problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$ | (\frac{1}{3},1] | 1/8 |
Which of the following is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$?
$\mathrm{(A) \ } -x\qquad \mathrm{(B) \ } x\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \sqrt{\frac{x}{2}}\qquad \mathrm{(E) \ } x\sqrt{-1}$ | \textbf{(A)}-x | 1/8 |
For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots?
$\textbf{(A)}\ \text{none}\qquad\textbf{(B)}\ -2<K<1\qquad\textbf{(C)}\ -2\sqrt{2}< K < 2\sqrt{2}\qquad$
$\textbf{(D)}\ K>1\text{ or }K<-2\qquad\textbf{(E)}\ \text{all}$ | \textbf{(E)}\ | 1/8 |
Let $m \circ n=(m+n) /(m n+4)$. Compute $((\cdots((2005 \circ 2004) \circ 2003) \circ \cdots \circ 1) \circ 0)$. | 1/12 | 2/8 |
Given that the students are numbered from 01 to 70, determine the 7th individual selected by reading rightward starting from the number in the 9th row and the 9th column of the random number table. | 44 | 1/8 |
There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.\] Find $n_1 + n_2 + \cdots + n_r$. | 21 | 7/8 |
The slope angle of the tangent line to the curve $y= \sqrt {x}$ at $x= \frac {1}{4}$ is ______. | \frac {\pi}{4} | 6/8 |
Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (x, 1)$,
1. If $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$ forms an acute angle, find the range of $x$.
2. Find the value of $x$ when $(\overrightarrow{a}+2\overrightarrow{b}) \perp (2\overrightarrow{a}-\overrightarrow{b})$. | \frac{7}{2} | 7/8 |
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$ , there exist 3 numbers $a$ , $b$ , $c$ among them satisfying $abc=m$ . | 11 | 6/8 |
Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?
$\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfrac{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}$ | \textbf{(A)}\p<\frac{1}{8} | 1/8 |
In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board? | 16 | 1/8 |
A quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent line \(\ell\) to this circle is drawn. A circle \(\omega\) passes through points \(A\) and \(B\) and is tangent to the line \(\ell\) at point \(P\). The line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is the point of tangency to the circle \(\omega\). | 1 | 1/8 |
Among the following propositions, the true one is marked by \_\_\_\_\_\_.
\\((1)\\) The negation of the proposition "For all \\(x > 0\\), \\(x^{2}-x \leqslant 0\\)" is "There exists an \\(x > 0\\) such that \\(x^{2}-x > 0\\)."
\\((2)\\) If \\(A > B\\), then \\(\sin A > \sin B\\).
\\((3)\\) Given a sequence \\(\{a_{n}\}\\), "The sequence \\(a_{n}\\), \\(a_{n+1}\\), \\(a_{n+2}\\) forms a geometric sequence" is a necessary and sufficient condition for "\\(a_{n+1}^{2} = a_{n}a_{n+2}\\)."
\\((4)\\) Given the function \\(f(x) = \lg x + \frac{1}{\lg x}\\), the minimum value of the function \\(f(x)\\) is \\(2\\). | (1) | 6/8 |
In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area of its circumscribed sphere is ______. | 14\pi | 6/8 |
In trapezoid \(ABCD\), the base \(AD\) is equal to 16, the sum of the side \(AB\) and the diagonal \(BD\) is equal to 40, and the angle \(CBD\) is \(60^\circ\). The ratio of the areas of triangles \(ABO\) and \(BOC\), where \(O\) is the intersection point of the diagonals, is 2. Find the area of the trapezoid. | 126\sqrt{3} | 4/8 |
Calculate:<br/>$(1)-3+8-15-6$;<br/>$(2)-35\div \left(-7\right)\times (-\frac{1}{7})$;<br/>$(3)-2^{2}-|2-5|\div \left(-3\right)$;<br/>$(4)(\frac{1}{2}+\frac{5}{6}-\frac{7}{12})×(-24)$;<br/>$(5)(-99\frac{6}{11})×22$. | -2190 | 7/8 |
Let \( g \) be a natural number, and suppose \( g^{4} + g^{3} + g^{2} + g + 1 \) is a perfect square. Find the sum of all such \( g \). | 3 | 1/8 |
Given positive integers \(a, b, c\) that satisfy
\[ 1 < a < b < c, \quad a + b + c = 111, \quad b^2 = ac, \]
find \(b\). | 36 | 3/8 |
Using systematic sampling, extract a sample of size 12 from a population of 123 individuals. The sampling interval is ______. | 10 | 7/8 |
Suppose rectangle $F O L K$ and square $L O R E$ are on the plane such that $R L=12$ and $R K=11$. Compute the product of all possible areas of triangle $R K L$. | 414 | 7/8 |
Let \(\left(x^{2}+2x-2\right)^{6}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\cdots+a_{12}(x+2)^{12}\), where \(a_{i} (i=0,1,2,\ldots,12)\) are real constants. Determine the value of \(a_{0}+a_{1}+2a_{2}+3a_{3}+\cdots+12a_{12}\). | 64 | 6/8 |
In triangle $AHI$, which is equilateral, lines $\overline{BC}$, $\overline{DE}$, and $\overline{FG}$ are all parallel to $\overline{HI}$. The lengths satisfy $AB = BD = DF = FH$. However, the point $F$ on line segment $AH$ is such that $AF$ is half of $AH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$. | \frac{3}{4} | 2/8 |
Find the minimum value of the distance $|AB|$ where point $A$ is the intersection of the line $y=a$ and the line $y=2x+2$, and point $B$ is the intersection of the line $y=a$ and the curve $y=x+\ln x$. | \frac{3}{2} | 7/8 |
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that
\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]
When does equality hold? | R\ge\frac{^2+b^2}{2\sqrt{2a^2+2b^2-^2}} | 4/8 |
Let \( ABC \) be a triangle in which \( \angle ABC = 60^\circ \). Let \( I \) and \( O \) be the incentre and circumcentre of \( ABC \), respectively. Let \( M \) be the midpoint of the arc \( BC \) of the circumcircle of \( ABC \), which does not contain the point \( A \). Determine \( \angle BAC \) given that \( MB = OI \). | 30 | 6/8 |
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$ , what is the largest possible area of the triangle? | 25 | 3/8 |
Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square. | \frac{\sqrt{5}-1}{4} | 1/8 |
Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 647 | 3/8 |
The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs? | 14400 | 7/8 |
If the graph of the function $f(x)=a^{x-2}-2a (a > 0, a \neq 1)$ always passes through the fixed point $\left(x\_0, \frac{1}{3}\right)$, then the minimum value of the function $f(x)$ on $[0,3]$ is equal to \_\_\_\_\_\_\_\_. | -\frac{1}{3} | 2/8 |
Given \( m+1 \) equally spaced horizontal lines and \( n+1 \) equally spaced vertical lines forming a rectangular grid with \( (m+1)(n+1) \) nodes, let \( f(m, n) \) be the number of paths from one corner to the opposite corner along the grid lines such that the path does not visit any node twice. Show that \( f(m, n) \leq 2^{mn} \). | f(,n)\le2^{mn} | 2/8 |
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 9x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 38 | 1/8 |
Independent trials are conducted, in each of which event \( A \) can occur with a probability of 0.001. What is the probability that in 2000 trials, event \( A \) will occur at least two and at most four times? | 0.541 | 4/8 |
For each polynomial $P(x)$ , define $$ P_1(x)=P(x), \forall x \in \mathbb{R}, $$ $$ P_2(x)=P(P_1(x)), \forall x \in \mathbb{R}, $$ $$ ... $$ $$ P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}. $$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$ , the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots? | Yes | 3/8 |
Let $x$ be the least real number greater than $1$ such that $\sin(x)= \sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20$ | \textbf{(B)}13 | 1/8 |
In an acute-angled triangle \(ABC\), the angle \(\angle ACB = 75^\circ\), and the height drawn from the vertex of this angle is equal to 1.
Find the radius of the circumscribed circle if it is known that the perimeter of the triangle \(ABC\) is \(4 + \sqrt{6} - \sqrt{2}\). | \sqrt{6}-\sqrt{2} | 3/8 |
Evaluate the expression $\frac {1}{3 - \frac {1}{3 - \frac {1}{3 - \frac{1}{4}}}}$. | \frac{11}{29} | 1/8 |
nic $\kappa$ y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic $\kappa$ y can label at least $dn^2$ cells of an $n\times n$ square.
*Proposed by Mihir Singhal and Michael Kural*
| \frac{2}{3} | 3/8 |
Determine the number of relatively prime dates in the month with the second fewest relatively prime dates. | 11 | 5/8 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$. | \frac{100}{101} | 1/8 |
Call a positive integer "mild" if its base-3 representation never contains the digit 2. How many values of \( n \) (where \( 1 \leq n \leq 1000 \)) have the property that both \( n \) and \( n^{2} \) are mild? | 7 | 5/8 |
A set of finitely many points \( M \) on the plane satisfies:
(a) Any three points are not collinear;
(b) Each point in \( M \) can be colored either red or blue in such a way that every triangle formed by any three points of the same color has at least one point of the other color inside it.
What is the maximum possible number of points in \( M \)? | 8 | 4/8 |
$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most? | 7 | 1/8 |
Given the function $y=2\sin \left(3x+ \dfrac{\pi}{4}\right)$, determine the shift required to obtain its graph from the graph of the function $y=2\sin 3x$. | \dfrac{\pi}{12} | 7/8 |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1019 | 3/8 |
1. In triangle \( \triangle ABC \), \( AB = 12 \), \( AC = 16 \), \( M \) is the midpoint of \( BC \). Points \( E \) and \( F \) are on sides \( AC \) and \( AB \) respectively. The line \( EF \) intersects \( AM \) at \( G \). If \( AE = 2AF \), find the ratio \(\frac{EG}{GF}\).
2. \( E \) is a point outside a circle. The chords \( EAB \) and \( EDC \) form an angle of \( 40^{\circ} \). If \( AB = BC = CD \), find \( \angle ACD \). | 60 | 2/8 |
Let the sequence \(\{a_n\}\) be defined as follows:
\[
\begin{aligned}
& a_1 = 1, \quad a_2 = 3, \\
& a_n = 3a_{n-1} - a_{n-2} \quad (n \in \mathbb{Z}_+, n \geq 3)
\end{aligned}
\]
Is there a positive integer \(n\) such that \(2^{2016} \| a_n\), i.e., \(2^{2016} \mid a_n\) and \(2^{2017} \nmid a_n\)? If such an \(n\) exists, find the smallest positive integer \(n\); if not, provide a justification. | 3\cdot2^{2013} | 1/8 |
6 boys and 4 girls are each assigned as attendants to 5 different buses, with 2 attendants per bus. Assuming that boys and girls are separated, and the buses are distinguishable, how many ways can the assignments be made? | 5400 | 7/8 |
The angle bisector of angle \(ABC\) forms an angle with its sides that is three times smaller than the adjacent angle to \(ABC\). Find the measure of angle \(ABC\). | 72 | 7/8 |
The first 14 terms of the sequence $\{a_n\}$ are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38. According to this pattern, find $a_{16}$. | 46 | 7/8 |
Let $k>1$ be a given positive integer. A set $S$ of positive integers is called *good* if we can colour the set of positive integers in $k$ colours such that each integer of $S$ cannot be represented as sum of two positive integers of the same colour. Find the greatest $t$ such that the set $S=\{a+1,a+2,\ldots ,a+t\}$ is *good* for all positive integers $a$ .
*A. Ivanov, E. Kolev* | 2k-1 | 1/8 |
Given $\triangle PQR$ with $\overline{RS}$ bisecting $\angle R$, $PQ$ extended to $D$ and $\angle n$ a right angle, then:
$\textbf{(A)}\ \angle m = \frac {1}{2}(\angle p - \angle q) \qquad \textbf{(B)}\ \angle m = \frac {1}{2}(\angle p + \angle q) \qquad \textbf{(C)}\ \angle d =\frac{1}{2}(\angle q+\angle p)\qquad \textbf{(D)}\ \angle d =\frac{1}{2}\angle m\qquad \textbf{(E)}\ \text{none of these is correct}$ | \textbf{(B)}\\angle=\frac{1}{2}(\anglep+\angleq) | 1/8 |
Define \(\phi^{!}(n)\) as the product of all positive integers less than or equal to \(n\) and relatively prime to \(n\). Compute the number of integers \(2 \leq n \leq 50\) such that \(n\) divides \(\phi^{!}(n) + 1\). | 30 | 1/8 |
Define a function $g$ from the positive integers to the positive integers with the following properties:
(i) $g$ is increasing.
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.
Compute all possible values of $g(88).$ | 7744 | 2/8 |
The number $5\,41G\,507\,2H6$ is divisible by $40.$ Determine the sum of all distinct possible values of the product $GH.$ | 225 | 2/8 |
Suppose that $\triangle{ABC}$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP=1$, $BP=\sqrt{3}$, and $CP=2$. What is $s$?
$\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5+\sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$ | \textbf{(B)}\sqrt{7} | 1/8 |
On the surface of a regular tetrahedron with an edge length of 1, nine points are marked.
Prove that among these points, there are two points such that the distance between them (in space) does not exceed 0.5. | 11 | 1/8 |
Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$ [list=a]
[*]Determine the general formula for $a_n.$ [*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$ [/list] | a_n=\frac{n}{2^n} | 1/8 |
Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______. | 5 - 2\sqrt{3} | 5/8 |
Given \( a = 19911991 \cdots \cdots 1991 \) (a repeated sequence of 1991), what is the remainder when \( a \) is divided by 13? | 8 | 3/8 |
What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell? | 12 | 1/8 |
A contest has six problems worth seven points each. On any given problem, a contestant can score either 0,1 , or 7 points. How many possible total scores can a contestant achieve over all six problems? | 28 | 6/8 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $a + c = 2b$ and $A - C = \frac{\pi}{3}$, find $\sin B$. | \frac{\sqrt{39}}{8} | 5/8 |
Given points O(0,0) and A(1,1), the slope angle of line OA is $\boxed{\text{answer}}$. | \frac{\pi}{4} | 2/8 |
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. | 12 | 2/8 |
In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds. | 110 | 1/8 |
Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | 24\pi | 7/8 |
In the tetrahedron \(KLMN\), the edge lengths are given as \(KL = MN = 9\), \(KM = LN = 15\), and \(KN = LM = 16\). The points \(P, Q, R,\) and \(S\) are the centers of the circles inscribed in the triangles \(KLM\), \(KLN\), \(KMN\), and \(LMN\), respectively. Find the volume of the tetrahedron \(PQRS\). Round the answer to two decimal places if necessary. | 4.85 | 1/8 |
There is a bag with 100 red and 100 blue balls. Balls are drawn one by one, randomly and without replacement, until all 100 red balls have been drawn. Determine the expected number of balls remaining in the bag. | \frac{100}{101} | 3/8 |
Four people, A, B, C, and D, are playing a table tennis tournament (there are no draws, each pair of players will have a match). After the tournament, the results are as follows: A won 2 matches, B won 1 match. How many matches did C win at most? | 3 | 6/8 |
Let \(ABCDEF\) be a regular hexagon and let point \(O\) be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn't exist any equilateral triangle with vertices of all the same color? | 6 | 1/8 |
Businessmen Ivanov, Petrov, and Sidorov decided to create a car company. Ivanov bought 70 identical cars for the company, Petrov bought 40 identical cars, and Sidorov contributed 44 million rubles to the company. It is known that Ivanov and Petrov can share the money among themselves in such a way that each of the three businessmen's contributions to the business is equal. How much money is Ivanov entitled to receive? Provide the answer in million rubles. | 12 | 5/8 |
In parallelogram $EFGH$, $EF = 5z + 5$, $FG = 4k^2$, $GH = 40$, and $HE = k + 20$. Determine the values of $z$ and $k$ and find $z \times k$. | \frac{7 + 7\sqrt{321}}{8} | 3/8 |
A father has $n(>4)$ children. What is the probability that there exists a child among them who has a younger brother, an older brother, a younger sister, and an older sister? (It is assumed that the probability of a person being male or female is equally likely at any given age.) | 1-\frac{n-2}{2^{(n-3)}} | 1/8 |
Let $S$ be a subset of $\{1,2,3, \ldots, 12\}$ such that it is impossible to partition $S$ into $k$ disjoint subsets, each of whose elements sum to the same value, for any integer $k \geq 2$. Find the maximum possible sum of the elements of $S$. | 77 | 2/8 |
Jack Sparrow needed to distribute 150 piastres across 10 purses. After placing a certain amount of piastres in the first purse, he placed more piastres in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse? | 16 | 4/8 |
Given the circumcenter \(O\) of \(\triangle ABC\), and \(3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \mathbf{0}\), find the value of \(\cos \angle ABC\). | \frac{\sqrt{5}}{5} | 7/8 |
How should a rook move on a chessboard to visit each square exactly once and make the fewest number of turns? | 14 | 7/8 |
Noelle needs to follow specific guidelines to earn homework points: For each of the first ten homework points she wants to earn, she needs to do one homework assignment per point. For each homework point from 11 to 15, she needs two assignments; for each point from 16 to 20, she needs three assignments and so on. How many homework assignments are necessary for her to earn a total of 30 homework points? | 80 | 7/8 |
Let $S$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{2}}{2} \le x \le \frac{\sqrt{3}}{2}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in S$ such that $z^n = 1.$ | 16 | 1/8 |
Given set \( A = \{0, 1, 2, 3, 4, 5, 9\} \), and \( a, b \in A \) where \( a \neq b \). The number of functions of the form \( y = -a x^2 + (4 - b)x \) whose vertex lies in the first quadrant is ___.
| 21 | 7/8 |
Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3. | -5050 | 5/8 |
For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\varphi(n)+4 \tau(n)=n$. For example, the number 44 is good because $\varphi(44)+4 \tau(44)=44$. Find the sum of all good positive integers $n$. | 172 | 1/8 |
Given the sequence $\{a_{n}\}$ defined by:
$$
a_{1}=2, \quad a_{n+1}=-\frac{(S_{n}-1)^{2}}{S_{n}} \quad (\text{for } n \in \mathbf{Z}_{+}),
$$
where $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$.
(1) Prove that $\left\{\frac{1}{S_{n}-1}\right\}$ forms an arithmetic sequence.
(2) For any $n$, it holds that
$$
\prod_{i=1}^{n}(S_{i}+1) \geqslant k n,
$$
find the maximum value of $k$. | 3 | 7/8 |
It is known that in the pyramid \(ABCD\) with vertex \(D\), the sum \(\angle ABD + \angle DBC = \pi\). Find the length of segment \(DL\), where \(L\) is the base of the bisector \(BL\) of triangle \(ABC\), given:
\[
AB = 9, \quad BC = 6, \quad AC = 5, \quad DB = 1.
\] | 7 | 5/8 |
Let's call a non-empty (finite or infinite) set $A$, consisting of real numbers, complete if for any real numbers $a$ and $b$ (not necessarily distinct and not necessarily belonging to $A$), such that $a+b$ belongs to $A$, the number $ab$ also belongs to $A$. Find all complete sets of real numbers. | \mathbb{R} | 4/8 |
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies.
[i] | n^2 - n + 1 | 1/8 |
A sequence of numbers is defined using the relation
$$
a_{n} = -a_{n-1} + 6a_{n-2}
$$
where \( a_1 = 2 \) and \( a_2 = 1 \). Find \( a_{100} + 3a_{99} \). | 7\cdot2^{98} | 1/8 |
Given the vector $$\overrightarrow {a_{k}} = (\cos \frac {k\pi}{6}, \sin \frac {k\pi}{6} + \cos \frac {k\pi}{6})$$ for k=0, 1, 2, …, 12, find the value of $$\sum\limits_{k=0}^{11} (\overrightarrow {a_{k}} \cdot \overrightarrow {a_{k+1}})$$. | 9\sqrt{3} | 2/8 |
A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$. | 0.3 | 6/8 |
The vertices of a regular hexagon are labeled \(\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)\). For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real \(\theta\)), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge? | 14 | 2/8 |
The minimum positive period and maximum value of the function $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{x}{3}$ are respectively $3\pi$ and $\sqrt{2}$. | \sqrt{2} | 7/8 |
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$ . | 13 | 4/8 |
In the tetrahedron \(ABCD\), \(\angle ACB = \angle CAD = 90^{\circ}\) and \(CA = CB = AD / 2\) and \(CD \perp AB\). What is the angle between the faces \(ACB\) and \(ACD\)? | 60 | 7/8 |
Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$ , $BC=AD=10$ , and $CA=DB=17$ . The intersection of these two triangular regions has area $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 59 | 7/8 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $x_{1}>0, x_{n+1}=\sqrt{5} x_{n}+2 \sqrt{x_{n}^{2}+1}$ for $n \in \mathbf{N}^{*}$, prove that among $x_{1}, x_{2}, \cdots, x_{2016}$, there are at least 672 irrational numbers. | 672 | 1/8 |
Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids. | 134 | 1/8 |
Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$. | 36^\circ | 2/8 |
Let $s_1, s_2, s_3$ be the respective sums of $n, 2n, 3n$ terms of the same arithmetic progression with $a$ as the first term and $d$
as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on:
$\textbf{(A)}\ a\text{ }\text{and}\text{ }d\qquad \textbf{(B)}\ d\text{ }\text{and}\text{ }n\qquad \textbf{(C)}\ a\text{ }\text{and}\text{ }n\qquad \textbf{(D)}\ a, d,\text{ }\text{and}\text{ }n\qquad \textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n$ | \textbf{(B)}\n | 1/8 |
Rotate a square around a line that lies on one of its sides to form a cylinder. If the volume of the cylinder is $27\pi \text{cm}^3$, then the lateral surface area of the cylinder is _________ $\text{cm}^2$. | 18\pi | 7/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.