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Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1 to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1? (For example, Raashan and Ted may each decide to give $1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0, Sylvia will have $2, and Ted will have $1, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $1 to, and the holdings will be the same at the end of the second round.) | \frac{1}{4} | 1/8 |
Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let \( n \) be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and \( m \) be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair \((n, m)\). | (3,4) | 1/8 |
Given positive integers \( x, y, z \) such that \( x + 2xy + 3xyz = 115 \), find \( x + y + z \). | 10 | 1/8 |
Let \( x \) and \( y \) be positive integers. Can \( x^{2} + 2y \) and \( y^{2} + 2x \) both be squares? | No | 3/8 |
Inside triangle \(ABC\) there are three circles \(k_1, k_2, k_3\), each of which is tangent to two sides of the triangle and to its incircle \(k\). The radii of \(k_1, k_2, k_3\) are 1, 4, and 9. Determine the radius of \(k\). | 11 | 5/8 |
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$ , find the minimum value of $p$ . | 5 | 7/8 |
From a uniform straight rod, a piece with a length of $s=80 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod shift as a result? | 40\\mathrm{} | 1/8 |
The axial section $SAB$ of a conical frustum is an equilateral triangle with side length 2. $O$ is the center of the base, and $M$ is the midpoint of $SO$. The moving point $P$ is on the base of the conical frustum (including the circumference). If $AM \perp MP$, find the length of the locus formed by point $P$. | \frac{\sqrt{7}}{2} | 2/8 |
A plate is bounded by the parabola \( y^{2} = x \) and its chord passing through the focus perpendicularly to the axis of the parabola. Find the mass of the plate if at each point the surface density is inversely proportional to the distance from the point to the directrix of the parabola. | 2K(1-\frac{\pi}{4}) | 1/8 |
Let $O_1$ , $O_2$ be two circles with radii $R_1$ and $R_2$ , and suppose the circles meet at $A$ and $D$ . Draw a line $L$ through $D$ intersecting $O_1$ , $O_2$ at $B$ and $C$ . Now allow the distance between the centers as well as the choice of $L$ to vary. Find the length of $AD$ when the area of $ABC$ is maximized. | \frac{2R_1R_2}{\sqrt{R_1^2+R_2^2}} | 1/8 |
Let ABCD be a trapezoid with $AB \parallel CD, AB = 5, BC = 9, CD = 10,$ and $DA = 7$ . Lines $BC$ and $DA$ intersect at point $E$ . Let $M$ be the midpoint of $CD$ , and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$ ). If $EN^2 = \tfrac ab$ for relatively prime positive integers $a$ and $b$ , compute $100a + b$ . | 90011 | 6/8 |
Triangle $ABC$ has $\angle BAC=90^\circ$ . A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , $AC=4$ , and $AX=\tfrac{9}{4}$ , compute the length of $AO$ . | 39/32 | 7/8 |
There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when $N$ is divided by $1000$.
| 310 | 2/8 |
Given a parallelogram \\(ABCD\\) where \\(AD=2\\), \\(∠BAD=120^{\\circ}\\), and point \\(E\\) is the midpoint of \\(CD\\), if \\( \overrightarrow{AE} \cdot \overrightarrow{BD}=1\\), then \\( \overrightarrow{BD} \cdot \overrightarrow{BE}=\\) \_\_\_\_\_\_. | 13 | 7/8 |
Leon has cards with digits from 1 to 7. How many ways are there to glue two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9? | 36 | 2/8 |
**p1.** Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is.**p2.** Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors?**p3.** Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls?**p4.** Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$ . $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order.**p5.** Steven has $100$ boxes labeled from $ 1$ to $100$ . Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$ . The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have?**p6.** In acute $\vartriangle ABC$ , $AB=4$ . Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$ . Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$ . Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$ . If $AP = 6$ , compute $AC$ .**p7.** Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$ , two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have?**p8.** Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile?**p9.** Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number?**p10.** Mike wrote on the board the numbers $1, 2, ..., n$ . At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible?
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | -2 | 2/8 |
Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$, and $T$, respectively. Line $\ell$ divides region $\mathcal{R}$ into two regions with areas in the ratio $1: 2$. Suppose that $AU = 84$, $AN = 126$, and $UB = 168$. Then $DA$ can be represented as $m\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. | 69 | 1/8 |
There are three pairwise distinct natural numbers $a$, $b$, $c$. Prove that the numbers $2023 + a - b$, $2023 + b - c$, and $2023 + c - a$ cannot be three consecutive natural numbers. | 3 | 2/8 |
The length of the hypotenuse of an isosceles right triangle is 40. A circle with a radius of 9 touches the hypotenuse at its midpoint. Find the length of the segment cut off by this circle on one of the legs of the triangle. | \sqrt{82} | 7/8 |
The perimeter of a trapezoid inscribed in a circle is 40. Find the midsegment of the trapezoid. | 10 | 1/8 |
Suppose that $a$ is a multiple of 4 and $b$ is a multiple of 8. Which of the following statements are true?
A. $a+b$ must be even.
B. $a+b$ must be a multiple of 4.
C. $a+b$ must be a multiple of 8.
D. $a+b$ cannot be a multiple of 8.
Answer by listing your choices in alphabetical order, separated by commas. For example, if you think all four are true, then answer $\text{A,B,C,D}$ | \text{A,B} | 7/8 |
The Smith family has four girls aged 5, 5, 5, and 12, and two boys aged 13 and 16. What is the mean (average) of the ages of the children? | 9.33 | 1/8 |
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. | 19 | 7/8 |
Let $ABC$ be a right triangle with hypotenuse $AC$. Let $B^{\prime}$ be the reflection of point $B$ across $AC$, and let $C^{\prime}$ be the reflection of $C$ across $AB^{\prime}$. Find the ratio of $[BCB^{\prime}]$ to $[BC^{\prime}B^{\prime}]$. | 1 | 5/8 |
Let $x,$ $y,$ $z$ be positive real number such that $xyz = \frac{2}{3}.$ Compute the minimum value of
\[x^2 + 6xy + 18y^2 + 12yz + 4z^2.\] | 18 | 4/8 |
A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood small blue hexagon' in exactly 2 ways? | 44 | 6/8 |
Arthur Gombóc lives at 1 Édes Street, and the chocolate shop is at the other end of the street, at number $n$. Each day, Arthur performs the following fitness routine: he starts from house number 2. If he is standing in front of house number $k$ (where $1 < k < n$), he flips an expired but fair chocolate coin. If it lands heads, he moves to house number $(k-1)$, and if it lands tails, he moves to house number $(k+1)$. If he reaches the chocolate shop, he goes in and eats a chocolate ball, then moves to house number $(n-1)$. If he reaches home, the exercise is over. On average, how many chocolate balls does Arthur eat each day? | 1 | 3/8 |
Quadrilateral \(ABCD\) with mutually perpendicular diagonals \(AC\) and \(BD\) is inscribed in a circle. Find its radius, given that \(AB = 4\) and \(CD = 2\). | \sqrt{5} | 5/8 |
In triangle \(ABC\) with a \(120^\circ\) angle at vertex \(A\), the bisectors \(AA_1\), \(BB_1\), and \(CC_1\) are drawn. Find the angle \(C_1 A_1 B_1\). | 90 | 5/8 |
Let $I$ be the set of all points in the plane such that both their x-coordinate and y-coordinate are irrational numbers, and let $R$ be the set of points whose both coordinates are rational. What is the maximum number of points from $R$ that can lie on a circle with an irrational radius and a center that belongs to $I$? | 2 | 5/8 |
On a long bench, a boy and a girl were sitting. One by one, 20 more children approached them, and each sat between two children already sitting there. We call a girl "brave" if she sat between two neighboring boys, and a boy "brave" if he sat between two neighboring girls. When everyone was seated, it turned out that boys and girls were sitting on the bench alternately. How many of them were brave? | 10 | 1/8 |
The velocity \( v \) of a point moving in a straight line changes over time \( t \) according to the law \( v(t) = t \sin 2t \). Determine the distance \( s \) traveled by the point from the start of the motion until time \( t = \frac{\pi}{4} \) units of time. | 0.25 | 1/8 |
Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\cdots+1998+1999-2000-2001$. | 2001 | 3/8 |
Given complex numbers \( x \) and \( y \), find the maximum value of \(\frac{|3x+4y|}{\sqrt{|x|^{2} + |y|^{2} + \left|x^{2}+y^{2}\right|}}\). | \frac{5\sqrt{2}}{2} | 2/8 |
Given a function \( f(x) \) defined on \(\mathbf{R}\), where \(f(1)=1\) and for any \( x \in \mathbf{R} \)
\[
\begin{array}{l}
f(x+5) \geq f(x) + 5 \\
f(x+1) \leq f(x) + 1
\end{array}
\]
If \( g(x) = f(x) + 1 - x \), then \( g(2002) = \) | 1 | 7/8 |
Calculate
\[\prod_{n = 1}^{13} \frac{n(n + 2)}{(n + 4)^2}.\] | \frac{3}{161840} | 3/8 |
Let's call a natural number "remarkable" if it is the smallest among natural numbers with the same sum of digits as it. What is the sum of the digits of the 2001st remarkable number? | 2001 | 5/8 |
In triangle \(ABC\), the height \(BD\) is equal to 11.2 and the height \(AE\) is equal to 12. Point \(E\) lies on side \(BC\) and \(BE : EC = 5 : 9\). Find side \(AC\). | 15 | 7/8 |
At the Lacsap Hospital, Emily is a doctor and Robert is a nurse. Not including Emily, there are five doctors and three nurses at the hospital. Not including Robert, there are $d$ doctors and $n$ nurses at the hospital. What is the product of $d$ and $n$? | 12 | 4/8 |
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
*2019 CCA Math Bonanza Individual Round #5* | 40 | 7/8 |
Show that
\[ \prod_{i=1}^n \text{lcm}(1, 2, 3, \ldots, \left\lfloor \frac{n}{i} \right\rfloor) = n! \] | n! | 7/8 |
A dormitory of a certain high school senior class has 8 people. In a health check, the weights of 7 people were measured to be 60, 55, 60, 55, 65, 50, 50 (in kilograms), respectively. One person was not measured due to some reasons, and it is known that the weight of this student is between 50 and 60 kilograms. The probability that the median weight of the dormitory members in this health check is 55 is __. | \frac{1}{2} | 7/8 |
In how many ways can we choose two different integers between -100 and 100 inclusive, so that their sum is greater than their product? | 10199 | 1/8 |
Given a rectangle, which is not a square, with the numerical value of its area equal to three times its perimeter, prove that one of its sides is greater than 12. | 12 | 7/8 |
Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$. | \frac{13}{7} | 5/8 |
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$. | 17 | 1/8 |
The expression
$$
\left(\begin{array}{lllll}
1 & 1 & 1 & \cdots & 1
\end{array}\right)
$$
is written on a board, with 2013 ones in between the outer parentheses. Between each pair of consecutive ones you may write either "+" or ")(". What is the maximum possible value of the resulting expression? | 3^{671} | 1/8 |
Given a large box containing two smaller boxes, each of which in turn contains two even smaller boxes, and so on for $n$ levels. Each of the $2^n$ smallest boxes contains a coin, with some coins initially showing heads (national emblem side) up and others showing tails (number side) up. You are allowed to flip any one box (of any size) along with all the contents inside it simultaneously. Prove that by making no more than $n$ flips, the number of coins showing heads up can be made equal to the number of coins showing tails up. | n | 1/8 |
A tangent to a circle inscribed in an equilateral triangle with a side length of $a$ intersects two of its sides. Find the perimeter of the resulting smaller triangle. | a | 6/8 |
All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?
$\mathrm{(A)}\ 10 \qquad \mathrm{(B)}\ 11 \qquad \mathrm{(C)}\ 12 \qquad \mathrm{(D)}\ 13 \qquad \mathrm{(E)}\ 14$ | \mathrm{(D)}\13 | 1/8 |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible? | 25 | 4/8 |
Consider pairs of sequences of positive real numbers \( a_1 \geq a_2 \geq a_3 \geq \cdots \) and \( b_1 \geq b_2 \geq b_3 \geq \cdots \), and the sums \( A_n = a_1 + \cdots + a_n \), \( B_n = b_1 + \cdots + b_n \) for \( n = 1, 2, \ldots \). For any pair, define \( c_i = \min \{a_i, b_i\} \) and \( C_n = c_1 + \cdots + c_n \) for \( n = 1, 2, \ldots \).
(a) Does there exist a pair \( \left(a_i\right)_{i \geq 1},\left(b_i\right)_{i \geq 1} \) such that the sequences \( \left(A_n\right)_{n \geq 1} \) and \( \left(B_n\right)_{n \geq 1} \) are unbounded while the sequence \( \left(C_n\right)_{n \geq 1} \) is bounded?
(b) Does the answer to question (a) change by assuming additionally that \( b_i = \frac{1}{i} \) for \( i = 1, 2, \ldots \)?
Justify your answer. | No | 1/8 |
Select two distinct integers, $m$ and $n$, randomly from the set $\{3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $3mn - m - n$ is a multiple of $5$? | \frac{2}{9} | 1/8 |
Xiao Zhang has three watches. The first watch runs 2 minutes fast every hour, the second watch runs 6 minutes fast, and the third watch runs 16 minutes fast. If the minute hands of the three watches are currently all pointing in the same direction, after how many hours will the three minute hands point in the same direction again? | 30 | 7/8 |
In triangle \(ABC\), \(AB = 6\), \(BC = 7\), and \(CA = 8\). Let \(D\), \(E\), and \(F\) be the midpoints of sides \(BC\), \(AC\), and \(AB\), respectively. Also, let \(O_A\), \(O_B\), and \(O_C\) be the circumcenters of triangles \(AFD\), \(BDE\), and \(CEF\), respectively. Find the area of triangle \(O_A O_B O_C\). | \frac{21\sqrt{15}}{16} | 2/8 |
Find the sum of all possible $n$ such that $n$ is a positive integer and there exist $a, b, c$ real numbers such that for every integer $m$ , the quantity $\frac{2013m^3 + am^2 + bm + c}{n}$ is an integer. | 2976 | 6/8 |
Let $k$ be the coefficient of similarity transformation centered at the origin. Does the point $A$ belong to the image of the plane $a$?
$A(2 ; 0 ;-1)$
$a: x - 3y + 5z - 1 = 0$
$k = -1$ | 0 | 1/8 |
Given 6 digits: $0, 1, 2, 3, 4, 5$. Find the sum of all four-digit even numbers that can be written using these digits (each digit may be repeated in the number). | 1769580 | 7/8 |
How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)? | 19 | 4/8 |
Given a sequence $\{a_n\}$ satisfies $a_n + (-1)^{n+1}a_{n+1} = 2n - 1$, find the sum of the first 40 terms, $S_{40}$. | 780 | 4/8 |
Let \( f(z) = z^2 + mz + n \) where \( m, n \in \mathbf{C} \). For all \( |z| = 1 \), it holds that \( |f(z)| = 1 \). Find the value of \( m + n \). | 0 | 7/8 |
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);[/asy] | 130 | 7/8 |
Determine the values of \(a\) for which there exist real numbers \(x\) and \(y\) that satisfy the equation \(\sqrt{2xy + a} = x + y + 17\). | \ge-\frac{289}{2} | 3/8 |
In the city built are $2019$ metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than $k$ lines should be made. It turned out that the order of the mayor is not feasible. What is the largest $k$ it could to happen? | 1008 | 7/8 |
In a particular country, the state of Sunland issues license plates with a format of one letter, followed by three digits, and then two letters (e.g., A123BC). Another state, Moonland, issues license plates where the format consists of two digits, followed by two letters, and then two more digits (e.g., 12AB34). Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates Sunland can issue compared to Moonland, given that Sunland always uses the letter 'S' as the starting letter in their license plates. | 6084000 | 5/8 |
The integers that can be expressed as a sum of three distinct numbers chosen from the set $\{4,7,10,13, \ldots,46\}$. | 37 | 1/8 |
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] | 4 | 1/8 |
In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ | \frac{8\sqrt{2}}{9} | 4/8 |
Given 9 points in space, where no 4 points are coplanar, find the smallest natural number $n$, such that in any connection of $n$ line segments between the given 9 points, where each segment is colored either red or blue, there always exists a monochromatic triangle. | 33 | 2/8 |
Find the number of sides of a regular polygon if for four of its consecutive vertices \(A, B, C, D\) the following equality holds:
\[ \frac{1}{AB} = \frac{1}{AC} + \frac{1}{AD} \] | 7 | 6/8 |
29 boys and 15 girls came to the ball. Some boys danced with some girls (no more than once per pair). After the ball, each person told their parents how many times they danced. What is the greatest number of distinct counts that the children could report? | 29 | 1/8 |
Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of paper has the original sheet been cut into? | 37 | 6/8 |
First, the boat traveled 10 km downstream, and then twice that distance on a lake the river flows into. The entire trip took 1 hour. Find the boat's own speed, given that the river's current speed is 7 km/h. | 28 | 7/8 |
Given a tetrahedron \( P-ABC \) with its four vertices on the surface of sphere \( O \), where \( PA = PB = PC \) and \( \triangle ABC \) is an equilateral triangle with side length 2. \( E \) and \( F \) are the midpoints of \( AC \) and \( BC \) respectively, and \( \angle EPF = 60^\circ \). Determine the surface area of sphere \( O \). | 6\pi | 6/8 |
While waiting for customers, a watermelon seller weighed 20 watermelons (each weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) by balancing a watermelon on one side of the scale with one or two weights (possibly identical) on the other side of the scale. The seller recorded the weights he used on a piece of paper. What is the minimum number of different numbers that could appear in his notes if the weight of each weight is an integer number of kilograms? | 6 | 1/8 |
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$ , let
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\]
For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$ . Find the maximal number of inequivalent elements in $\mathbb{E}_p$ . | 2p+6 | 1/8 |
Given the odd function $f(x)$ is increasing on the interval $[3, 7]$ and its minimum value is 5, determine the nature of $f(x)$ and its minimum value on the interval $[-7, -3]$. | -5 | 4/8 |
For $a>0$ , let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$ , which is tangent at point $A(a,a^{2})$ , around $A$ by $-\frac{\pi}{6}$ .
Let $B$ be the other intersection of $l$ and $y=x^{2}$ . Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$ .
(2) Let $S(a)$ be the area of the region bounded by $OC$ , $CA$ and $y=x^{2}$ . Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$ . Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$ . | 4 | 1/8 |
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$ | 3 | 6/8 |
In the Cartesian coordinate plane $xOy$, the ellipse $C: \frac{x^2}{3} + y^2 = 1$ has its upper vertex at $A$. A line $l$ intersects the ellipse $C$ at points $P$ and $Q$, but does not pass through point $A$, and $\overrightarrow{AP} \cdot \overrightarrow{AQ} = 0$.
(1) Does the line $l$ pass through a fixed point? If so, find the coordinates of the fixed point; if not, explain the reason.
(2) Tangent lines to the ellipse are drawn at points $P$ and $Q$. These tangents intersect at point $B$. Determine the range of the area of $\triangle BPQ$. | [\frac{9}{4},+\infty) | 1/8 |
We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$ . Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$ . Over all such choices of $a, b, c$ , what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\] | 1 | 1/8 |
Given that $-9, a_1, a_2, -1$ form an arithmetic sequence and $-9, b_1, b_2, b_3, -1$ form a geometric sequence, find the value of $b_2(a_2 - a_1)$. | -8 | 7/8 |
Given a positive integer \( n (n \geq 2) \). There are \( 2n \) positive real numbers \( a_{1}, a_{2}, \cdots, a_{2n} \) that satisfy:
\[ \sum_{k=1}^{n} a_{2k-1} \cdot \sum_{k=1}^{n} a_{2k} = \prod_{k=1}^{n} a_{2k-1} + \prod_{k=1}^{n} a_{2k}. \]
Find the minimum value of \( S = \sum_{k=1}^{2n} \frac{a_{k}^{n-1}}{a_{k+1}} \), where \( a_{2n+1} = a_{1} \). | n^3 | 4/8 |
A zoo houses five different pairs of animals, each pair consisting of one male and one female. To maintain a feeding order by gender alternation, if the initial animal fed is a male lion, how many distinct sequences can the zookeeper follow to feed all the animals? | 2880 | 7/8 |
The minimum positive period of $y=\tan(4x+ \frac{\pi}{3})$ is $\pi$. | \frac{\pi}{4} | 5/8 |
In the triangle \( \triangle ABC \), \( \angle C = 90^{\circ} \), and \( CB > CA \). Point \( D \) is on \( BC \) such that \( \angle CAD = 2 \angle DAB \). If \( \frac{AC}{AD} = \frac{2}{3} \) and \( \frac{CD}{BD} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, then what is \( m + n \)?
(49th US High School Math Competition, 1998) | 14 | 5/8 |
A fair $6$ sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?
$\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac5{12}\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac7{12}\qquad\textbf{(E) }\dfrac56$ | \textbf{(D)}\frac{7}{12} | 1/8 |
Determine the number of ways to arrange the letters of the word COMBINATION. | 4,\!989,\!600 | 3/8 |
In the four-sided pyramid \(SABCD\):
- The lateral faces \(SAB, SBC, SCD, SDA\) have areas of 9, 9, 27, 27 respectively;
- The dihedral angles at the edges \(AB, BC, CD, DA\) are equal;
- The quadrilateral \(ABCD\) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \(SABCD\). | 54 | 2/8 |
We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$ . With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$ . In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$ . We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$ , thus defining the sets $M_1,M_2,...,M_n$ . Find the minimum value of $n$ . | 8 | 6/8 |
Let \(a_{1}, a_{2}, \cdots, a_{100}\) be positive real numbers such that \(a_{i} \geqslant a_{101-i}\) for \(i = 1, 2, \cdots, 50\). Define \(x_{k} = \frac{k a_{k+1}}{a_{1}+a_{2}+\cdots+a_{k}}\) for \(k = 1, 2, \cdots, 99\). Prove that \(x_{1} x_{2}^{2} \cdots x_{99}^{99} \leqslant 1\). | 1 | 1/8 |
Let the line \( y = x + \sqrt{2} \) intersect the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( a > b > 0 \)) at points \( M \) and \( N \), and suppose \( OM \perp ON \) (where \( O \) is the origin). If \( MN = \sqrt{6} \), find the equation of the ellipse. | \frac{x^2}{4+2\sqrt{2}}+\frac{y^2}{4-2\sqrt{2}}=1 | 5/8 |
In the coordinate plane, a square $K$ with vertices at points $(0,0)$ and $(10,10)$ is given. Inside this square, illustrate the set $M$ of points $(x, y)$ whose coordinates satisfy the equation
$$
[x] < [y]
$$
where $[a]$ denotes the integer part of the number $a$ (i.e., the largest integer not exceeding $a$; for example, $[10]=10,[9.93]=9,[1 / 9]=0,[-1.7]=-2$). What portion of the area of square $K$ does the area of set $M$ constitute? | 0.45 | 2/8 |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. What is the probability that $x^2 + y^2 < y$? | \frac{\pi}{32} | 7/8 |
It is known that ship $A$ is located at $80^{\circ}$ north by east from lighthouse $C$, and the distance from $A$ to $C$ is $2km$. Ship $B$ is located at $40^{\circ}$ north by west from lighthouse $C$, and the distance between ships $A$ and $B$ is $3km$. Find the distance from $B$ to $C$ in $km$. | \sqrt {6}-1 | 3/8 |
For certain natural numbers \( n \), the numbers \( 2^{n} \) and \( 5^{n} \) have the same leading digit. What could be the possible leading digits? | 3 | 3/8 |
There are 2009 piles, each containing 2 stones. You are allowed to take the largest pile with an even number of stones (if there are multiple such piles, you can choose any one of them) and transfer exactly half of its stones to any other pile. What is the maximum number of stones that can be obtained in a single pile through these operations? | 2010 | 1/8 |
Let $a_0 = 2,$ $b_0 = 3,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\]for all $n \ge 0.$ Then $b_8 = \frac{3^m}{2^n}$ for some integers $m$ and $n.$ Enter the ordered pair $(m,n).$ | (3281,3280) | 7/8 |
What is the base-10 integer 515 when expressed in base 6? | 2215_6 | 1/8 |
A function $f$ is defined recursively by $f(1)=f(2)=1$ and \[f(n)=f(n-1)-f(n-2)+n\]for all integers $n \geq 3$. What is $f(2018)$? | 2017 | 7/8 |
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