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The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) }19\qquad \textbf{(B) }31\qquad \textbf{(C) }271\qquad \textbf{(D) }319\qquad \textbf{(E) }511\qquad$ | \textbf{(C)}271 | 1/8 |
Let \( \triangle ABC \) be a triangle with \( AB = 3 \), \( AC = 8 \), \( BC = 7 \). \( M \) and \( N \) are the midpoints of \( \overline{AB} \) and \( \overline{AC} \), respectively. Point \( T \) is selected on side \( BC \) such that \( AT = TC \). The circumcircles of triangles \( BAT \) and \( MAN \) intersect at \( D \). Compute \( DC \). | \frac{7\sqrt{3}}{3} | 7/8 |
The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minute hand at moment $Y$ will be where the hour hand is at moment $X$. How many interesting moments will there be from 00:01 to 12:01? | 143 | 3/8 |
$ (a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $ a_n \not \equal{} 0$ , $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$ and $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$ . So $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$ | 42 | 5/8 |
A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.
How many meters of wire were there originally in the bundle? | 54 | 5/8 |
Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$. | \frac{2865 - 120\sqrt{165}}{49} | 1/8 |
Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\%$ profit and then buys it back from Mr. $Y$ at a $15\%$ loss. | 2160 | 7/8 |
Given $F(x) = \int_{0}^{x} (t^{2} + 2t - 8) \, dt$, where $x > 0$.
1. Determine the intervals of monotonicity for $F(x)$.
2. Find the maximum and minimum values of the function $F(x)$ on the interval $[1, 3]$. | -\frac{28}{3} | 7/8 |
Find all integers \( n > 1 \) such that any prime divisor of \( n^6 - 1 \) is a divisor of \((n^3 - 1)(n^2 - 1)\). | 2 | 5/8 |
The base of a rectangular parallelepiped is a square with a side length of \(2 \sqrt{3}\). The diagonal of a lateral face forms an angle of \(30^\circ\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped. | 72 | 7/8 |
Prove that the sum of an odd number of unit vectors passing through the same point \( O \) and lying in the same half-plane whose border passes through \( O \) has length greater than or equal to 1. | 1 | 5/8 |
Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$ , such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines. | 2 | 4/8 |
Let \( n \in \mathbf{Z}_{+} \), and
$$
\begin{array}{l}
a, b, c \in \{x \mid x \in \mathbf{Z} \text{ and } x \in [1,9]\}, \\
A_{n} = \underbrace{\overline{a a \cdots a}}_{n \text{ digits}}, B_{n} = \underbrace{b b \cdots b}_{2n \text{ digits}}, C_{n} = \underbrace{c c \cdots c}_{2n \text{ digits}}.
\end{array}
$$
The maximum value of \(a + b + c\) is ( ), given that there exist at least two \(n\) satisfying \(C_{n} - B_{n} = A_{n}^{2}\). | 18 | 1/8 |
What is the minimum number of cells that need to be colored in a square with a side length of 65 cells (a $65 \times 65$ square, with a total of 4225 cells) so that from any uncolored cell, it is impossible to reach another uncolored cell with a knight's move in chess? | 2112 | 7/8 |
Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus. | 110 | 1/8 |
There are five positive integers that are divisors of each number in the list $$48, 144, 24, 192, 216, 120.$$ Find the sum of these positive integers. | 16 | 1/8 |
Given the ellipse \( C \):
\[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad (a>b>0), \]
which passes through the point \( P\left(3, \frac{16}{5}\right) \) and has an eccentricity of \( \frac{3}{5} \). Draw a line \( l \) with slope \( k \) passing through the right focus of the ellipse \( C \), intersecting the ellipse at points \( A \) and \( B \). Let the slopes of \( PA \) and \( PB \) be \( k_{1} \) and \( k_{2} \) respectively.
(1) Find the standard equation of the ellipse.
(2) If \( k_{1} + k_{2} = 0 \), find the real number \( k \). | \frac{3}{5} | 7/8 |
Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow.
The guard fired an arrow with an initial velocity of $100 \dfrac{\text{ft}}{\text{s}}$ . At the same time, Edward started running away with an acceleration of $1 \dfrac{\text{ft}}{\text{s}^2}$ . Assuming that air resistance causes the arrow to decelerate at $1 \dfrac{\text{ft}}{\text{s}^2}$ , and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\dfrac{\text{ft}}{\text{s}}$ )? | 75\, | 1/8 |
Given that none of the terms in the sequence \(\left\{a_{n}\right\}\) are zero. Let the sum of the first \(n\) terms of the sequence \(\left\{a_{n}\right\}\) be \(S_{n}\), and the sum of the first \(n\) terms of the sequence \(\left\{a_{n}^{2}\right\}\) be \(T_{n}\). It is known that
\[ S_{n}^{2}+4 S_{n}-3 T_{n}=0\ \ (n \in \mathbf{Z}_{+}). \]
Find the sum of the first \(n\) terms of the sequence \(\left\{\frac{(n-1)}{n(n+1)} a_{n}\right\}\). | \frac{2^{n+1}}{n+1}-2 | 5/8 |
Given an ellipse with its center at the origin \( O \), foci on the \( x \)-axis, and eccentricity \( \frac{\sqrt{3}}{2} \) which passes through the point \(\left(\sqrt{2}, \frac{\sqrt{2}}{2}\right)\). Suppose a line \( l \) that does not pass through the origin \( O \) intersects the ellipse at points \( P \) and \( Q \), and the slopes of lines \( OP \), \( PQ \), and \( OQ \) form a geometric sequence. Find the possible range of values for the area of triangle \( \triangle OPQ \). | (0,1) | 1/8 |
Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let $a,b,c$ be integers not divisible by $p$ . Prove that there are at most $1+\sqrt {2p}$ positive integers $n$ such that $n<p$ and $p$ divides $a^n+b^n+c^n$ .
| 1+\sqrt{2p} | 1/8 |
$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards.
A) 600
B) 720
C) 840
D) 960
E) 1080 | 720 | 1/8 |
Let $a$ be an integer such that $x^2 - x + a$ divides $x^{13} + x + 90$. Find the value of $a$. | 2 | 4/8 |
Let set $M$ consist of an odd number of elements. For each element $x$ in set $M$, there is a uniquely determined subset $H_{x} \subseteq M$ corresponding to $x$, satisfying the conditions:
(1) For each $x \in M$, $x \in H_{x}$;
(2) For any $x, y \in M$, $y \in H_{x}$ if and only if $x \in H_{y}$.
Prove that there is at least one $H_{x}$ with an odd number of elements. | 1 | 7/8 |
Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .
*2021 CCA Math Bonanza Lightning Round #2.4* | 14 | 5/8 |
In triangle $ABC$ with an acute angle at vertex $A$, an angle bisector $AE$ and an altitude $BH$ are drawn. It is known that $\angle AEB = 45^{\circ}$. Find the angle $EHC$. | 45 | 1/8 |
When $s$ and $t$ range over all real numbers, what is the minimum value of $(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}$? | 2 | 3/8 |
A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\begin{cases} ax+by=2 \\\\ 2x+y=3\\end{cases}$.
(I) Find the probability that the system of equations has only one solution.
(II) If each solution of the system of equations corresponds to a point $P(x,y)$ in the Cartesian plane, find the probability that point $P$ lies in the fourth quadrant. | \frac{7}{12} | 4/8 |
Person A and person B independently attempt to decrypt a password. Their probabilities of successfully decrypting the password are $\dfrac{1}{3}$ and $\dfrac{1}{4}$, respectively. Calculate:
$(1)$ The probability that exactly one of them decrypts the password.
$(2)$ If the probability of decrypting the password needs to be $\dfrac{99}{100}$, what is the minimum number of people like B required? | 17 | 6/8 |
The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is: | C=10+3(P-1) | 4/8 |
Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$ , $$ f\left(m^2+n^2\right)=f(m)^2+f(n)^2. $$ (a) Calculate $f(k)$ for $0\le k\le12$ .
(b) Calculate $f(n)$ for any natural number $n$ . | f(n)=n | 1/8 |
Let $\mathcal{C}$ be the hyperbola $y^2 - x^2 = 1$. Given a point $P_0$ on the $x$-axis, we construct a sequence of points $(P_n)$ on the $x$-axis in the following manner: let $\ell_n$ be the line with slope 1 passing passing through $P_n$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_n$ and $\mathcal C$ onto the $x$-axis. (If $P_n = 0$, then the sequence simply terminates.)
Find the number of starting positions $P_0$ on the $x$-axis such that $P_0 = P_{2008}$. Your answer should use exponential notation, in simplest form. | 2^{2008}-2 | 3/8 |
Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . The incircle of $ABC$ meets $BC$ at $D$ . Line $AD$ meets the circle through $B$ , $D$ , and the reflection of $C$ over $AD$ at a point $P\neq D$ . Compute $AP$ .
*2020 CCA Math Bonanza Tiebreaker Round #4* | 2\sqrt{145} | 5/8 |
Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$? | \frac{2+\sqrt{3}}{3} | 3/8 |
Find the number of solutions to $\sin x = \lg x$. | 3 | 5/8 |
Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain?
(An *arithmetic progression* is a set of the form $\{a,a+d,\dots,a+kd\}$ , where $a,d,k$ are positive integers, and $k\geqslant 2$ ; thus an arithmetic progression has at least three elements, and successive elements have difference $d$ , called the *common difference* of the arithmetic progression.) | n^2 | 6/8 |
Determine the number of pairs of positive integers $x,y$ such that $x\le y$ , $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$ . | 16384 | 7/8 |
Prove that the circumcenter $O$, centroid $G$, and orthocenter $H$ of triangle $\triangle ABC$ are collinear and that $OG:GH = 1:2$. | OG:GH=1:2 | 6/8 |
Let $S$ be the set of points $(a, b)$ with $0 \leq a, b \leq 1$ such that the equation $x^{4}+a x^{3}-b x^{2}+a x+1=0$ has at least one real root. Determine the area of the graph of $S$. | \frac{1}{4} | 5/8 |
The point $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\circ$ about the $y$-axis, and reflected through the $xz$-plane. Find the coordinates of the point now. | (-1,1,1) | 7/8 |
It is known that \(a^{2} + b = b^{2} + c = c^{2} + a\). What values can the expression \(a\left(a^{2} - b^{2}\right) + b\left(b^{2} - c^{2}\right) + c\left(c^{2} - a^{2}\right)\) take? | 0 | 7/8 |
Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? | \frac{2\sqrt{3}}{3} | 1/8 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 630 = 0$ has integral solutions? | 160 | 7/8 |
What is the probability that Hannah gets fewer than 4 heads if she flips 12 coins? | \frac{299}{4096} | 7/8 |
For a group of children, it holds that in every trio of children from the group, there is a boy named Adam, and in every quartet, there is a girl named Beata.
What is the maximum number of children that can be in such a group, and what are their names? | 5 | 3/8 |
Olave sold 108 apples at a constant rate over 6 hours. If she continues to sell apples at the same rate, how many apples will she sell in the next 1 hour and 30 minutes?
(A) 27
(B) 33
(C) 45
(D) 36
(E) 21 | 27 | 7/8 |
For all real numbers \( x \), let
\[ f(x) = \frac{1}{\sqrt[2011]{1 - x^{2011}}}. \]
Evaluate \( (f(f(\ldots(f(2011)) \ldots)))^{2011} \), where \( f \) is applied 2010 times. | 2011^{2011} | 6/8 |
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)? | 15 | 2/8 |
Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$. | 360 | 1/8 |
Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( BC = 8 \) and \( AC = 4 \). Find the length of side \( AB \) if the length of the vector \( 4 \overrightarrow{OA} - \overrightarrow{OB} - 3 \overrightarrow{OC} \) is 10. | 5 | 5/8 |
Find the smallest \( a \in \mathbf{N}^{*} \) such that the following equation has real roots:
$$
\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0 .
$$ | 6 | 2/8 |
In an acute-angled triangle \(ABC\), median \(BM\) and altitude \(CH\) are drawn. It is given that \(BM = CH = \sqrt{3}\), and \(\angle MBC = \angle ACH\). Find the perimeter of triangle \(ABC\). | 6 | 2/8 |
The number of passengers on the bus remained constant from departure to the second stop. Then, it was observed that at each stop, as many people boarded as had boarded at the two previous stops, and as many people disembarked as had boarded at the previous stop. The tenth stop from departure is the final one, and at this stop, 55 passengers disembarked.
How many passengers were on the bus between the seventh and eighth stops? | 21 | 1/8 |
Let $S$ be a subset of $\{1, 2, . . . , 500\}$ such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in $S$ . | 306 | 3/8 |
You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square? | 36 | 3/8 |
The altitude drawn from the vertex of the right angle to the hypotenuse divides the triangle into two triangles, each of which is inscribed with a circle. Find the angles and the area of the triangle formed by the legs of the original triangle and the line passing through the centers of these circles, given that the height of the original triangle is \( h \). | \frac{^2}{2} | 1/8 |
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$ , $ AB\equal{}CD$ , $ \angle ACD\equal{}\angle CBD$ . Find angle $ \angle ACD$ | 30 | 1/8 |
On another island of knights, liars, and normal people, the king held opposite views and gave his daughter different paternal advice: "Dear, I don't want you to marry any knight or liar. I would like you to marry a solid, normal person with a good reputation. You should not marry a knight, because all knights are hypocrites. You should not marry a liar either, because all liars are treacherous. No, no matter what, a decent normal person would be just right for you!"
Assume you are an inhabitant of this island and a normal person. Your task is to convince the king that you are a normal person.
a) How many true statements will you need for this?
b) How many false statements will you need for the same purpose?
(In both cases, the task is to find the minimum number of statements.). | 1 | 4/8 |
A regular dodecagon \( Q_1 Q_2 \dotsb Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (4,0) \) and \( Q_7 \) at \( (2,0) \). If \( Q_n \) is the point \( (x_n,y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).
\] | 531440 | 7/8 |
Inside a convex $n$-gon there are 100 points positioned in such a way that no three of these $n+100$ points are collinear. The polygon is divided into triangles, each having vertices among any 3 of the $n+100$ points. For what maximum value of $n$ can no more than 300 triangles be formed? | 102 | 7/8 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \):
\[ f(f(f(x) + f(y))) = f(x) + y. \] | f(x)=x | 5/8 |
A convex quadrilateral is determined by the points of intersection of the curves \(x^{4} + y^{4} = 100\) and \(xy = 4\); determine its area. | 4\sqrt{17} | 3/8 |
Given that there is a gathering attended by 1982 people, and among any group of 4 people, at least 1 person knows the other 3. How many people, at minimum, must know all the attendees at this gathering? | 1979 | 1/8 |
For any positive integer \( n \), let \( f(n) \) denote the number of 1's in the base-2 representation of \( n \). For how many values of \( n \) with \( 1 \leq n \leq 2002 \) do we have \( f(n) = f(n+1) \)? | 501 | 5/8 |
Find the smallest positive integer \( n \) such that any simple graph with 10 vertices and \( n \) edges, with edges colored in two colors, always contains a monochromatic triangle or a monochromatic quadrilateral. | 31 | 3/8 |
Given the function \( f(x) = \left|\log_{2} x\right| \), if the real numbers \( a \) and \( b \) (where \( a < b \)) satisfy \( f(a) = f(b) \), then find the range of values for \( a + 2014b \). | (2015,+\infty) | 2/8 |
Let \( S_{n} = 1 + 2 + \cdots + n \). How many of \( S_{1}, S_{2}, \cdots, S_{2015} \) are multiples of 2015?
| 8 | 4/8 |
A rectangular table of dimensions \( x \) cm \(\times 80\) cm is covered with identical sheets of paper of size \( 5 \) cm \(\times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters? | 77 | 7/8 |
Find a positive integer that is divisible by 21 and has a square root between 30 and 30.5. | 903 | 4/8 |
A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at 1 meter per second. Suddenly someone turns on the turntable; it spins at $30 \mathrm{rpm}$. Consider the set $S$ of points the mouse can reach in his car within 1 second after the turntable is set in motion. What is the area of $S$, in square meters? | \frac{\pi}{6} | 7/8 |
Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occupied given that all planets are distinct. | 981 | 4/8 |
Given that in quadrilateral ABCD, $\angle A : \angle B : \angle C : \angle D = 1 : 3 : 5 : 6$, express the degrees of $\angle A$ and $\angle D$ in terms of a common variable. | 144 | 2/8 |
Xiaohua wrote a computer program. After one minute of running the program, some bubbles first appear on the computer screen. Thereafter, new bubbles will appear at every whole minute, in the same quantity as the first minute. Half a minute after the 11th appearance of the bubbles, one bubble bursts. Subsequently, every minute, more bubbles will burst, with the quantity increasing by 1 each minute (i.e., half a minute after the 12th appearance of bubbles, 2 bubbles will burst, etc.). At a certain point, the total number of burst bubbles exactly equals the total number of bubbles that have appeared on the screen, meaning that all bubbles have disappeared. What is the maximum number of bubbles that can appear on the screen simultaneously during the entire process? | 1026 | 1/8 |
Let \( n \) be a strictly positive integer. Let \( m_{n} \) be the largest real number satisfying the following property: "Let \( x_{0}, x_{1}, \ldots, x_{n} \) be mutually distinct integers. For any monic polynomial \( F(x) \) of degree \( n \) with integer coefficients, at least one of the numbers \( \left|F\left(x_{0}\right)\right|, \left|F\left(x_{1}\right)\right|, \ldots,\left|F\left(x_{n}\right)\right| \) is greater than or equal to \( m_{n} \)."
Find the problem corresponding to the integer part of \( m_{9} \). | 708 | 1/8 |
Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with
every possible height-weight combination.
The USAMTS wants to field a competitive team, so there are some strict requirements.
- If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
- If person $P$ is on the team, then no one whose weight is the same as $P$ ’s height can also be on the team.
Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed? | 128 | 1/8 |
The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$. | 67 | 5/8 |
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? | 2\sqrt{2}+2 | 1/8 |
The sum of an infinite geometric series is \( 16 \) times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio? | -\frac{1}{4} | 1/8 |
A and B agreed to meet at a restaurant for a meal. Due to the restaurant being very busy, A arrived first and took a waiting number and waited for B. After a while, B also arrived but did not see A, so he took another waiting number. While waiting, B saw A, and they took out their waiting numbers and discovered that the digits of these two numbers are reversed two-digit numbers, and the sum of the digits of both numbers (for example, the sum of the digits of 23 is $2+3=5$) is 8, and B's number is 18 greater than A's. What is A's number? | 35 | 4/8 |
In triangle \( ABC \), given \( a^{2} + b^{2} + c^{2} = 2\sqrt{3} \, ab \, \sin C \), find \( \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \). | \frac{3\sqrt{3}}{8} | 7/8 |
Define the sequence \(x_{i}\) by \(x_{1}=a\) and \(x_{i+1}=2x_{i}+1\). Define the sequence \(y_{i}=2^{x_{i}}-1\). Determine the largest integer \(k\) such that \(y_{1}, \ldots, y_{k}\) are all prime numbers. | 2 | 5/8 |
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\{b_{n}\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \gt 1$.
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;
$(2)$ Let all terms of sequences $\{a_{n}\}$ and $\{b_{n}\}$ form sets $A$ and $B$ respectively. Arrange the elements of $A\cup B$ in ascending order to form a new sequence $\{c_{n}\}$. Find $T_{20}=c_{1}+c_{2}+c_{3}+\cdots +c_{20}$. | 660 | 4/8 |
Let \( Q_1 \) be a regular \( t \)-gon and \( Q_2 \) be a regular \( u \)-gon \((t \geq u \geq 3)\) such that each interior angle of \( Q_1 \) is \( \frac{60}{59} \) as large as each interior angle of \( Q_2 \). What is the largest possible value of \( u \)? | 119 | 5/8 |
Given 4 points \( A, B, C, D \) on a plane, with the distances \( AB = 1, BC = 9, CA = 9, AD = 7 \). Prove that at least one of the lengths \( BD \) or \( CD \) is not an integer. | 1 | 1/8 |
Given an obtuse triangle \( \triangle ABC \) with the following conditions:
1. The lengths of \( AB \), \( BC \), and \( CA \) are positive integers.
2. The lengths of \( AB \), \( BC \), and \( CA \) do not exceed 50.
3. The lengths of \( AB \), \( BC \), and \( CA \) form an arithmetic sequence with a positive common difference.
Determine the number of obtuse triangles that satisfy the above conditions, and identify the side lengths of the obtuse triangle with the largest perimeter. | 157 | 1/8 |
Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the remaining dogs into these groups? | 18480 | 7/8 |
The polynomial \( f(x) = x^{2007} + 17 x^{2006} + 1 \) has distinct zeroes \( r_1, \ldots, r_{2007} \). A polynomial \( P \) of degree 2007 has the property that \( P\left( r_j + \frac{1}{r_j} \right) = 0 \) for \( j = 1, \ldots, 2007 \). Determine the value of \( P(1) / P(-1) \). | \frac{289}{259} | 2/8 |
The inscribed circle of the right triangle \(ABC\) (with the right angle at \(C\)) touches the sides \(AB\), \(BC\), and \(CA\) at the points \(C_1\), \(A_1\), and \(B_1\) respectively. The altitudes of the triangle \(A_1B_1C_1\) intersect at the point \(D\). Find the distance between the points \(C\) and \(D\), given that the lengths of the legs of the triangle \(ABC\) are 3 and 4. | 1 | 7/8 |
The expression $\frac{k^{2}}{1.001^{k}}$ reaches its maximum value with which natural number $k$? | 2001 | 6/8 |
Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X. | 200 | 7/8 |
For testing a certain product, there are 6 different genuine items and 4 different defective items. The test continues until all defective items are identified. If all defective items are exactly identified by the 5th test, how many possible testing methods are there? | 576 | 5/8 |
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$ | (B)\4 | 1/8 |
The sides of a triangle with area \( T \) are \( a, b, c \). Prove that
$$
T^{2} \leq \frac{a b c(a+b+c)}{16}
$$ | T^2\le\frac{(+b+)}{16} | 5/8 |
The nonzero numbers \( a \), \( b \), and \( c \) satisfy the equations \( a^{2}(b+c-a) = b^{2}(a+c-b) = c^{2}(b+a-c) \). What is the maximum value that the expression \(\frac{2b + 3c}{a}\) can take? | 5 | 2/8 |
Prove that if the $k$-th, $n$-th, and $p$-th terms of an arithmetic progression form three consecutive terms of a geometric progression, then its common ratio is $\frac{n-p}{k-n}$. | \frac{n-p}{k-n} | 7/8 |
Given the sets $M={x|m\leqslant x\leqslant m+ \frac {7}{10}}$ and $N={x|n- \frac {2}{5}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, find the minimum value of the "length" of the set $M\cap N$. (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.) | \frac{1}{10} | 7/8 |
In triangle \( \triangle ABC \), \(AB = AC\) and \(\angle BAC = 100^\circ\). Point \(D\) is on the extension of side \(AB\) such that \(AD = BC\). Find the measure of \(\angle BCD\). | 10 | 3/8 |
At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting? | 36 | 7/8 |
In the staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight congruent sides marked with a tick mark have length 1 foot. If the region has area 53 square feet, what is the number of feet in the perimeter of the region? [asy]
size(120);
draw((5,7)--(0,7)--(0,0)--(9,0)--(9,3)--(8,3)--(8,4)--(7,4)--(7,5)--(6,5)--(6,6)--(5,6)--cycle);
label("9 ft",(4.5,0),S);
draw((7.85,3.5)--(8.15,3.5)); draw((6.85,4.5)--(7.15,4.5)); draw((5.85,5.5)--(6.15,5.5)); draw((4.85,6.5)--(5.15,6.5));
draw((8.5,2.85)--(8.5,3.15)); draw((7.5,3.85)--(7.5,4.15)); draw((6.5,4.85)--(6.5,5.15)); draw((5.5,5.85)--(5.5,6.15));
[/asy] | 32 | 6/8 |
To prevent the spread of the novel coronavirus, individuals need to maintain a safe distance of at least one meter between each other. In a certain meeting room with four rows and four columns of seats, the distance between adjacent seats is more than one meter. During the epidemic, for added safety, it is stipulated that when holding a meeting in this room, there should not be three people seated consecutively in any row or column. For example, the situation shown in the first column of the table below does not meet the condition (where "$\surd $" indicates a seated person). According to this rule, the maximum number of participants that can be accommodated in this meeting room is ____.
| | | | |
|-------|-------|-------|-------|
| $\surd $ | | | |
| $\surd $ | | | |
| $\surd $ | | | | | 11 | 1/8 |
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