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Mia is researching a yeast population. There are 50 yeast cells present at 10:00 a.m. and the population triples every 5 minutes. Assuming none of the yeast cells die, how many yeast cells are present at 10:18 a.m. the same day? | 1350 | 4/8 |
In a right triangle $PQR$, where $\angle P = 90^\circ$, suppose $\cos Q = \frac{4}{5}$. The length of side $PQ$ (adjacent to $\angle Q$) is $12$. What is the length of $PR$ (hypotenuse)? | 15 | 2/8 |
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} | 1/8 |
Let $[x]$ denote the largest integer not greater than the real number $x$. Define $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2016}}{8}\right]$. Find the remainder when $A$ is divided by 50. | 42 | 7/8 |
Given a rectangular parallelepiped \(A B C D A_{1} B_{1} C_{1} D_{1}\). Points \(K\) and \(P\) are the midpoints of edges \(B B_{1}\) and \(A_{1} D_{1}\), respectively. Point \(H\) is the midpoint of edge \(C C_{1}\). Point \(E\) lies on edge \(B_{1} C_{1}\) and \(B_{1} E : E C_{1} = 1 : 3\). Is it true that the line \(K P\) intersects the lines \(A E\) and \(A_{1} H\)? | True | 4/8 |
A radio system consisting of 1000 components (each with a failure rate of $\lambda_{i} = 10^{-6}$ failures/hour) has been tested and accepted by the customer. Determine the probability of the system operating without failure over the interval $t_{1} < (t = t_{1} + \Delta t) < t_{2}$, where $\Delta t = 1000$ hours. | 0.367879 | 1/8 |
Calculate the definite integral:
$$
\int_{0}^{\operatorname{arctg} \frac{1}{3}} \frac{8+\operatorname{tg} x}{18 \sin ^{2} x+2 \cos ^{2} x} \, dx
$$ | \frac{\pi}{3}+\frac{\ln2}{36} | 7/8 |
We have some identical sweet candies that we distribute into three non-empty heaps in such a way that the number of candies in each heap is different. How many candies do we have if the number of possible different distributions in this way is exactly one more than the number of candies? | 18 | 4/8 |
Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$ . The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$ , and the angle bisector of $\angle B$ intersects the side $AC$ at $E$ .
Find the measure of $\measuredangle{BED}$ . | 45 | 6/8 |
$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$ g(x)+g(y)=g(x+y)-xy. $$ If $g(23)=0$ , what is the sum of all possible values of $g(35)$ ? | 210 | 7/8 |
There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2017$ and $y^3 - 3x^2y = 2016$. Compute $\left(2 - \frac{x_1}{y_1}\right)\left(2 - \frac{x_2}{y_2}\right)\left(2 - \frac{x_3}{y_3}\right)$. | \frac{26219}{2016} | 2/8 |
The squares of a $3 \times 3$ grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. How many such labelings are possible? | 2448 | 1/8 |
About 40% of students in a certain school are nearsighted, and about 30% of the students in the school use their phones for more than 2 hours per day, with a nearsighted rate of about 50% among these students. If a student who uses their phone for no more than 2 hours per day is randomly selected from the school, calculate the probability that the student is nearsighted. | \frac{5}{14} | 7/8 |
Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called *good* if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called *very good* if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of *good* elements in $M$ and $v$ denote the number of *very good* elements in $M.$ Prove that
\[v^2+v \leq g \leq n^2-n.\] | v^2+v\le\len^2-n | 2/8 |
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number | 6 | 4/8 |
In triangle $ABC$, $\angle C=90^{\circ}, \angle B=30^{\circ}, AC=2$, $M$ is the midpoint of $AB$. Fold triangle $ACM$ along $CM$ such that the distance between points $A$ and $B$ becomes $2\sqrt{2}$. Find the volume of the resulting triangular pyramid $A-BCM$. | \frac{2 \sqrt{2}}{3} | 6/8 |
Given that \( m \) and \( n \) are known positive integers, and the number of digits of \( m \) in decimal notation is \( d \), where \( d \leq n \). Find the sum of all digits in the decimal representation of \((10^n - 1)m\). | 9n | 3/8 |
A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the distance between points \( A \) and \( B \)? | 40 | 7/8 |
The perimeter of triangle \(ABC\) is \(2p\). What is the maximum length that a segment parallel to \(BC\) and tangent to the inscribed circle of \(ABC\) can have inside the triangle? For what triangle(s) is this value achieved? | \frac{p}{4} | 5/8 |
If \( a^3 + b^3 + c^3 = 3abc = 6 \) and \( a^2 + b^2 + c^2 = 8 \), find the value of \( \frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a} \). | -8 | 2/8 |
In the trapezoid \( ABCD \) with bases \( AD \) and \( BC \), the side \( AB \) is equal to 2. The angle bisector of \( \angle BAD \) intersects the line \( BC \) at point \( E \). A circle is inscribed in triangle \( ABE \), touching side \( AB \) at point \( M \) and side \( BE \) at point \( H \). Given that \( MH = 1 \), find the angle \( \angle BAD \). | 120 | 1/8 |
Given $A=\{x|x^{3}+3x^{2}+2x > 0\}$, $B=\{x|x^{2}+ax+b\leqslant 0\}$ and $A\cap B=\{x|0 < x\leqslant 2\}$, $A\cup B=\{x|x > -2\}$, then $a+b=$ ______. | -3 | 7/8 |
Given that acute angles $\alpha$ and $\beta$ satisfy $\sin\alpha=\frac{4}{5}$ and $\cos(\alpha+\beta)=-\frac{12}{13}$, determine the value of $\cos \beta$. | -\frac{16}{65} | 1/8 |
How many distinct four-digit positive integers are there such that the product of their digits equals 18? | 36 | 5/8 |
We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$ . What is the sum of the three smallest metallic integers?
*Proposed by Lewis Chen* | 165 | 1/8 |
The sides of triangle \(ABC\) are divided by points \(M, N\), and \(P\) such that \(AM : MB = BN : NC = CP : PA = 1 : 4\). Find the ratio of the area of the triangle bounded by lines \(AN, BP\), and \(CM\) to the area of triangle \(ABC\). | 3/7 | 4/8 |
If $x = 151$ and $x^3y - 3x^2y + 3xy = 3423000$, what is the value of $y$? | \frac{3423000}{3375001} | 1/8 |
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 | 7/8 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sin ^{2}A+\cos ^{2}B+\cos ^{2}C=2+\sin B\sin C$.<br/>$(1)$ Find the measure of angle $A$;<br/>$(2)$ If $a=3$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$, find the maximum length of segment $AD$. | \frac{\sqrt{3}}{2} | 7/8 |
A circle passing through the vertex $A$ of triangle $ABC$ is tangent to side $BC$ at point $M$ and intersects sides $AC$ and $AB$ respectively at points $L$ and $K$, different from vertex $A$. Find the ratio $AC: AB$, given that the length of segment $LC$ is twice the length of segment $KB$, and the ratio $CM: BM = 3: 2$. | \frac{9}{8} | 7/8 |
A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours? | 225 | 4/8 |
There are 15 cards, each with 3 different Chinese characters. No two cards have the exact same set of Chinese characters, and in any set of 6 cards, there are always at least 2 cards that share a common Chinese character. What is the maximum number of different Chinese characters that can be on these 15 cards? | 35 | 1/8 |
Solve the following equations using appropriate methods:
$(1)\left(3x-1\right)^{2}=9$.
$(2)x\left(2x-4\right)=\left(2-x\right)^{2}$. | -2 | 2/8 |
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
$\textbf{(A) }2\sqrt{3}\qquad \textbf{(B) }4\qquad \textbf{(C) }3\sqrt{2}\qquad \textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }5\qquad$ | \textbf{(D)}\2\sqrt{5} | 1/8 |
Given that \( S_n \) and \( T_n \) are the sums of the first \( n \) terms of the arithmetic sequences \( \{a_n\} \) and \( \{b_n\} \), respectively, and that
\[
\frac{S_n}{T_n} = \frac{2n + 1}{4n - 2} \quad (n = 1, 2, \ldots),
\]
find the value of
\[
\frac{a_{10}}{b_3 + b_{18}} + \frac{a_{11}}{b_6 + b_{15}}.
\] | \frac{41}{78} | 7/8 |
A line that passes through the two foci of the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola. | \frac{\sqrt{5} + 1}{2} | 1/8 |
The line \( L \) crosses the \( x \)-axis at \((-8,0)\). The area of the shaded region is 16. Find the slope of the line \( L \). | \frac{1}{2} | 3/8 |
Three numbers $a, b,$ and $c$ were written on a board. They were erased and replaced with the numbers $a-1, b+1,$ and $c^2$. After this, it turned out that the same numbers were on the board as initially (possibly in a different order). What values can the number $a$ take, given that the sum of the numbers $a, b,$ and $c$ is 2008? If necessary, round your answer to the nearest hundredth. | 1004 | 1/8 |
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
| 408 | 6/8 |
A group of cyclists decided to practice on a small road before the competition. They all ride at a speed of 35 km/h. Then one of the cyclists suddenly breaks away from the group and, moving at a speed of 45 km/h, travels 10 km, then turns back and, without reducing speed, rejoins the rest (who continue to move at the same speed, not paying attention to him).
How much time has passed since the cyclist broke away from the group until he rejoined them? | \frac{1}{4}\\(or\15\) | 1/8 |
A stone general is a chess piece that moves one square diagonally upward on each move; that is, it may move from the coordinate \((a, b)\) to either of the coordinates \((a-1, b+1)\) or \((a+1, b+1)\). How many ways are there for a stone general to move from \((5,1)\) to \((4,8)\) in seven moves on a standard 8 by 8 chessboard? | 35 | 6/8 |
Let \( f(x) \) and \( g(x) \) have continuous derivatives on \([0,1]\), and \( f(0) = 0 \), \( f^{\prime}(x) \geq 0 \), \( g^{\prime}(x) \geq 0 \). Prove that for any \( a \in [0,1] \), the following inequality holds:
\[ \int_{0}^{a} g(x) f^{\prime}(x) \, \mathrm{d}x + \int_{0}^{1} f(x) g^{\prime}(x) \, \mathrm{d} x \geq f(a) g(1). \] | \int_{0}^{}(x)f^{\}(x)\,\mathrm{}x+\int_{0}^{1}f(x)^{\}(x)\,\mathrm{}x\gef()(1) | 1/8 |
Given that $\overrightarrow{a}=(2,3)$, $\overrightarrow{b}=(-4,7)$, and $\overrightarrow{a}+\overrightarrow{c}=\overrightarrow{0}$, find the projection of $\overrightarrow{c}$ on the direction of $\overrightarrow{b}$. | -\frac{\sqrt{65}}{5} | 4/8 |
In the plane \(Oxy\), the coordinates of point \(A\) are given by the equation:
\[ 5a^2 - 4ay + 8x^2 - 4xy + y^2 + 12ax = 0 \]
and the parabola with its vertex at point \(B\) is given by the equation:
\[ ax^2 - 2a^2 x - ay + a^3 + 3 = 0 \]
Find all values of the parameter \(a\) for which points \(A\) and \(B\) lie on the same side of the line \(2x - y = 5\) (points \(A\) and \(B\) do not lie on this line). | (-\frac{5}{2},-\frac{1}{2})\cup(0,3) | 5/8 |
You are given a square $n \times n$ . The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$ , find the largest value of $m$ for which it is possible.
*Proposed by Oleksiy Masalitin, Fedir Yudin* | 2n-1 | 1/8 |
Given a sphere resting on a flat surface and a 1.5 m tall post, the shadow of the sphere is 15 m and the shadow of the post is 3 m, determine the radius of the sphere. | 7.5 | 1/8 |
A circle $\omega$ with center $O$ is circumscribed around triangle $ABC$. Circle $\omega_{1}$ is tangent to line $AB$ at point $A$ and passes through point $C$, and circle $\omega_{2}$ is tangent to line $AC$ at point $A$ and passes through point $B$. A line is drawn through point $A$ that intersects circle $\omega_{1}$ a second time at point $X$ and circle $\omega_{2}$ a second time at point $Y$. Point $M$ is the midpoint of segment $XY$. Find the angle $OMX$. | 90 | 3/8 |
The bases of two cones, sharing a common vertex, lie in the same plane. The difference in their volumes is \( V \). Find the volume of the smaller cone if the tangents drawn to the circle of its base from an arbitrary point on the circumference of the base of the larger cone form an angle \( \alpha \). | V\tan^2\frac{\alpha}{2} | 5/8 |
Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$ . Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$ . Tangent form $D$ touches $\Omega$ in $E$ . FInd $\angle BEC$ . | 110 | 3/8 |
Compute
\[ e^{2 \pi i/17} + e^{4 \pi i/17} + e^{6 \pi i/17} + \dots + e^{32 \pi i/17}. \] | -1 | 7/8 |
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths of 10 and 24 units respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. Determine the area of quadrilateral $GHCD$. | 153.75 | 1/8 |
A sequence \(\left\{a_{n}\right\}_{n \geq 1}\) of positive reals is defined by the rule \(a_{n+1} a_{n-1}^{5} = a_{n}^{4} a_{n-2}^{2}\) for integers \(n > 2\) together with the initial values \(a_{1} = 8\), \(a_{2} = 64\), and \(a_{3} = 1024\). Compute
\[
\sqrt{a_{1}+\sqrt{a_{2}+\sqrt{a_{3}+\cdots}}}
\] | 3\sqrt{2} | 1/8 |
In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters? | 5 | 2/8 |
What is the smallest number that can be written as a sum of $2$ squares in $3$ ways? | 325 | 7/8 |
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio. | -\frac{1}{2} | 7/8 |
Find the value of $y$ if $y$ is positive and $y \cdot \lfloor y \rfloor = 132$. Express your answer as a decimal. | 12 | 1/8 |
In how many ways can two people be chosen from ten for two different positions? | 90 | 7/8 |
The probability of an event occurring in each of the independent trials is 0.8. How many trials need to be conducted in order to expect the event to occur at least 75 times with a probability of 0.9? | 100 | 7/8 |
Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$ . Find the number of elements in $S$ that do not divide $\frac{L}{2016}$ .
*Proposed by Yannick Yao* | 44 | 7/8 |
\[\log_{10} x + \log_{\sqrt{10}} x + \log_{\sqrt[3]{10}} x + \ldots + \log_{\sqrt[1]{10}} x = 5.5\] | \sqrt[10]{10} | 1/8 |
How many pairs of positive integer solutions \((x, y)\) satisfy \(\frac{1}{x+1} + \frac{1}{y} + \frac{1}{(x+1) y} = \frac{1}{1991}\)? | 64 | 4/8 |
Nikolai invented a game for himself: he rearranges the digits in the number 2015 and then places a multiplication sign between any two digits. Neither of the resulting numbers should start with a zero. He then calculates the value of this expression. For example: \( 150 \times 2 = 300 \) or \( 10 \times 25 = 250 \). What is the maximum number he can get as a result of such a calculation? | 1050 | 7/8 |
Reflect a point in space across the faces of a given tetrahedron and consider the sphere determined by the four reflected images. Let the center of this sphere be $P$ and its radius be $r$. Prove that if the same steps are performed with $P$, the radius of the new sphere will also be $r$. | r | 3/8 |
The difference between two numbers A and B is 144. Number A is 14 less than three times number B. What is the value of number A? | 223 | 4/8 |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | 60 | 2/8 |
Given the hyperbola \( x^2 - \frac{y^2}{4} = 1 \):
1. Two perpendicular rays are drawn from the center \( O \) and intersect the hyperbola at points \( A \) and \( B \). Find the equation of the locus of the midpoint \( P \) of segment \( AB \).
2. Find the distance from the center \( O \) of the hyperbola to the line \( AB \). | \frac{2\sqrt{3}}{3} | 1/8 |
Given that $b_1, b_2, b_3, b_4$ are positive integers, the polynomial $g(z)=(1-z)^{b_1}\left(1-z^2\right)^{b_2}\left(1-z^3\right)^{b_3}\left(1-z^4\right)^{b_4}$ simplifies to $1-2z$ when terms higher than the 4th degree are ignored. Additionally, $\alpha$ is the largest root of the polynomial $f(x)=x^3-b_4 x^2+b_2$. Find the remainder when $\left[\alpha^{1995}\right]$ is divided by 9, where $[x]$ denotes the greatest integer less than or equal to $x$. | 5 | 1/8 |
Given a pyramid \(S-ABC\) with height \(SO = 3\) and a base whose side length is 6, a perpendicular is drawn from point A to its opposite face \(SBC\) with the foot of the perpendicular being \(O'\). On the line segment \(AO'\), a point \(P\) is selected such that \(\frac{AP}{PO'} = 8\). Find the area of the cross-section through point \(P\) and parallel to the base. | \sqrt{3} | 5/8 |
From the 4 digits 0, 1, 2, 3, select 3 digits to form a three-digit number without repetition. How many of these three-digit numbers are divisible by 3? | 10 | 7/8 |
A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider? ("Opposite" means "symmetric with respect to the center of the cube"). | No | 2/8 |
Mike bought two books from the Canadian Super Mathematics Company. He paid full price for a \$33 book and received 50% off the full price of a second book. In total, he saved 20% on his purchase. In dollars, how much did he save? | 11 | 7/8 |
In a $3 \times 3$ square grid, the numbers $1, 2, 3, \ldots, 9$ are placed. Then, in each circle (Fig. 2), the arithmetic mean of the four surrounding numbers is recorded. After that, the arithmetic mean of these four resulting numbers is calculated. What is the largest number that can be obtained? | 6.125 | 1/8 |
Let the function $f(x)=2\tan \frac{x}{4}\cdot \cos^2 \frac{x}{4}-2\cos^2\left(\frac{x}{4}+\frac{\pi }{12}\right)+1$.
(Ⅰ) Find the smallest positive period and the domain of $f(x)$;
(Ⅱ) Find the intervals of monotonicity and the extremum of $f(x)$ in the interval $[-\pi,0]$; | -\frac{\sqrt{3}}{2} | 3/8 |
The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$ , $x_{n+1}=1-x_1x_2x_3*...*x_{100}$ . Prove that $x_{100}>0.99$ . | x_{100}>0.99 | 3/8 |
Find all functions \( f: \mathbb{R} \longrightarrow \mathbb{R} \) such that \( f \) is monotone and there exists an \( n \in \mathbb{N} \) such that for all \( x \):
\[ f^{n}(x) = -x \] | f(x)=-x | 1/8 |
795. Calculate the double integral \(\iint_{D} x y \, dx \, dy\), where region \(D\) is:
1) A rectangle bounded by the lines \(x=0, x=a\), \(y=0, y=b\);
2) An ellipse \(4x^2 + y^2 \leq 4\);
3) Bounded by the line \(y=x-4\) and the parabola \(y^2=2x\). | 90 | 6/8 |
Let \(ABC\) be a triangle with centroid \(G\). Determine, with proof, the position of the point \(P\) in the plane of \(ABC\) such that
\[ AP \cdot AG + BP \cdot BG + CP \cdot CG \]
is a minimum, and express this minimum value in terms of the side lengths of \(ABC\). | \frac{^2+b^2+^2}{3} | 3/8 |
Equilateral $\triangle A B C$ has side length 6. Let $\omega$ be the circle through $A$ and $B$ such that $C A$ and $C B$ are both tangent to $\omega$. A point $D$ on $\omega$ satisfies $C D=4$. Let $E$ be the intersection of line $C D$ with segment $A B$. What is the length of segment $D E$? | \frac{20}{13} | 4/8 |
Each integer 1 through 9 is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer? | 0 | 6/8 |
Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$ (2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2. $$ | 100 | 6/8 |
The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much should each resident contribute to collect exactly the required rent? | 10.54 | 7/8 |
Round 1278365.7422389 to the nearest hundred. | 1278400 | 6/8 |
Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions: - $S$ is non-empty. - No subset of $S$ has the property that the sum of its elements is 10. | 34 | 1/8 |
The sum \(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}\) is represented as a fraction with the denominator \(45! = 1 \cdot 2 \cdots 45\). How many zeros (in decimal notation) does the numerator of this fraction end with? | 8 | 4/8 |
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? | 5 | 1/8 |
If the integer part of $\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\_\_\_\_\_\_$. | \sqrt{10} - 3 | 2/8 |
Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A)+\cos(3B)+\cos(3C)=1.$ Two sides of the triangle have lengths 10 and 13. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}.$ Find $m.$ | 399 | 3/8 |
The sequence \(a_{n} = b[\sqrt{n+c}] + d\) is given, where the successive terms are
\[
1, 3, 3, 3, 5, 5, 5, 5, 5, \cdots
\]
In this sequence, each positive odd number \(m\) appears exactly \(m\) times consecutively. The integers \(b\), \(c\), and \(d\) are to be determined. Find the value of \(b+c+d\). | 2 | 7/8 |
Let \( k \) be a natural number. Let \( S \) be a set of \( n \) points in the plane such that:
- No three points of \( S \) are collinear
- For every point \( P \) in \( S \), there exists a real number \( r \) such that there are at least \( k \) points at a distance \( r \) from \( P \).
Show that:
$$
k < \frac{1}{2} + \sqrt{2n}
$$ | k<\frac{1}{2}+\sqrt{2n} | 7/8 |
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$ . | q | 7/8 |
The area of an equilateral triangle is $50 \sqrt{12}$. If its perimeter is $p$, find $p$.
The average of $q, y, z$ is 14. The average of $q, y, z, t$ is 13. Find $t$.
If $7-24 x-4 x^{2} \equiv K+A(x+B)^{2}$, where $K, A, B$ are constants, find $K$.
If $C=\frac{3^{4 n} \cdot 9^{n+4}}{27^{2 n+2}}$, find $C$. | 9 | 6/8 |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\in\mathbb{N}^*)$, and the sequence $\{a_{2n-1}\}$ is increasing while $\{a_{2n}\}$ is decreasing, then $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}=$ ______. | -\frac{1}{2} | 2/8 |
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$. | 1700 | 3/8 |
Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a smooth function such that \( f^{\prime}(x)^{2} = f(x) f^{\prime \prime}(x) \) for all \( x \). Suppose \( f(0) = 1 \) and \( f^{(4)}(0) = 9 \). Find all possible values of \( f^{\prime}(0) \). | \\sqrt{3} | 1/8 |
Given the numbers \(\log _{\sqrt{2 x-3}}(x+1)\), \(\log _{2 x^{2}-3 x+5}(2 x-3)^{2}\), \(\log _{x+1}\left(2 x^{2}-3 x+5\right)\), find the values of \(x\) for which two of these numbers are equal and the third is smaller by 1. | 4 | 1/8 |
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$ . Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to $17$ . | 30 | 1/8 |
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is | \frac{2}{x+1} | 7/8 |
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$. | \mathbb{Z} | 4/8 |
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$
| 149 | 7/8 |
Let
$$
\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots-\frac{1}{1}}}}=\frac{m}{n}
$$
where \(m\) and \(n\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \(m^2 + mn - n^2\). | -1 | 1/8 |
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