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Determine the area of the region of the circle defined by $x^2 + y^2 - 8x + 16 = 0$ that lies below the $x$-axis and to the left of the line $y = x - 4$. | 4\pi | 1/8 |
The lengths of the sides of triangle \(ABC\) are \(a\), \(b\), and \(c\) (with \(AB = c\), \(BC = a\), \(CA = b\) and \(a < b < c\)). Points \(B_1\) and \(A_1\) are marked on rays \(BC\) and \(AC\) respectively such that \(BB_1 = AA_1 = c\). Points \(C_2\) and \(B_2\) are marked on rays \(CA\) and \(BA\) respectively such that \(CC_2 = BB_2 = a\). Find the ratio \(A_1B_1 : C_2B_2\). | \frac{}{} | 1/8 |
Below is a portion of the graph of a quadratic function, $y=p(x)=dx^2 + ex + f$:
The value of $p(12)$ is an integer. The graph's axis of symmetry is $x = 10.5$, and the graph passes through the point $(3, -5)$. Based on this, what is the value of $p(12)$? | -5 | 1/8 |
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\frac{12}5\sqrt2$. What is the volume of the tetrahedron? | \frac{24}{5} | 4/8 |
Find (in terms of $n \geq 1$) the number of terms with odd coefficients after expanding the product: $\prod_{1 \leq i<j \leq n}\left(x_{i}+x_{j}\right)$ | n! | 1/8 |
Alex and Katy play a game on an \(8 \times 8\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the number of Ks on the grid at the end of the game. What is the highest score Katy can be sure to get if she plays well, no matter what Alex does? | 32 | 1/8 |
Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \cdot zyxw$? | 11 | 1/8 |
Given a tree with $n$ vertices, where $n \geq 2$. Each vertex is assigned a number $x_{1}, x_{2}, \ldots, x_{n}$, and each edge is labeled with the product of the numbers at its endpoints. Let $S$ denote the sum of the numbers on all edges. Prove that $\sqrt{n-1}\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right) \geq 2 S$. | \sqrt{n-1}(x_1^2+x_2^2+\ldots+x_n^2)\ge2S | 2/8 |
A complex number \(\omega\) satisfies \(\omega^{5}=2\). Find the sum of all possible values of \(\omega^{4} + \omega^{3} + \omega^{2} + \omega + 1\). | 5 | 7/8 |
The lateral sides \( AB \) and \( CD \) of trapezoid \( ABCD \) are 8 and 10 respectively, and the base \( BC \) is 2. The angle bisector of \(\angle ADC\) passes through the midpoint of the side \( AB \). Find the area of the trapezoid. | 40 | 3/8 |
The greatest common divisor of two positive integers is $(x+7)$ and their least common multiple is $x(x+7)$, where $x$ is a positive integer. If one of the integers is 56, what is the smallest possible value of the other one? | 294 | 1/8 |
Given that \(a^{3} - a - 1 = 0\), where \(a + \sqrt{2}\) is a root of a polynomial with integer coefficients, find the polynomial with the highest degree coefficient equal to 1 (and with the lowest possible degree) that satisfies the given condition. | x^6-8x^4-2x^3+13x^2-10x-1 | 6/8 |
Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\). | 110 | 5/8 |
Find the envelope of a family of straight lines that form a triangle with a right angle and have an area of $a^{2} / 2$. | xy=\frac{^2}{4} | 1/8 |
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format) | 3:00 | 4/8 |
Determine all composite positive integers $n$ with the following property: Let $1 = d_1 < d_2 < \ldots < d_k = n$ be all the positive divisors of $n$. Then:
$$
\left(d_{2} - d_{1}\right) : \left(d_{3} - d_{2}\right) : \cdots : \left(d_{k} - d_{k-1}\right) = 1 : 2 : \cdots : (k - 1).
$$ | 4 | 1/8 |
In a $4 \times 4$ grid of 16 small squares, fill in 2 $a$'s and 2 $b$'s, with each square containing at most one letter. If the same letter is neither in the same row nor in the same column, how many different arrangements are there? | 3960 | 3/8 |
Given that the diagonals $AC$ and $BD$ of quadrilateral $ABCD$ intersect at point $P$, a line through $P$ intersects the lines containing the sides $AB$, $BC$, $CD$, and $DA$ at points $E$, $M$, $F$, and $N$, respectively, and that $PE = PF$. Prove that $PM = PN$. | PM=PN | 6/8 |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n=n^2$ ($n\in\mathbb{N}^*$).
1. Find $a_n$;
2. The function $f(n)$ is defined as $$f(n)=\begin{cases} a_{n} & \text{, $n$ is odd} \\ f(\frac{n}{2}) & \text{, $n$ is even}\end{cases}$$, and $c_n=f(2^n+4)$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of the sequence ${c_n}$, denoted as $T_n$.
3. Let $\lambda$ be a real number, for any positive integers $m$, $n$, $k$ that satisfy $m+n=3k$ and $m\neq n$, the inequality $S_m+S_n>\lambda\cdot S_k$ always holds, find the maximum value of the real number $\lambda$. | \frac{9}{2} | 1/8 |
An ordered pair of sets $(A, B)$ is good if $A$ is not a subset of $B$ and $B$ is not a subset of $A$. How many ordered pairs of subsets of $\{1,2, \ldots, 2017\}$ are good? | 4^{2017}-2\cdot3^{2017}+2^{2017} | 5/8 |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2? | \frac{1}{663} | 1/8 |
Simplify $(- \sqrt {3})^{2}\;^{- \frac {1}{2}}$. | \frac{\sqrt{3}}{3} | 7/8 |
Suppose you have a sphere tangent to the $xy$ -plane with its center having positive $z$ -coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$ -plane, it gives the conic section $y=x^2$ . If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$ . | 3 | 1/8 |
In $\triangle ABC$, the sides corresponding to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $b = a \sin C + c \cos A$,
(1) Find the value of $A + B$;
(2) If $c = \sqrt{2}$, find the maximum area of $\triangle ABC$. | \frac{1 + \sqrt{2}}{2} | 7/8 |
All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?
[asy]
label("A",(0,0),N);
label("B",(10,0),N);
label("C",(20,0),N);
label("D",(30,0),N);
label("E",(40,0),N);
label("Row 1",(-10,-7),W);
label("2",(10,-12),N);
label("3",(20,-12),N);
label("4",(30,-12),N);
label("5",(40,-12),N);
label("Row 2",(-10,-24),W);
label("9",(0,-29),N);
label("8",(10,-29),N);
label("7",(20,-29),N);
label("6",(30,-29),N);
label("Row 3",(-10,-41),W);
label("10",(10,-46),N);
label("11",(20,-46),N);
label("12",(30,-46),N);
label("13",(40,-46),N);
label("Row 4",(-10,-58),W);
label("17",(0,-63),N);
label("16",(10,-63),N);
label("15",(20,-63),N);
label("14",(30,-63),N);
label("Row 5",(-10,-75),W);
label("18",(10,-80),N);
label("19",(20,-80),N);
label("20",(30,-80),N);
label("21",(40,-80),N);
[/asy] | \text{B} | 5/8 |
Determine the remainder when $$\sum_{i=0}^{2015}\left\lfloor\frac{2^{i}}{25}\right\rfloor$$ is divided by 100, where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. | 14 | 7/8 |
Prove that if \(a, b, c\) are the side lengths of a triangle with a perimeter of 2, then \(a^{2}+b^{2}+c^{2}<2(1-abc)\). | ^2+b^2+^2<2(1-abc) | 7/8 |
How many different three-letter sets of initials are possible using the letters $A$ through $G$? | 343 | 7/8 |
A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
$\textbf{(A) } 22 \qquad\textbf{(B) } 24 \qquad\textbf{(C) } 25 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 28$ | \textbf{(B)}24 | 1/8 |
The sequence is defined as \( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\frac{k}{a_{k}} \) for \( k=1,2, \cdots, n-1 \). Determine the value of \( n \) for which \( a_{n}=0 \). | 201 | 7/8 |
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}, BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$? | 194 | 4/8 |
Estimate the population of Nisos in the year 2050. | 2000 | 1/8 |
Seven children sit in a circle. The teacher distributes pieces of candy to the children such that:
- Every child gets at least one piece of candy.
- No two children receive the same number of pieces of candy.
- The numbers of candy pieces given to any two adjacent children have a common factor other than 1.
- There is no prime dividing every child's number of candy pieces.
What is the smallest number of pieces of candy that the teacher must have ready for the children? | 44 | 1/8 |
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.”
Claire says, “Now I know your favorite number!” What is Cat’s favorite number? | 25 | 1/8 |
Let $ABC$ be a triangle, let the $A$ -altitude meet $BC$ at $D$ , let the $B$ -altitude meet $AC$ at $E$ , and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$ . Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$ , then the area of $ABC$ can be written as $a+\sqrt{b}$ , where $a$ and $b$ are positive integers. What is $a+b$ ?
*2021 CCA Math Bonanza Individual Round #12* | 112 | 4/8 |
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches?
$\textbf{(A)}\hspace{.05in}36\qquad\textbf{(B)}\hspace{.05in}40\qquad\textbf{(C)}\hspace{.05in}64\qquad\textbf{(D)}\hspace{.05in}72\qquad\textbf{(E)}\hspace{.05in}88$ | \textbf{(E)}\88 | 1/8 |
Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February. | 142.00 | 1/8 |
Let $a_1$ , $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$ .
Call a nonempty contiguous subsequence of this sequence *good* if the product of the integers in it leaves a remainder of $1$ when divided by $101$ .
In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$ .
*Proposed by Yang Liu* | 19320 | 3/8 |
In the geometric sequence $\{a_{n}\}$, $a_{5}a_{8}=6$, $a_{3}+a_{10}=5$, then $\frac{a_{20}}{a_{13}}=$ \_\_\_\_\_\_. | \frac{2}{3} | 5/8 |
For what real values of \( k \) do \( 1988x^2 + kx + 8891 \) and \( 8891x^2 + kx + 1988 \) have a common zero? | \10879 | 2/8 |
Suppose $r^{}_{}$ is a real number for which
$\left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546.$
Find $\lfloor 100r \rfloor$. (For real $x^{}_{}$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x^{}_{}$.) | 743 | 7/8 |
Given that
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
find the greatest integer that is less than $\frac N{100}$.
| 137 | 7/8 |
Maria is 54 inches tall, and Samuel is 72 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is each person in centimeters? Additionally, what is the difference in their heights in centimeters? | 45.72 | 5/8 |
For which values of the parameter \( p \) does the inequality
$$
\left(1+\frac{1}{\sin x}\right)^{3} \geq \frac{p}{\tan^2 x}
$$
hold for any \( 0 < x < \frac{\pi}{2} \)? | p\le8 | 1/8 |
Given $f(x)=\cos x\cdot\ln x$, $f(x_{0})=f(x_{1})=0(x_{0}\neq x_{1})$, find the minimum value of $|x_{0}-x_{1}|$ ___. | \dfrac {\pi}{2}-1 | 7/8 |
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds. | 9 | 7/8 |
In the Cartesian coordinate system $xOy$, the equation of circle $C$ is $(x- \sqrt {3})^{2}+(y+1)^{2}=9$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of circle $C$;
$(2)$ The line $OP$: $\theta= \frac {\pi}{6}$ ($p\in R$) intersects circle $C$ at points $M$ and $N$. Find the length of segment $MN$. | 2 \sqrt {6} | 6/8 |
Let the set \( A = \left\{\frac{1}{2}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{15}, \frac{1}{32}\right\} \) have non-empty subsets \( A_{1}, A_{2}, \cdots, A_{63} \). Denote the product of all elements in the subset \( A_{i} \) (where \( i=1,2,\cdots,63 \)) as \( p_{i} \) (the product of elements in a single-element subset is the element itself). Then,
\[
p_{1} + p_{2} + \cdots + p_{63} =
\] | \frac{79}{65} | 6/8 |
(1) Use the Horner's method to calculate the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 79x^3 - 8x^2 + 35x + 12$ when $x = -4$, find the value of $v_3$.
(2) Convert the hexadecimal number $210_{(6)}$ into a decimal number. | 78 | 7/8 |
The base of the trapezoid \( ABCD \) is \( AB \). On the leg \( AD \), point \( P \), and on the leg \( CB \), point \( Q \) are such that \( AP:PD=CQ:QB \). The line \( PQ \) intersects \( AC \) at \( R \) and \( BD \) at \( S \). Prove that \( PR = SQ \). Is the statement still true if \( P \) and \( Q \) are on the extensions of the legs? | PR=SQ | 2/8 |
Let \( X \) be the set of non-negative reals. \( f: X \to X \) is bounded on the interval \([0, 1]\) and satisfies \( f(x) f(y) \leq x^2 f\left(\frac{y}{2}\right) + y^2 f\left(\frac{x}{2}\right) \) for all \( x, y \). Show that \( f(x) \leq x^2 \). | f(x)\lex^2 | 2/8 |
Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors. | 23 | 7/8 |
Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | 280 | 7/8 |
Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum? | 10 | 7/8 |
A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \mathrm{~cm}$, and the son's step length is $60 \mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What distance did they travel? | 1440 | 7/8 |
The sequence \(a_n\) is defined by \(a_1 = 20\), \(a_2 = 30\), and \(a_{n+1} = 3a_n - a_{n-1}\). Find all \(n\) for which \(5a_{n+1} \cdot a_n + 1\) is a perfect square. | 3 | 1/8 |
Let \( x \) be the smallest positive integer that simultaneously satisfies the following conditions: \( 2x \) is the square of an integer, \( 3x \) is the cube of an integer, and \( 5x \) is the fifth power of an integer. Find the prime factorization of \( x \). | 2^{15}\cdot3^{20}\cdot5^{24} | 3/8 |
Seven points are evenly spaced out on a circle and connected as shown below to form a 7-pointed star. What is the sum of the angle measurements of the seven tips of the star, in degrees? One such angle is marked as $\alpha$ below.
[asy]
dotfactor=4;
draw(Circle((0,0),1));
real x = 2*pi/7;
pair A,B,C,D,E,F,G;
A=(cos(4*x), sin(4*x));
B=(cos(3*x), sin(3*x));
C=(cos(2*x), sin(2*x));
D=(cos(x), sin(x));
E=(cos(5*x), sin(5*x));
F=(cos(6*x), sin(6*x));
G=(cos(7*x), sin(7*x));
dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot((0,0));
label("$A$",A,W); label("$B$",B,W); label("$C$",C,N); label("$D$",D,N); label("$E$",G,ENE); label("$F$",F,SE); label("$G$",E,S);
draw(A--C--G--E--B--D--F--cycle); label("$\alpha$",C, - 1.5*dir(C));
[/asy] | 540 | 6/8 |
Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. | 682 | 5/8 |
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$ . Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$ . Find the minimum possible value of sum $P_A +P_B$ .
PS. It is a variation of [JBMO Shortlist 2019 A3 ](https://artofproblemsolving.com/community/c6h2267998p17621980) | 402 | 1/8 |
The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers? | 4 | 2/8 |
Petra had natural numbers from 1 to 9 written down. She added two of these numbers, erased them, and wrote the resulting sum in place of the erased addends. She then had eight numbers left, which she was able to divide into two groups with the same product.
Determine the largest possible product of these groups. | 504 | 1/8 |
Calculate \(\sin (\alpha+\beta)\), given \(\sin \alpha+\cos \beta=\frac{1}{4}\) and \(\cos \alpha+\sin \beta=-\frac{8}{5}\). | \frac{249}{800} | 5/8 |
The distance from point $A$ to point $B$ by rail is 88 km, while by river it is 108 km. The train departs from $A$ 1 hour later than the ship and arrives in $B$ 15 minutes earlier. Find the average speed of the train, given that it is 40 km/h faster than the average speed of the ship. | 88\, | 1/8 |
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$ . Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$ . Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$ , where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$ . | 7 | 1/8 |
Given a sequence of positive integers $a_1, a_2, a_3, \ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \ldots$), let $b_j = k_1 + k_2 + \ldots + k_j$ ($j=1, 2, 3, \ldots$),
define $g(m) = b_1 + b_2 + \ldots + b_m - 100m$ ($m=1, 2, 3, \ldots$).
(I) Given $k_1 = 40, k_2 = 30, k_3 = 20, k_4 = 10, k_5 = \ldots = k_{100} = 0$, calculate $g(1), g(2), g(3), g(4)$;
(II) If the maximum term in $a_1, a_2, a_3, \ldots, a_{100}$ is 50, compare the values of $g(m)$ and $g(m+1)$;
(III) If $a_1 + a_2 + \ldots + a_{100} = 200$, find the minimum value of the function $g(m)$. | -100 | 1/8 |
Regular decagon \( ABCDEFGHIJ \) has its center at \( K \). Each of the vertices and the center are to be associated with one of the digits \( 1 \) through \( 10 \), with each digit used exactly once, in such a way that the sums of the numbers on the lines \( AKF \), \( BKG \), \( CKH \), \( DKI \), and \( EKJ \) are all equal. Find the number of valid ways to associate the numbers. | 3840 | 1/8 |
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
$\textbf{(A)}\ 210 \qquad \textbf{(B)}\ 220 \qquad \textbf{(C)}\ 230 \qquad \textbf{(D)}\ 240 \qquad \textbf{(E)}\ 250$ | \textbf{(D)}\240 | 1/8 |
Find the number of ordered pairs $(a,b)$ of complex numbers such that
\[a^3 b^5 = a^7 b^2 = 1.\] | 29 | 4/8 |
Joshua chooses five distinct numbers. In how many different ways can he assign these numbers to the variables $p, q, r, s$, and $t$ so that $p<s, q<s, r<t$, and $s<t$? | 8 | 2/8 |
A monkey is climbing up and down a ladder with $n$ rungs. Each time it ascends 16 rungs or descends 9 rungs. If it can climb from the ground to the top rung and then return to the ground, what is the minimum value of $n$? | 24 | 1/8 |
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6
| x - 2y + 3z| \le 6
| x - 2y - 3z| \le 6
| x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$ . | 6 | 1/8 |
Given that \(\operatorname{tg} \alpha = m\), find
\[ A = \sin^{2}\left(\frac{\pi}{4} + \alpha\right) - \sin^{2}\left(\frac{\pi}{6} - \alpha\right) - \cos \frac{5 \pi}{12} \cdot \sin \left(\frac{5 \pi}{12} - 2 \alpha\right) \] | \frac{2m}{1+^2} | 5/8 |
Determine which of the following expressions has the largest value: $4^2$, $4 \times 2$, $4 - 2$, $\frac{4}{2}$, or $4 + 2$. | 16 | 7/8 |
In \(\triangle ABC\), \(AB = 2AC\) and \(\angle BAC = 112^\circ\). Points \(P\) and \(Q\) are on \(BC\) such that \(AB^2 + BC \cdot CP = BC^2\) and \(3AC^2 + 2BC \cdot CQ = BC^2\). Find \(\angle PAQ\). | 22 | 1/8 |
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i] | 8 | 1/8 |
Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most 5 such that $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)^{2}=(a+b+c+d)(a-b+c-d)\left((a-c)^{2}+(b-d)^{2}\right)$ | 49 | 1/8 |
Let \( n \geq 1 \) be an integer, and \( x_{1}, \ldots, x_{2n} \) be positive real numbers. Define
\[
S=\left(\frac{1}{2n} \sum_{i=1}^{2n}\left(x_{i}+2\right)^{n}\right)^{\frac{1}{n}}, \quad S^{\prime}=\left(\frac{1}{2n} \sum_{i=1}^{2n}\left(x_{i}+1\right)^{n}\right)^{\frac{1}{n}}, \quad P=\left(\prod_{i=1}^{2n} x_{i}\right)^{\frac{1}{n}}
\]
Show that if \( S \geq P \), then \( S^{\prime} \geq \frac{3}{4} P \). | S'\ge\frac{3}{4}P | 1/8 |
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is\[f(2) + f(3) + f(4) + f(5)+ f(6)?\]$(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$ | \textbf{(A)}8 | 1/8 |
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:
- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$ .
- It is divisible by $11$ .
What is the sum of all possible numbers of cows that the herder has? | 726 | 5/8 |
15 balls numbered 1 through 15 are placed in a bin. Joe produces a list of four numbers by performing the following sequence four times: he chooses a ball, records the number, and places the ball back in the bin. Finally, Joe chooses to make a unique list by selecting 3 numbers from these 4, and forgetting the order in which they were drawn. How many different lists are possible? | 202500 | 1/8 |
What is the largest number, all of whose digits are either 5, 3, or 1, and whose digits add up to $15$? | 555 | 1/8 |
Let $A,B,C,D$ , be four different points on a line $\ell$ , so that $AB=BC=CD$ . In one of the semiplanes determined by the line $\ell$ , the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$ . | 60 | 3/8 |
Tanya was 16 years old 19 months ago, and Misha will be 19 years old in 16 months. Who is older? Explain your answer. | Mishaisolder | 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 \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 1025.\]
Find $n_1 + n_2 + \cdots + n_r$. | 17 | 1/8 |
Diameter $\overline{AB}$ of a circle with center $O$ is $10$ units. $C$ is a point $4$ units from $A$, and on $\overline{AB}$. $D$ is a point $4$ units from $B$, and on $\overline{AB}$. $P$ is any point on the circle. Then the broken-line path from $C$ to $P$ to $D$:
$\textbf{(A)}\ \text{has the same length for all positions of }{P}\qquad\\ \textbf{(B)}\ \text{exceeds }{10}\text{ units for all positions of }{P}\qquad \\ \textbf{(C)}\ \text{cannot exceed }{10}\text{ units}\qquad \\ \textbf{(D)}\ \text{is shortest when }{\triangle CPD}\text{ is a right triangle}\qquad \\ \textbf{(E)}\ \text{is longest when }{P}\text{ is equidistant from }{C}\text{ and }{D}.$ | \textbf{(E)} | 1/8 |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18$
| 16 | 1/8 |
Find all functions $f : R \to R$ satisfying the conditions:
1. $f (x + 1) \ge f (x) + 1$ for all $x \in R$
2. $f (x y) \ge f (x)f (y)$ for all $x, y \in R$ | f(x)=x | 1/8 |
Let \(\mathcal{P}\) be a parabola with focus \(F\) and directrix \(\ell\). A line through \(F\) intersects \(\mathcal{P}\) at two points \(A\) and \(B\). Let \(D\) and \(C\) be the feet of the altitudes from \(A\) and \(B\) onto \(\ell\), respectively. Given that \(AB = 20\) and \(CD = 14\), compute the area of \(ABCD\). | 140 | 3/8 |
Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number).
$261 \times 345$
$234 \times 345$
$256 \times 345$
$562 \times 345$ | $234 \times 345$ | 2/8 |
Define $ a \circledast b = a + b-2ab $ . Calculate the value of $$ A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014} $$ | \frac{1}{2} | 7/8 |
In this problem only, assume that $s_{1}=4$ and that exactly one board square, say square number $n$, is marked with an arrow. Determine all choices of $n$ that maximize the average distance in squares the first player will travel in his first two turns. | n=4 | 2/8 |
A box contains a collection of triangular, square, and rectangular tiles. There are 32 tiles in the box, consisting of 114 edges in total. Each rectangle has 5 edges due to a small notch cut on one side. Determine the number of square tiles in the box. | 10 | 1/8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $A > B$, $\cos C= \frac {5}{13}$, and $\cos (A-B)= \frac {3}{5}$.
(1) Find the value of $\cos 2A$;
(2) If $c=15$, find the value of $a$. | 2 \sqrt {65} | 7/8 |
Given sets $A=\{x,\frac{y}{x},1\}$ and $B=\{{x}^{2},x+y,0\}$, if $A=B$, then $x^{2023}+y^{2024}=\_\_\_\_\_\_.$ | -1 | 6/8 |
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
$\textbf{(A)}\ 17\qquad\textbf{(B)}\ 19\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 23$ | \textbf{(E)}\23 | 1/8 |
The hexagon with the R is colored red. Each hexagon is colored either red, yellow or green, such that no two hexagons with a common side are colored the same color. In how many different ways can the figure be colored?
[asy]
path a=(0,0)--(10,0)--(15,8.7)--(10,17.3)--(0,17.3)--(-5,8.7)--cycle;
draw(a);
draw(shift((15,8.7))*a);
draw(shift((15,-8.7))*a);
draw(shift((30,17.3))*a);
draw(shift((30,0))*a);
draw(shift((30,-17.3))*a);
draw(shift((45,26))*a);
draw(shift((45,8.7))*a);
draw(shift((45,-8.7))*a);
draw(shift((60,17.3))*a);
draw(shift((60,0))*a);
draw(shift((60,-17.3))*a);
draw(shift((75,8.7))*a);
draw(shift((75,-8.7))*a);
label("$R$",(5,10),S);
[/asy] | 2 | 1/8 |
Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=a_{2}=1, a_{3}=2, \text{ and } 3a_{n+3}=4a_{n+2}+a_{n+1}-2a_{n}, \text{ for } n=1,2, \cdots,$ find the general term of the sequence $\left\{a_{n}\right\}$. | a_n=\frac{1}{25}+\frac{3}{5}n-\frac{27}{50}(-\frac{2}{3})^n | 1/8 |
Find the volume of the pyramid \( PABCD \), which has a quadrilateral \( ABCD \) with sides 5, 5, 10, and 10 as the base, its shorter diagonal being \( 4\sqrt{5} \), and all the lateral faces inclined at an angle of \( 45^\circ \) to the base. | \frac{500}{9} | 5/8 |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-104)^{2}
$$
If the result is a non-integer, round it to the nearest integer. | 49608 | 7/8 |
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