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A billiard table is in the shape of a $2 \times 1$ rectangle, with pockets located at the corners and midpoints of the long sides. What is the minimum number of balls that need to be placed inside the rectangle so that each pocket is collinear with some two balls? | 4 | 1/8 |
For how many three-digit whole numbers does the sum of the digits equal $25$?
$\text{(A)}\ 2 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$ | (C)\6 | 1/8 |
Three lathes \( A, B, C \) each process the same type of standard parts at a certain work efficiency. Lathe \( A \) starts 10 minutes earlier than lathe \( C \), and lathe \( C \) starts 5 minutes earlier than lathe \( B \). After lathe \( B \) has been working for 10 minutes, the number of standard parts processed by lathes \( B \) and \( C \) is the same. After lathe \( C \) has been working for 30 minutes, the number of standard parts processed by lathes \( A \) and \( C \) is the same. How many minutes after lathe \( B \) starts will it have processed the same number of standard parts as lathe \( A \)? | 15 | 6/8 |
Prove that the sequence defined by \( a_0 = 2 \) and given terms \( 3, 6, 14, 40, 152, 784, \ldots \) with the general term \( a_n = (n+4) a_{n-1} - 4n a_{n-2} + (4n-8) a_{n-3} \) is the sum of two well-known sequences. | a_n=n!+2^n | 6/8 |
Circle $\omega$ is inscribed in unit square $PLUM$ , and points $I$ and $E$ lie on $\omega$ such that $U,I,$ and $E$ are collinear. Find, with proof, the greatest possible area for $\triangle PIE$ . | \frac{1}{4} | 2/8 |
A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers. | 432 | 2/8 |
Numbers from 1 to 200 are placed in random order on a circle so that the distances between adjacent numbers on the circle are equal.
For any given number, the following is true: if we consider the 99 numbers situated clockwise from it and the 99 numbers situated counterclockwise from it, both groups will contain an equal number of numbers that are smaller than the given number.
What number is opposite the number 113? | 114 | 1/8 |
Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$ , and \[ \int_0^{\infty} f^2(x)dx
<\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy] | \int_0^{\infty}^2(x)\,dx=\int_0^{\infty}f^2(x)\,dx | 1/8 |
Divide the numbers 1 through 10 into three groups such that the difference between any two numbers within the same group does not appear in that group. If one of the groups is {2, 5, 9}, then the sum of all the numbers in the group that contains 10 is $\qquad$. | 22 | 3/8 |
Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}⊥(\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$. | -\frac{1}{2} | 7/8 |
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]
| 594 | 1/8 |
How many consecutive "0"s are there at the end of the product \(5 \times 10 \times 15 \times 20 \times \cdots \times 2010 \times 2015\)? | 398 | 7/8 |
Solve
\[\arcsin (\sin x) = \frac{x}{2}.\]Enter all the solutions, separated by commas. | -\frac{2 \pi}{3}, 0, \frac{2 \pi}{3} | 1/8 |
For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?
$\textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 54$ | \textbf{(D)}\48 | 1/8 |
Given that the sides of a right-angled triangle are positive integers, and the perimeter of the triangle is equal to the area of the triangle, find the length of the hypotenuse. | 13 | 7/8 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{4}+1\right)\left(n^{2}-1\right)}-\sqrt{n^{6}-1}}{n}$$ | -\frac{1}{2} | 6/8 |
If \( a > 0 \) and \( b > 0 \), and \( \arcsin a + \arcsin b = \frac{\pi}{2} \), with \( m = \log_2 \frac{1}{a} + \log_2 \frac{1}{b} \), what is the range of values for \( m \)? | [1,+\infty) | 1/8 |
In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be "Selecting a part from the $100$ parts at random, and the part is qualified", and event $B$ be "Selecting a part from the $100$ parts at random, and the part is produced by A". Then, the probability of $A$ given $B$ is \_\_\_\_\_\_. | \dfrac{7}{8} | 2/8 |
Given the probability of failing on the first attempt and succeeding on the second attempt to guess the last digit of a phone number is requested, calculate the probability of this event. | \frac{1}{10} | 5/8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $c=a\cos B+b\sin A$.
(1) Find $A$;
(2) If $a=2$ and $b=c$, find the area of $\triangle ABC$. | \sqrt{2}+1 | 1/8 |
Petya's watch runs 5 minutes fast per hour, and Masha's watch runs 8 minutes slow per hour. At 12:00, they set their watches to the accurate school clock and agreed to meet at the skating rink at 6:30 PM according to their respective watches. How long will Petya wait for Masha if each arrives at the skating rink exactly at 6:30 PM according to their own watch? | 1.5 | 4/8 |
Given vectors $$\overrightarrow {m}=( \sqrt {3}\sin x+\cos x,1), \overrightarrow {n}=(\cos x,-f(x)), \overrightarrow {m}\perp \overrightarrow {n}$$.
(1) Find the monotonic intervals of $f(x)$;
(2) Given that $A$ is an internal angle of $\triangle ABC$, and $$f\left( \frac {A}{2}\right)= \frac {1}{2}+ \frac { \sqrt {3}}{2},a=1,b= \sqrt {2}$$, find the area of $\triangle ABC$. | \frac { \sqrt {3}-1}{4} | 1/8 |
An equilateral triangle \( \triangle ABC \) is divided into 4 smaller triangles \( \triangle ADE \), \( \triangle BDF \), \( \triangle DEF \), and \( \triangle CEF \) by its 3 medians. On each of the 9 sides of these smaller triangles, take the midpoint \( K, L, M, N, O, P, Q, R, S \). Each of these 15 points is colored either red or blue. Prove that there must be 3 points of the same color which form the vertices of an equilateral triangle. | 3 | 1/8 |
Find the greatest number \( A \) for which the following statement is true.
No matter how we pick seven real numbers between 1 and \( A \), there will always be two numbers among them whose ratio \( h \) satisfies \( \frac{1}{2} \leq h \leq 2 \). | 64 | 4/8 |
A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor. | 841 | 6/8 |
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$ , $b$ , and $c$ . If $P(1, 2, 3) = 1$ , compute $P(2, 4, 8)$ .
Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$ . | 56 | 5/8 |
Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1,2, \ldots, a b$, putting the numbers $1,2, \ldots, b$ in the first row, $b+1, b+2, \ldots, 2 b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $i j$ in the cell in row $i$ and column $j$. Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is 1200. Compute $a+b$. | 21 | 4/8 |
Given complex numbers \( z_{1} \) and \( z_{2} \) such that \( \left|z_{1}\right| = 2 \) and \( \left|z_{2}\right| = 3 \). If the angle between their corresponding vectors is \( 60^\circ \), find \( \frac{\left|z_{1} + z_{2}\right|}{\left|z_{1} - z_{2}\right|} \). | \frac{\sqrt{133}}{7} | 5/8 |
Let \([x]\) be the largest integer not greater than \(x\), for example, \([2.5] = 2\). If \(a = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots + \frac{1}{2004^{2}}\) and \(S = [a]\), find the value of \(a\). | 1 | 6/8 |
Let
$$
\mu(k)=\left\{\begin{array}{ll}
0 & k \text{ has a square factor,} \\
(-1)^{r} & k \text{ does not have a square factor and has } r \text{ distinct prime factors.}
\end{array}\right.
$$
For a positive integer \( n \), let \( P_{n}(x) = \sum_{k \mid n} \mu(k) x^{\frac{n}{k}} \).
Given a monic polynomial \( f(x) \) with integer coefficients of degree \( m \), let its \( m \) roots in the complex field be \( z_{1}, z_{2}, \ldots, z_{m} \). Prove that for any positive integer \( n \), we have
$$
n \mid P_{n}(z_{1}) + P_{n}(z_{2}) + \ldots + P_{n}(z_{m})
$$ | n | 1/8 |
In the Cartesian coordinate system $xOy$, given points $M(-1,2)$ and $N(1,4)$, point $P$ moves along the x-axis. When the angle $\angle MPN$ reaches its maximum value, the x-coordinate of point $P$ is ______. | 1 | 7/8 |
Given \( n \) is a positive integer, determine how many different triangles can be formed by the following numbers:
$$
\lg 12, \lg 75, \lg \left(n^{2}-16n+947\right)
$$ | 5 | 6/8 |
Given $$(5x- \frac {1}{ \sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. | 375 | 3/8 |
Winnie the Pooh decided to give Piglet a birthday cake in the shape of a regular hexagon. On his way, he got hungry and cut off 6 pieces from the cake, each containing one vertex and one-third of a side of the hexagon (see the illustration). As a result, he gave Piglet a cake weighing 900 grams. How many grams of the cake did Winnie the Pooh eat on the way? | 112.5 | 7/8 |
Determine the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\] | 3124 | 7/8 |
Four integers are marked on a circle. At each step, we simultaneously replace each number by the difference between this number and the next number on the circle, in a given direction (that is, the numbers \(a, b, c, d\) are replaced by \(a-b, b-c, c-d, d-a\)). Is it possible after 1996 such steps to have numbers \(a, b, c, d\) such that the numbers \(|bc - ad|,|ac - bd|,|ab - cd|\) are primes? | No | 3/8 |
Zhang Jie wrote 26 consecutive odd numbers starting from 27, and Wang Qiang wrote 26 consecutive natural numbers starting from 26. Then they both summed their respective 26 numbers. What is the difference between these two sums in yuan? | 351 | 6/8 |
Given two lines $l_1: y=a$ and $l_2: y= \frac {18}{2a+1}$ (where $a>0$), $l_1$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points A and B, and $l_2$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points C and D. Let the projection lengths of line segments AC and BD on the x-axis be $m$ and $n$ respectively. When $a= \_\_\_\_\_\_$, $\frac {n}{m}$ reaches its minimum value. | \frac {5}{2} | 6/8 |
Given 9 points in space, no four of which are coplanar, a line segment is drawn between every pair of points. These line segments can be colored blue or red or left uncolored. Find the minimum value of \( n \) such that, for any coloring of any \( n \) of these line segments with either red or blue, the set of these \( n \) line segments necessarily contains a triangle with all edges the same color. | 33 | 1/8 |
Given that \( f \) is a real-valued function on the set of all real numbers such that for any real numbers \( a \) and \( b \),
\[
f(a f(b)) = a b,
\]
find the value of \( f(2011) \). | 2011 | 2/8 |
In rhombus \(ABCD\), \(\angle DAB = 60^\circ\). Fold \(\triangle ABD\) along \(BD\) to obtain \(\triangle A_1BD\). If the dihedral angle \(A_1 - BD - C\) is \(60^\circ\), then find the cosine of the angle formed by the skew lines \(DA_1\) and \(BC\). | \frac{1}{8} | 1/8 |
Define $p(n)$ to be th product of all non-zero digits of $n$ . For instance $p(5)=5$ , $p(27)=14$ , $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\] | 103 | 7/8 |
In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw \( k \) lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.
Find the minimal value of \( k \) such that the goal is attainable for every possible configuration of 4027 points. | 2013 | 4/8 |
Given that the domain of the function $f(x)$ is $R$, $f(2x+2)$ is an even function, $f(x+1)$ is an odd function, and when $x\in [0,1]$, $f(x)=ax+b$. If $f(4)=1$, find the value of $\sum_{i=1}^3f(i+\frac{1}{2})$. | -\frac{1}{2} | 6/8 |
The sum $1+1+4$ of the digits of the number 114 divides the number itself. What is the largest number less than 900 that satisfies this property? | 888 | 6/8 |
Points $P_{1}, P_{2}, \cdots, P_{6}, \cdots$ are taken on $\triangle ABC$ such that $P_{1}, P_{4}, P_{7}, \cdots$ are on $AC$, $P_{2}, P_{5}, P_{8}, \cdots$ are on $AB$, and $P_{3}, P_{6}, P_{9}, \cdots$ are on $BC$. Additionally, $AP_{1} = AP_{2}$, $BP_{2} = BP_{3}$, $CP_{3} = CP_{4}$, $AP_{4} = AP_{5}$, $BP_{5} = BP_{6}$, and so on. Find the distance $\left|P_{5} P_{2003}\right|$. | 0 | 1/8 |
Let $E$ be the midpoint of side $BC$ of the triangle $ABC$. Given the angles $BAE = \alpha_{1}$, $EAC = \alpha_{2}$, and $AEB = \delta$, prove the following relationship:
$\operatorname{cotg} \alpha_{2}-\operatorname{cotg} \alpha_{1}=2 \operatorname{cotg} \delta$. | \cot\alpha_{2}-\cot\alpha_{1}=2\cot\delta | 3/8 |
Given that there are 5 people standing in a row, calculate the number of ways for person A and person B to stand such that there is exactly one person between them. | 36 | 3/8 |
Two cubes with an edge length of $a$ share a segment that connects the centers of two opposite faces. One cube is rotated by $45^{\circ}$ relative to the other. Find the volume of the intersection of these cubes. | 2a^3(\sqrt{2}-1) | 1/8 |
Prove that the value of the numerical expression \(11^{11} + 12^{12} + 13^{13}\) is divisible by 10. | 10 | 1/8 |
A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices. | 333 | 5/8 |
The expressions $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$ and $ abc$ are called the elementary symmetric expressions on the three letters $ a, b, c;$ symmetric because if we interchange any two letters, say $ a$ and $ c,$ the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let $ S_k(n)$ denote the elementary expression on $ k$ different letters of order $ n;$ for example $ S_4(3) \equal{} abc \plus{} abd \plus{} acd \plus{} bcd.$ There are four terms in $ S_4(3).$ How many terms are there in $ S_{9891}(1989)?$ (Assume that we have $ 9891$ different letters.) | \binom{9891}{1989} | 1/8 |
Given $e^{i \theta} = \frac{3 + i \sqrt{8}}{5}$, find $\cos 4 \theta$. | -\frac{287}{625} | 1/8 |
Points \( A, B, \) and \( C \) are distinct points on circle \( O \), with \(\angle AOB = 120^\circ\). Point \( C \) lies on the minor arc \( \overset{\frown}{AB} \) (not coinciding with points \( A \) or \( B \)). Given that \(\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB} (\lambda, \mu \in \mathbb{R})\), determine the range of values of \(\lambda + \mu\). | (1,2] | 6/8 |
The set of all positive integers can be divided into two disjoint subsets of positive integers \(\{f(1), f(2), \cdots, f(n), \cdots\}, \{g(1), g(2), \cdots, g(n), \cdots\}\), where \(f(1)<f(2)<\cdots<f(n)<\cdots\) and \(g(1)< g(2)<\cdots<g(n)<\cdots\) and it is given that \(g(n)=f[f(n)]+1\) for \(n \geq 1\). Find \(f(240)\). | 388 | 5/8 |
A sequence of natural numbers $\{x_{n}\}$ is constructed according to the following rules:
$$
x_{1}=a, \quad x_{2}=b, \quad x_{n+2}=x_{n}+x_{n+1}, \quad n \geqslant 1 .
$$
It is known that one term in the sequence is 1000. What is the minimum possible value of $a+b$?
(Note: Recommended problem by the Soviet Ministry of Education, 1990) | 10 | 1/8 |
Let $A B C D$ be a square and $E$ be the point on segment $[B D]$ such that $E B = A B$. Define point $F$ as the intersection of lines $(C E)$ and $(A D)$. Find the value of the angle $\widehat{F E A}$. | 45 | 7/8 |
Six students are to be arranged into two classes, with two students in each class, and there are six classes in total. Calculate the number of different arrangement plans. | 90 | 1/8 |
Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is:
$\textbf{(A)}\ 21\frac{1}{3}\pi-32\sqrt{3}\qquad \textbf{(B)}\ 32\sqrt{3}+21\frac{1}{3}\pi\qquad \textbf{(C)}\ 32\sqrt{3}+42\frac{2}{3}\pi \qquad\\ \textbf{(D)}\ 16\sqrt {3} + 42\frac {2}{3}\pi \qquad \textbf{(E)}\ 42\frac {2}{3}\pi$ | \textbf{(B)}\32\sqrt{3}+21\frac{1}{3}\pi | 1/8 |
How many times in a day does the hour and minute hands of a correctly functioning clock form a $90^\circ$ angle? | 44 | 7/8 |
In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$, calculate the distance from point $F$ to the midpoint of segment $DE$. | 7.5 | 4/8 |
In the diagram, if $\angle PQR = 48^\circ$, what is the measure of $\angle PMN$? [asy]
size(6cm);
pair p = (0, 0); pair m = dir(180 - 24); pair n = dir(180 + 24); pair r = 1.3 * dir(24); pair q = 2 * 1.3 * Cos(48) * dir(-24);
label("$M$", m, N); label("$R$", r, N); label("$P$", p, 1.5 * S); label("$N$", n, S); label("$Q$", q, SE);
draw(m--q--r--n--cycle);
add(pathticks(m--p, s=4));
add(pathticks(n--p, s=4));
add(pathticks(r--p, 2, spacing=0.9, s=4));
add(pathticks(r--q, 2, spacing=0.9, s=4));
[/asy] | 66^\circ | 7/8 |
According to the results of a football tournament, it is known that in every match, one of the teams scored either twice as many or half as many goals as its opponent. Could the total number of goals scored be 2020? | No | 7/8 |
On a segment of length 1, several intervals are marked. It is known that the distance between any two points, belonging to the same or different marked intervals, is not equal to 0.1. Prove that the sum of the lengths of the marked intervals does not exceed 0.5. | 0.5 | 7/8 |
In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI=2$ and $LD=3$, then $IC=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 13 | 2/8 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for any real numbers \( x \) and \( y \),
\[
f(x+y) = \max (f(x), y) + \min (f(y), x) .
\] | f(x)=x | 1/8 |
The sequence \( 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots \) is formed as follows: write down infinitely many '1's, insert a '2' between the first and the second '1's, insert two '2's between the second and the third '1's, insert three '2's between the third and the fourth '1's, and so on. If \( a_{n} \) denotes the \( n \)-th term of the sequence, find the value of \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2013} a_{2014} \). | 7806 | 1/8 |
Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$. | \frac{\pi}{2} | 6/8 |
Five distinct digits from 1 to 9 are given. Arnaldo forms the largest possible number using three of these 5 digits. Then, Bernaldo writes the smallest possible number using three of these 5 digits. What is the units digit of the difference between Arnaldo's number and Bernaldo's number? | 0 | 7/8 |
Numbers \(1^{2}, 2^{2}, \ldots, 8^{2}\) were placed at the vertices of a cube (one number per vertex). The product of the numbers at the ends of each edge was calculated. Find the maximum possible sum of all these products. | 9420 | 2/8 |
Let \( x \) be a real number satisfying \( x^{2} - \sqrt{6} x + 1 = 0 \). Find the numerical value of \( \left| x^{4} - \frac{1}{x^{4}} \right|. | 4\sqrt{2} | 1/8 |
Will stands at a point \(P\) on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of \(n^{\circ}\) and \((n+1)^{\circ}\) with the tangent at \(P\), where \(n\) is a positive integer less than 90. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach \(P\) again. (\(P\) is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room? | 28 | 1/8 |
Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$ . What is the maximum possible value for $n$ ?
*Proposed by James Lin* | 5 | 5/8 |
Find the minimum value of the function
$$
f(x)=\sqrt{15-12 \cos x}+\sqrt{4-2 \sqrt{3} \sin x}+\sqrt{7-4 \sqrt{3} \sin x}+\sqrt{10-4 \sqrt{3} \sin x-6 \cos x}.
$$ | 6 | 2/8 |
The probability of hitting the target at least once by a shooter in three shots is 0.875. Find the probability of hitting the target in a single shot. | 0.5 | 3/8 |
Find the volume of the region in space defined by
\[|x - y + z| + |x - y - z| \le 10\]and $x, y, z \ge 0$. | 125 | 1/8 |
(a) A natural number $n$ is less than 120. What is the largest remainder that the number 209 can give when divided by $n$?
(b) A natural number $n$ is less than 90. What is the largest remainder that the number 209 can give when divided by $n$? | 69 | 6/8 |
Three numbers $x, y,$ and $z$ are nonzero and satisfy the equations $x^{2}-y^{2}=y z$ and $y^{2}-z^{2}=x z$. Prove that $x^{2}-z^{2}=x y$. | x^2-z^2=xy | 2/8 |
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? | 1 | 7/8 |
It is known that none of the digits of a three-digit number is zero, and the sum of all possible two-digit numbers composed of the digits of this number is equal to the number itself. Find the largest such three-digit number. | 396 | 7/8 |
Given a circle with center \( O \) and radius 1. From point \( A \), tangents \( AB \) and \( AC \) are drawn. Point \( M \) lies on the circle such that the quadrilaterals \( OBMC \) and \( ABMC \) have equal areas. Find \( MA \). | 1 | 6/8 |
Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. | 972538 | 2/8 |
If \( A \) is the area of a square inscribed in a circle of diameter 10, find \( A \).
If \( a+\frac{1}{a}=2 \), and \( S=a^{3}+\frac{1}{a^{3}} \), find \( S \).
An \( n \)-sided convex polygon has 14 diagonals. Find \( n \).
If \( d \) is the distance between the 2 points \( (2,3) \) and \( (-1,7) \), find \( d \). | 5 | 7/8 |
Determine the smallest positive integer $n$ such that $4n$ is a perfect square and $5n$ is a perfect cube. | 25 | 6/8 |
Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$ , then we have
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
$n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \hline
$p(n)$ & $1$ & $2$ & $5$ & $15$ & $52$ & $203$ & $877$ & $4140$ & $21147$ & $115975$ \hline
\end{tabular}
Define a *splitting* of the $n$ distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has $8$ candies, what is the sum of the number of groups over all splittings he can use?
*2020 CCA Math Bonanza Lightning Round #3.4* | 17007 | 7/8 |
The age of each of Paulo's three children is an integer. The sum of these three integers is 12 and their product is 30. What is the age of each of his three children? | 1,5,6 | 1/8 |
A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, f(3) &=& 3, f(2n) &=& f(n), f(4n+1) &=& 2f(2n+1)-f(n), f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$ . Determine the number of positive integers $n$ , less than or equal to 1988, for which $f(n) = n$ . | 92 | 1/8 |
When flipping a coin, if we denote heads as $Z$ and tails as $F$, then a sequence of coin flips can be represented as a sequence composed of $Z$ and $F$. We can count the occurrences of heads followed by tails $(F Z)$, heads followed by heads $(Z Z)$, etc. For example, the sequence
\[ ZZFFZZZZFZZFFFF \]
is the result of flipping a coin 15 times. In this sequence, there are 5 occurrences of $ZZ$, 3 occurrences of $ZF$, 2 occurrences of $FZ$, and 4 occurrences of $FF$. How many sequences of 15 coin flips have exactly 2 occurrences of $ZZ$, 3 occurrences of $ZF$, 4 occurrences of $FZ$, and 5 occurrences of $FF$? | 560 | 2/8 |
Given a quadratic function $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \in \mathbb{R}$, the range of $f(x)$ is $[0, +\infty)$.
(1) Find the expression for $f(x)$.
(2) Let $g(x) = f(x) - 2kx$, where $k \in \mathbb{R}$.
(i) If $g(x)$ is monotonic on $x \in [-2, 2]$, find the range of the real number $k$.
(ii) If the minimum value of $g(x)$ on $x \in [-2, 2]$ is $g(x)_{\text{min}} = -15$, find the value of $k$. | k = 6 | 7/8 |
Swap the digit in the hundreds place with the digit in the units place of a three-digit number while keeping the digit in the tens place unchanged. The new number obtained is equal to the original number. How many such numbers are there? How many of these numbers are divisible by 4? | 20 | 4/8 |
In right triangle $ABC$ with $\angle B = 90^\circ$, we have $AB = 8$ and $AC = 6$. Find $\cos C$. | \frac{4}{5} | 1/8 |
Two runners are running in the same direction on a circular track at constant speeds. At a certain moment, runner $A$ is 10 meters ahead of runner $B$, but after $A$ runs 22 meters, runner $B$ catches up.
How many points on the track are there where $B$ can later lap $A$? | 5 | 7/8 |
In triangle $ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE =
10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE$, $\cot \angle CBE$, $\cot \angle DBC$ form an arithmetic progression. What is the area of triangle $ABC$?
[asy]
pair A,B,C,D,E;
A=(0,0);
B=(4,8);
C=(8,0);
E=(10,0);
D=(4,0);
draw(A--B--E--cycle,linewidth(0.7));
draw(C--B--D,linewidth(0.7));
label("$B$",B,N);
label("$A$",A,S);
label("$D$",D,S);
label("$C$",C,S);
label("$E$",E,S);
[/asy] | \frac{50}{3} | 2/8 |
In a tetrahedron \(ABCD\), it is given that:
\[ AB = AC = 3, BD = BC = 4, BD \perp \text{plane } ABC. \]
Find the radius of the circumsphere of tetrahedron \(ABCD\). | \frac{\sqrt{805}}{10} | 5/8 |
30 people are arranged in six rows of five people each. Each of them is either a knight, who always tells the truth, or a liar, who always lies, and they all know who among them is a knight or a liar. A journalist asked each of them: "Is it true that there are at least 4 rows in which more than half are liars?" What is the maximum number of "yes" answers he could hear? | 21 | 3/8 |
A line passing through point $A$ intersects a circle with diameter $AB$ at point $K$, which is distinct from $A$, and intersects a circle with center at $B$ at points $M$ and $N$. Prove that $MK=KN$. | MK=KN | 7/8 |
Suppose $n \geq 2$ is a positive integer and $a_1, a_2, \ldots, a_n$ are non-negative real numbers such that $\sum_{i=1}^{n} a_i = 1$. Find the maximum value of $\left( \sum_{i=1}^{n} i^2 a_i \right) \cdot \left( \sum_{i=1}^{n} \frac{a_i}{i} \right)^{2}$. | \frac{4(n^2+n+1)^3}{27n^2(n+1)^2} | 1/8 |
Four normal students, A, B, C, and D, are to be assigned to work at three schools, School A, School B, and School C, with at least one person at each school. It is known that A is assigned to School A. What is the probability that B is assigned to School B? | \dfrac{5}{12} | 7/8 |
The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units, and the resulting graph corresponds to an even function. Find the minimum value of $m$ such that there exists $x \in \left[ 0,\frac{\pi}{2} \right]$ such that the inequality $f(x) \leqslant m$ holds. | -\frac{1}{2} | 7/8 |
Consider a $3 \times 3$ grid where each cell contains $-1$, $0$, or $1$. We consider the sum of the numbers in each column, each row, and each of the main diagonals. Show that among these sums, there are always two that are equal. | 2 | 6/8 |
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