document? | 18000 | 38.28125 |
28,884 | Given that $\sin \alpha = 2 \cos \alpha$, find the value of $\cos ( \frac {2015\pi}{2}-2\alpha)$. | - \frac {4}{5} | 57.03125 |
28,885 | Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$ , with $A$ closer to $B$ than $C$ , such that $2016 \cdot AB = BC$ . Line $XY$ intersects line $AC$ at $D$ . If circles $C_1$ and $C_2$ have radii of $20$ and $16$ , respectively, find $\sqrt{1+BC/BD}$ . | 2017 | 3.125 |
28,886 | Let $\triangle ABC$ have $\angle ABC=67^{\circ}$ . Point $X$ is chosen such that $AB = XC$ , $\angle{XAC}=32^\circ$ , and $\angle{XCA}=35^\circ$ . Compute $\angle{BAC}$ in degrees.
*Proposed by Raina Yang* | 81 | 3.125 |
28,887 | The greatest common divisor of 30 and some number between 70 and 80 is 10. What is the number, if the least common multiple of these two numbers is also between 200 and 300? | 80 | 12.5 |
28,888 | Suppose $b$ and $c$ are constants such that the quadratic equation $2ax^2 + 15x + c = 0$ has exactly one solution. If the value of $c$ is 9, find the value of $a$ and determine the unique solution for $x$. | -\frac{12}{5} | 0 |
28,889 | $a,b,c$ - are sides of triangle $T$ . It is known, that if we increase any one side by $1$ , we get new
a) triangle
b)acute triangle
Find minimal possible area of triangle $T$ in case of a) and in case b) | \frac{\sqrt{3}}{4} | 0.78125 |
28,890 | The angle of inclination of the line $x-y-1=0$ is ____. | \dfrac{\pi}{4} | 67.1875 |
28,891 | Let an ordered pair of positive integers $(m, n)$ be called *regimented* if for all nonnegative integers $k$ , the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016}-N$ .
*Proposed by Ashwin Sah* | 10086 | 0.78125 |
28,892 | If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth. | 0.16 | 55.46875 |
28,893 | The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:
1. The Falcons won more games than the Hawks.
2. The Warriors won more games than the Knights but fewer than the Royals.
3. The Knights won more than 25 games.
How many games did the Warriors win? The wins recorded are 20, 26, 30, 35, 40, and 45. | 35 | 67.96875 |
28,894 | Let \( p, q, r \) be the roots of the polynomial \( x^3 - 8x^2 + 14x - 2 = 0 \). Define \( t = \sqrt{p} + \sqrt{q} + \sqrt{r} \). Find \( t^4 - 16t^2 - 12t \). | -8 | 3.125 |
28,895 | Find a three-digit number equal to the sum of the tens digit, the square of the hundreds digit, and the cube of the units digit.
Find the number \(\overline{abcd}\) that is a perfect square, if \(\overline{ab}\) and \(\overline{cd}\) are consecutive numbers, with \(\overline{ab} > \(\overline{cd}\). | 357 | 1.5625 |
28,896 | Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], determine the range of the function g(x) on the interval [-2020,2020]. | [-4043,4041] | 10.15625 |
28,897 | The minimum positive period of the function $f(x)=2\sin\left(\frac{x}{3}+\frac{\pi}{5}\right)-1$ is ______, and the minimum value is ______. | -3 | 89.0625 |
28,898 | Given \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \). | \frac{1}{4} | 4.6875 |
28,899 | A school organized a trip to the Expo Park for all third-grade students and rented some large buses. Initially, the plan was to have 28 people on each bus. After all the students boarded, it was found that 13 students could not get on the buses. So, they decided to have 32 people on each bus, and this resulted in 3 empty seats on each bus. How many third-grade students does this school have? How many large buses were rented? | 125 | 13.28125 |
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