id
int64 20
101k
| problem
stringlengths 18
4.16k
| gt_ans
stringlengths 1
191
|
|---|---|---|
46,711 | $6 \cdot 84$ For a finite set $A$, there exists a function $f: N \rightarrow A$ with the following property: if $|i-j|$ is a prime number, then $f(i) \neq f(j), N=\{1,2, \cdots\}$, find the minimum number of elements in the finite set $A$.
| 4 |
46,736 | 9. (16 points) Let $a_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}\left(n \in \mathbf{Z}_{+}\right)$. Find the smallest positive real number $\lambda$, such that for any $n \geqslant 2$, we have
$$
a_{n}^{2}<\lambda \sum_{k=1}^{n} \frac{a_{k}}{k} .
$$ | 2 |
46,740 | Example 2 Let $0<x<\frac{9}{2}$. Find the range of the function
$$
y=\left[1+\frac{1}{\lg \left(\sqrt{x^{2}+10}+x\right)}\right]\left[1+\frac{1}{\lg \left(\sqrt{x^{2}+10}-x\right)}\right]
$$ | (9,+\infty) |
46,758 | 1. Find the locus of the intersection point of two perpendicular intersecting tangents to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. | x^{2}+y^{2}=a^{2}+b^{2} |
46,830 | Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have? | 2^{n+1} |
46,868 | 13. Given constants $a, b$ satisfy $a, b>0, a \neq 1$, and points $P(a, b), Q(b, a)$ are both on the curve $y=\cos (x+c)$, where $c$ is a constant. Then $\log _{a} b=$ $\qquad$ | 1 |
46,882 | Determine the largest real number $M$ such that for every infinite sequence $x_{0}, x_{1}, x_{2}, \ldots$ of real numbers that satisfies
a) $x_{0}=1$ and $x_{1}=3$,
b) $x_{0}+x_{1}+\cdots+x_{n-1} \geq 3 x_{n}-x_{n+1}$,
it holds that
$$
\frac{x_{n+1}}{x_{n}}>M
$$
for all $n \geq 0$. | 2 |
46,897 | Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as:
$u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$
a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit
b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$ | 5 |
46,910 | Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$, we have $$\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.$$ | \frac{2n-1}{(n-1)^2} |
46,942 | Let $S$ be the set of 81 points $(x, y)$ such that $x$ and $y$ are integers from $-4$ through $4$. Let $A$, $B$, and $C$ be random points chosen independently from $S$, with each of the 81 points being equally likely. (The points $A$, $B$, and $C$ do not have to be different.) Let $K$ be the area of the (possibly degenerate) triangle $ABC$. What is the expected value (average value) of $K^2$ ? | \frac{200}{3} |
46,950 | 1. Let $\angle A, \angle B, \angle C$ be the three interior angles of $\triangle ABC$, and the complex number
$$
\begin{array}{l}
z=\frac{\sqrt{65}}{5} \sin \frac{A+B}{2}+\mathrm{i} \cos \frac{A-B}{2}, \\
|z|=\frac{3 \sqrt{5}}{5} .
\end{array}
$$
Then the maximum value of $\angle C$ is $\qquad$ | \pi-\arctan \frac{12}{5} |
46,994 | In the Cartesian coordinate system $x O y$, with the origin $O$ as the center, two circles are drawn with radii $a$ and $b$ ($a > b > 0$). Point $Q$ is the intersection of the radius $O P$ of the larger circle with the smaller circle. A perpendicular line $A N \perp O x$ is drawn from point $P$, with the foot of the perpendicular being $N$. A perpendicular line $Q M \perp P N$ is drawn from point $Q$, with the foot of the perpendicular being $M$. Let the trajectory of point $M$ as the radius $O P$ rotates around point $O$ be the curve $E$.
(1) Find the equation of the curve $E$;
(2) Let $A$, $B$, and $C$ be three points on the curve $E$, and satisfy $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\mathbf{0}$. Find the area of $\triangle A B C$. | \frac{3 \sqrt{3} a b}{4} |
47,008 |
Problem 9.1. Let $p$ be a real parameter such that the equation $x^{2}-3 p x-p=0$ has real and distinct roots $x_{1}$ and $x_{2}$.
a) Prove that $3 p x_{1}+x_{2}^{2}-p>0$.
b) Find the least possible value of
$$
A=\frac{p^{2}}{3 p x_{1}+x_{2}^{2}+3 p}+\frac{3 p x_{2}+x_{1}^{2}+3 p}{p^{2}}
$$
When does equality obtain?
| 2 |
47,022 | 7. Given the set $M=\left\{(a, b) \mid\left(y^{2}+4\right) a^{2}-\right.$ $2(x y+b y+8) a+x^{2}+2 b x+2 b^{2}+12$ is a square of a linear expression in $x$ and $y$ $\}$. When $(a, b)$ takes all elements in the set $M$, the maximum distance from the point $(a, b)$ to the origin $O$ is $\qquad$ | \frac{2 \sqrt{21}}{3} |
47,049 | One. (15 points) As shown in Figure 1, $A$ is a fixed point between two parallel lines $l_{1}$ and $l_{2}$, and the distances from point $A$ to lines $l_{1}$ and $l_{2}$ are $A M=1, A N=\sqrt{3}$. Let the other two vertices $C$ and $B$ of $\triangle A B C$ move on $l_{1}$ and $l_{2}$ respectively, and satisfy $A B<A C, \frac{A B}{\cos B}=\frac{A C}{\cos C}$.
(1) Determine the shape of $\triangle A B C$ and prove the conclusion;
(2) Find the maximum value of $\frac{1}{A B}+\frac{\sqrt{3}}{A C}$. | \sqrt{2} |
47,062 | 1. Let the set $P_{n}=\{1,2, \cdots, n\}\left(n \in \mathbf{Z}_{+}\right)$. Denote $f(n)$ as the number of sets $A$ that simultaneously satisfy the following conditions:
(1) $A \subseteq P_{n}, \bar{A}=P_{n} \backslash A$;
(2) If $x \in A$, then $2 x \notin A$;
(3) If $x \in \bar{A}$, then $2 x \notin \bar{A}$.
Then $f(2018)=$ | 2^{1009} |
47,071 | Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$ | 0 |
47,074 | Six. (25 points) Given
$$
f(x)=\lg (x+1)-\frac{1}{2} \log _{3} x .
$$
(1) Solve the equation: $f(x)=0$;
(2) Find the number of subsets of the set
$$
M=\left\{n \mid f\left(n^{2}-214 n-1998\right) \geqslant 0, n \in \mathbf{Z}\right\}
$$
(Li Tiehan, problem contributor) | 4 |
47,090 | Example 12. Given $x=19^{94}-1, y=2^{m} \cdot 3^{n} \cdot 5^{l}$ $(m, n, l$ are non-negative integers, and $m+n+l \neq 0)$. Find the sum $S$ of all divisors of $x$ that are of the form $y$. | 1169 |
47,123 | 2. Solve the system of equations
$$
\left\{\begin{array}{l}
\left(1+4^{2 x-y}\right) 5^{1-2 x+y}=1+2^{2 x-y+1}, \\
y^{3}+4 x+1+\ln \left(y^{2}+2 x\right)=0 .
\end{array}\right.
$$
(1999, Vietnam Mathematical Olympiad) | x=0, y=-1 |
47,145 | 6. 84 For a finite set $A$, there exists a function $f: N \rightarrow A$ with the following property: if $|i-j|$ is a prime number, then $f(i) \neq f(j), N=\{1,2, \cdots\}$, find the minimum number of elements in the finite set $A$.
| 4 |
47,154 | 11. Given $f(x)=\frac{a x+1}{3 x-1}$, and the equation $f(x)=-4 x+8$ has two distinct positive roots, one of which is three times the other. Let the first $n$ terms of the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ be $S_{n}$ and $T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=f(n)(n=1,2, \cdots)$.
(1) If $g(n)=\frac{a_{n}}{b_{n}}(n=1,2, \cdots)$, find the maximum value of $g(n)$.
(2) If $a_{1}=\frac{5}{2}$, and the common difference of the sequence $\left\{b_{n}\right\}$ is 3, investigate whether there are equal terms in the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$. If so, find the general term formula of the sequence $\left\{c_{n}\right\}$ formed by these equal terms arranged in ascending order; if not, explain the reason. | \frac{5}{2} |
47,162 | Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.
Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning. | 131070 |
47,216 | A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ($k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$[size=75]1[/size] and $R$[size=75]2[/size] there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$[size=75]1[/size] with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$[size=75]2[/size] with the columns from $S$ there are odd number of $1's$.
Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table. | n |
47,240 | Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ that satisfy
$$
f(f(f(n)))+f(f(n))+f(n)=3 n
$$
for all $n \in \mathbb{Z}_{>0}$. | f(n)=n |
47,251 | $$
\begin{array}{l}
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+ \\
\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right) \\
=
\end{array}
$$ | 3.14159265 |
47,295 | 2. In the complex plane, non-zero complex numbers $z_{1}, z_{2}$ lie on a circle centered at $i$ with a radius of 1, the real part of $\overline{z_{1}} \cdot z_{2}$ is zero, and the principal value of the argument of $z_{1}$ is $\frac{\pi}{6}$. Then $z_{2}=$ $\qquad$ . | -\frac{\sqrt{3}}{2}+\frac{3}{2} i |
47,311 | The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point $C_n$. Find the sum
\[\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP) \] | 0 |
47,330 | The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m. | 825 |
47,335 | (1) For $ a>0,\ b\geq 0$, Compare
$ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$.
(2) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$. | 1 |
47,363 | Consider a set $X$ with $|X| = n\geq 1$ elements. A family $\mathcal{F}$ of distinct subsets of $X$ is said to have property $\mathcal{P}$ if there exist $A,B \in \mathcal{F}$ so that $A\subset B$ and $|B\setminus A| = 1$.
i) Determine the least value $m$, so that any family $\mathcal{F}$ with $|\mathcal{F}| > m$ has property $\mathcal{P}$.
ii) Describe all families $\mathcal{F}$ with $|\mathcal{F}| = m$, and not having property $\mathcal{P}$.
([i]Dan Schwarz[/i]) | 2^{n-1} |
47,378 | Example 5 Let $x, y, z \in \mathbf{R}^{+}$, and $x^{4}+y^{4}+z^{4}=1$, find
$$f(x, y, z)=\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}+\frac{z^{3}}{1-z^{8}}$$
the minimum value. | \frac{9}{8} \sqrt[4]{3} |
47,390 | 2. Let $2 n$ real numbers $a_{1}, a_{2}, \cdots, a_{2 n}$ satisfy the condition
$$
\begin{array}{c}
\sum_{i=1}^{2 n-1}\left(a_{i+1}-a_{i}\right)^{2}=1 . \\
\text { Find the maximum value of }\left(a_{n+1}+a_{n+2}+\cdots+a_{2 n}\right)-\left(a_{1}+a_{2}+\cdots+a_{n}\right) \text { . }
\end{array}
$$ | \sqrt{\frac{n\left(2 n^{2}+1\right)}{3}} |
47,391 | 14. A tangent line is drawn from the left focus $F$ of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ to the circle $x^{2}+y^{2}=9$, with the point of tangency being $T$. Extend $F T$ to intersect the right branch of the hyperbola at point $P$. If $M$ is the midpoint of segment $F P$, and $O$ is the origin, find the value of $|M O|-|M T|$. | 1 |
47,398 | Let $ABC$ be an acute triangle. Let$H$ and $D$ be points on $[AC]$ and $[BC]$, respectively, such that $BH \perp AC$ and $HD \perp BC$. Let $O_1$ be the circumcenter of $\triangle ABH$, and $O_2$ be the circumcenter of $\triangle BHD$, and $O_3$ be the circumcenter of $\triangle HDC$. Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$. | \frac{1}{4} |
47,411 | 8. $n$ chess players participate in a chess tournament, with each pair of players competing in one match. The rules are: the winner gets 1 point, the loser gets 0 points, and in the case of a draw, both players get 0.5 points. If it is found after the tournament that among any $m$ players, there is one player who has won against the other $m-1$ players, and there is also one player who has lost to the other $m-1$ players, this situation is said to have property $P(m)$.
For a given $m(m \geqslant 4)$, find the minimum value of $n$, denoted as $f(m)$, such that for any tournament situation with property $P(m)$, all $n$ players have distinct scores.
(Wang Jianwei provided) | 2m-3 |
47,412 | Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ \[f(3x-2)\leq f(x)\leq f(2x-1).\]
[i]Proposed by A. Golovanov[/i] | f(x) = c |
47,413 | There is a positive integer s such that there are s solutions to the equation
$64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$
where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$. Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$. | 100 |
47,435 | 8. Given that the function $f(x)$ is a decreasing function defined on $(-\infty, 3]$, and for $x \in \mathbf{R}$,
$$
f\left(a^{2}-\sin x\right) \leqslant f\left(a+1+\cos ^{2} x\right)
$$
always holds. Then the range of real number $a$ is $\qquad$ | \left[-\sqrt{2}, \frac{1-\sqrt{10}}{2}\right] |
47,439 | 5. Try to simplify $\sum_{k=0}^{n} \frac{(-1)^{k} C_{n}^{k}}{k^{3}+9 k^{2}+26 k+24}$ into the form $\frac{p(n)}{q(n)}$, where $p(n)$ and $q(n)$ are two polynomials with integer coefficients. | \frac{1}{2(n+3)(n+4)} |
47,452 | Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $, let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers:
\[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \]
Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $. | 2 - \frac{1}{p} |
47,482 | One. (35 points) The lines $l_{1}$ and $l_{2}$ are tangent to a circle at points $A$ and $B$, respectively. On lines $l_{1}$ and $l_{2}$, take 1993 points $A_{1}, A_{2}, \cdots, A_{1993}$ and $B_{1}, B_{2}, \cdots, B_{1993}$, such that $A A_{i}=(i+1) B B_{i} (i=1, 2, \cdots, 1993)$. If the extension of $A_{i} B_{i}$ intersects the extension of $A B$ at point $M_{i} (i=1, 2, \cdots, 1993)$, find:
$$
\frac{A_{1} B_{1}}{A_{1} M_{1}} \cdot \frac{A_{2} B_{2}}{A_{2} M_{2}} \cdot \cdots \cdot \frac{A_{1993} B_{1993}}{A_{1993} M_{1993}}=?
$$ | \frac{1}{1994} |
47,496 | 3. In tetrahedron $ABCD$,
\[
\begin{array}{l}
\angle ABC=\angle BAD=60^{\circ}, \\
BC=AD=3, AB=5, AD \perp BC,
\end{array}
\]
$M, N$ are the midpoints of $BD, AC$ respectively. Then the size of the acute angle formed by lines $AM$ and $BN$ is \qquad (express in radians or inverse trigonometric functions). | \arccos \frac{40}{49} |
47,521 | If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$
When does the equality occur?
[i]Proposed by Dorlir Ahmeti, Albania[/i] | 3 |
47,523 | Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 913 |
47,526 | Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$. Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$, what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of $k$?
[i]2018 CCA Math Bonanza Lightning Round #4.1[/i] | 1009k |
47,542 | Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively prime integers $a, b$. Find $a + b$. | 19 |
47,555 | Let $x_1 \dots, x_{42}$, be real numbers such that $5x_{i+1}-x_i-3x_ix_{i+1}=1$ for each $1 \le i \le 42$, with $x_1=x_{43}$. Find all the product of all possible values for $x_1 + x_2 + \dots + x_{42}$.
[i] Proposed by Michael Ma [/i] | 588 |
47,567 | Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer $n$ such that $2^n < S^{2007} < 2^{n+1}$. | 501 |
47,569 | Three. (25 points) If positive numbers $a, b, c$ satisfy
$$
\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)^{2}+\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)^{2}+\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)^{2}=3 \text {, }
$$
find the value of the algebraic expression
$$
\frac{b^{2}+c^{2}-a^{2}}{2 b c}+\frac{c^{2}+a^{2}-b^{2}}{2 c a}+\frac{a^{2}+b^{2}-c^{2}}{2 a b}
$$ | 1 |
47,581 | Let $l$ be the tangent line at the point $P(s,\ t)$ on a circle $C:x^2+y^2=1$. Denote by $m$ the line passing through the point $(1,\ 0)$, parallel to $l$. Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\ 0)$.
Note : if $m$ is the line $x=1$, then $P'$ is considered as $(1,\ 0)$.
Call $T$ the operation such that the point $P'(s',\ t')$ is obtained from the point $P(s,\ t)$ on $C$.
(1) Express $s',\ t'$ as the polynomials of $s$ and $t$ respectively.
(2) Let $P_n$ be the point obtained by $n$ operations of $T$ for $P$.
For $P\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$, plot the points $P_1,\ P_2$ and $P_3$.
(3) For a positive integer $n$, find the number of $P$ such that $P_n=P$. | 2^n - 1 |
47,588 | For a sequence $x_{1}, x_{2}, \ldots, x_{n}$ of real numbers, we define its price as
$$ \max _{1 \leqslant i \leqslant n}\left|x_{1}+\cdots+x_{i}\right| $$
Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_{1}$ such that $\left|x_{1}\right|$ is as small as possible; among the remaining numbers, he chooses $x_{2}$ such that $\left|x_{1}+x_{2}\right|$ is as small as possible, and so on. Thus, in the $i^{\text {th }}$ step he chooses $x_{i}$ among the remaining numbers so as to minimise the value of $\left|x_{1}+x_{2}+\cdots+x_{i}\right|$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$.
Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G \leqslant c D$. (Georgia) | 2 |
47,592 | 6 Given that $\lambda$ is a positive real number. Find the maximum value of $\lambda$ such that for all positive real numbers $u, v, w$ satisfying the condition
$$u \sqrt{v w}+v \sqrt{w u}+w \sqrt{u v} \geqslant 1$$
we have
$$u+v+w \geqslant \lambda .$$ | \sqrt{3} |
47,622 | * Four, the sequence $1,1,3,3,3^{2}, 3^{2}, \cdots, 3^{1992}, 3^{1992}$ consists of two 1s, two 3s, two $3^{2}$s, ..., two $3^{1992}$s arranged in ascending order. The sum of the terms in the sequence is denoted as $S$. For a given natural number $n$, if it is possible to select some terms from different positions in the sequence such that their sum is exactly $n$, it is called a selection scheme, and the number of all selection schemes with the sum $n$ is denoted as $f(n)$. Try to find:
$$
f(1)+f(2)+\cdots+f(s) \text { . }
$$ | 4^{1993}-1 |
47,640 | 7. In tetrahedron $ABCD$, $\angle ADB = \angle BDC = \angle CDA = 60^{\circ}$, $AD = BD = 3$, $CD = 2$. Then the volume of the circumscribed sphere of tetrahedron $ABCD$ is $\qquad$ | 4 \sqrt{3} \pi |
47,647 | 5. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and $P$ as any point on the ellipse not coinciding with the left or right vertices, points $I$ and $G$ are the incenter and centroid of $\triangle P F_{1} F_{2}$, respectively. When $I G$ is always perpendicular to the $x$-axis, the eccentricity of the ellipse is . $\qquad$ | \frac{1}{3} |
47,680 | 5. Let two ellipses be
$$
\frac{x^{2}}{t^{2}+2 t-2}+\frac{y^{2}}{t^{2}+t+2}=1
$$
and $\frac{x^{2}}{2 t^{2}-3 t-5}+\frac{y^{2}}{t^{2}+t-7}=1$
have common foci. Then $t=$ $\qquad$ . | 3 |
47,681 | Let there be two different media I and II on either side of the $x$-axis (Figure 2). The speed of light in media I and II is $c_{1}$ and $c_{2}$, respectively. Now, if light travels from point $A$ in medium I to point $B$ in medium II, what path should the light take to minimize the time of travel? | \frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{c_{1}}{c_{2}} |
47,692 | Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$ | 0 |
47,721 | Let $a$ be a positive real numbers. Let $t,\ u\ (t<u)$ be the $x$ coordinates of the point of intersections of the curves : $C_1:y=|\cos x|\ (0\leq x\leq \pi),\ C_2:y=a\sin x\ (0\leq x\leq \pi).$ Denote by $S_1$ the area of the part bounded by $C_1,\ C_2$ and $y$-axis in $0\leq x\leq t$, and by $S_2$ the area of the part bounded by $C_1,\ C_2$ in $t\leq x\leq u$. When $a$ moves over all positive real numbers, find the minimum value of $S_1+S_2$. | 2\sqrt{2} - 2 |
47,756 | Given is a triangle $ABC$ with the property that $|AB|+|AC|=3|BC|$. Let $T$ be the point on line segment $AC$ such that $|AC|=4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively, such that first, $KL \parallel BC$ and second, $KL$ is tangent to the incircle of $\triangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$.
 | \frac{2}{3} |
47,767 | Let $x,y,z$ be complex numbers such that\\
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\
\\
If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$, find $m+n$. | 16 |
47,781 | 15. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n}+a_{n+1}=n(-1)^{\frac{a(a+1)}{2}} \text {, }
$$
the sum of the first $n$ terms is $S_{n}, m+S_{2015}=-1007, a_{1} m>0$. Then the minimum value of $\frac{1}{a_{1}}+\frac{4}{m}$ is $\qquad$ . | 9 |
47,783 | 14. As shown in Figure 4, there is a sequence of curves $P_{0}, P_{1}, P_{2}, \cdots$, where it is known that the area of the equilateral triangle enclosed by $P_{0}$ is 1. $P_{\lambda+1}$ is obtained by performing the following operation on $P_{A}$: divide each side of $P_{k}$ into three equal parts, construct an equilateral triangle outward using the middle segment of each side as the base, and then remove the middle segment $(k=0,1,2, \cdots)$. Let $S_{n}$ be the area of the figure enclosed by the curve $P_{n}$.
(1) Find the general term formula for the sequence $\left\{S_{n}\right\}$;
(2) Find $\lim _{n \rightarrow \infty} S_{n}$. | \frac{8}{5} |
47,789 | For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote
\[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying
\[a_1+a_2+\cdots+a_{10}=20.\]
Compute the number of positive integer divisors of $N$.
[i]2021 CCA Math Bonanza Individual Round #14[/i] | 462 |
47,804 | 178 Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all \( x, y \in \mathbf{R} \),
\[
f\left((x+y)^{2}\right)=(x+y)(f(x)+f(y)).
\] | f(x)=kx |
47,810 | 2. Let $0 \leqslant \alpha, \beta < 2 \pi, \alpha, \beta \neq \frac{\pi}{3}, \frac{5 \pi}{3}$, and
$$
\frac{2 \sin \alpha - \sqrt{3}}{2 \cos \alpha - 1} + \frac{2 \sin \beta - \sqrt{3}}{2 \cos \beta - 1} = 0 \text{. }
$$
Then $\cot \frac{\alpha + \beta}{2} = $ $\qquad$ | -\frac{\sqrt{3}}{3} |
47,818 | Let $x_{1}, \ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1}+x_{i+2} \leq 1$ for all $i=1, \ldots, 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$ ). Find the maximal possible value of the sum
$$ S=\sum_{i=1}^{100} x_{i} x_{i+2} $$
(Russia) Answer. $\frac{25}{2}$. | \frac{25}{2} |
47,819 | An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants. | 130 |
47,835 | In a three-dimensional Euclidean space, by $\overrightarrow{u_1}$ , $\overrightarrow{u_2}$ , $\overrightarrow{u_3}$ are denoted the three orthogonal unit vectors on the $x, y$, and $z$ axes, respectively.
a) Prove that the point $P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3}$ , where $t$ takes all real values, describes a straight line (which we will denote by $L$).
b) What describes the point $Q(t) = (1-t^2)\overrightarrow{u_1} +(2-3t^2)\overrightarrow{u_2} +(2t^2 -1)\overrightarrow{u_3}$ if $t$ takes all the real values?
c) Find a vector parallel to $L$.
d) For what values of $t$ is the point $P(t)$ on the plane $2x+ 3y + 2z +1 = 0$?
e) Find the Cartesian equation of the plane parallel to the previous one and containing the point $Q(3)$.
f) Find the Cartesian equation of the plane perpendicular to $L$ that contains the point $Q(2)$. | x + 3y - 2z + 47 = 0 |
47,858 | 155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds. | 270 |
47,870 | Let $\{x\}$ denote the fractional part of $x$, which means the unique real $0\leq\{x\}<1$ such that $x-\{x\}$ is an integer. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $[m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and $b$ range along all real numbers. | \frac{7}{3} |
47,881 | Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$. | 4 |
47,907 | Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property:
$$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$ | f(x) = x + a |
47,908 | 12. Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ with the left vertex $A$ and the right focus $F$. Let $P$ be any point on the hyperbola in the first quadrant. If $\angle P F A=2 \angle F A P$ always holds, then the eccentricity $e$ of the hyperbola is | 2 |
47,933 | 32. Let $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$, find the maximum value of $A=\frac{\left(1-\sqrt{\tan \frac{\alpha}{2} \tan \frac{\beta}{2}}\right)^{2}}{\cot \alpha+\cot \beta}$. | 3-2 \sqrt{2} |
47,935 | ## Aufgabe 3 - 331223
Man ermittle alle diejenigen Paare $(m ; n)$ positiver ganzer Zahlen $m, n$, für die $1994^{m}-1993^{n}$ eine Quadratzahl ist.
| (1,1) |
47,945 | Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers. | \frac{41}{2} |
47,954 | In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if
there are two people $A$ and $ B$ such that A took $B$ and $ B $ took $A$. For each positive integer n, let $f (n)$ be the number of hidden friends with n people where there is no “marmalade”, i.e. $f (n)$ is equal to the number of permutations $\sigma$ of {$1, 2,. . . , n$} such that:
*$\sigma (i) \neq i$ for all $i=1,2,...,n$
* there are no $ 1 \leq i <j \leq n $ such that $ \sigma (i) = j$ and $\sigma (j) = i. $
Determine the limit
$\lim_{n \to + \infty} \frac{f(n)}{n!}$ | \exp\left(-\frac{3}{2}\right) |
47,955 | 2. Let $2 n$ real numbers $a_{1}, a_{2}, \cdots, a_{2 n}$ satisfy the condition $\sum_{i=1}^{2 n-1}\left(a_{i+1}-a_{i}\right)^{2}=1$, find the maximum value of $\left(a_{n+1}+\right.$ $\left.a_{n+2}+\cdots+a_{2 n}\right)-\left(a_{1}+a_{2}+\cdots+a_{n}\right)$. (2003 Western Mathematical Olympiad) | \sqrt{\frac{n\left(2 n^{2}+1\right)}{3}} |
47,973 | Given a positive integer $n(n>1)$, for a positive integer $m$, the set $S_{m}=\{1,2, \cdots, m n\}$. The family of sets $\mathscr{T}$ satisfies the following conditions:
(1) Each set in $\mathscr{T}$ is an $m$-element subset of $S_{m}$;
(2) Any two sets in $\mathscr{T}$ have at most one common element;
(3) Each element of $S_{m}$ appears in exactly two sets in $\mathscr{T}$.
Find the maximum value of $m$.
Try to solve the problem. | 2n-1 |
48,003 | 9. (16 points) For positive integers $n(n \geqslant 2)$, let
$$
a_{n}=\sum_{k=1}^{n-1} \frac{n}{(n-k) 2^{k-1}} \text {. }
$$
Find the maximum value in the sequence $\left\{a_{n}\right\}$. | \frac{10}{3} |
48,006 | Matrices $A$, $B$ are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix}\]
Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$. | \frac{\pi}{3} |
48,014 | Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
[i]Proposed by Robin Park[/i] | 22200 |
48,020 | 10. (15 points) From the 2015 positive integers 1, 2, $\cdots$, 2015, select $k$ numbers such that the sum of any two different numbers is not a multiple of 50. Find the maximum value of $k$.
untranslated part:
在 1,2 , $\cdots, 2015$ 这 2015 个正整数中选出 $k$ 个数,使得其中任意两个不同的数之和均不为 50 的倍数. 求 $k$ 的最大值.
translated part:
From the 2015 positive integers 1, 2, $\cdots$, 2015, select $k$ numbers such that the sum of any two different numbers is not a multiple of 50. Find the maximum value of $k$. | 977 |
48,035 | For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i] | \frac{1}{4} + \frac{1}{2} \ln 2 |
48,046 | Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$. | 10 |
48,078 | II. (50 points) Find the maximum value of the area of an inscribed triangle in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$. | \frac{3 \sqrt{3}}{4} a b |
48,090 | Find all non empty subset $ S$ of $ \mathbb{N}: \equal{} \{0,1,2,\ldots\}$ such that $ 0 \in S$ and exist two function $ h(\cdot): S \times S \to S$ and $ k(\cdot): S \to S$ which respect the following rules:
i) $ k(x) \equal{} h(0,x)$ for all $ x \in S$
ii) $ k(0) \equal{} 0$
iii) $ h(k(x_1),x_2) \equal{} x_1$ for all $ x_1,x_2 \in S$.
[i](Pierfrancesco Carlucci)[/i] | \{0\} |
48,109 | Let $a, b, c, d, e, f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$$
a b c+b c d+c d e+d e f+e f a+f a b
$$
and determine all 6-tuples $(a, b, c, d, e, f)$ for which this maximal value is achieved.
Answer: 8. | 8 |
48,114 | Two circles are said to be [i]orthogonal[/i] if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_1$ and $\omega_2$ with radii $10$ and $13$, respectively, are externally tangent at point $P$. Another circle $\omega_3$ with radius $2\sqrt2$ passes through $P$ and is orthogonal to both $\omega_1$ and $\omega_2$. A fourth circle $\omega_4$, orthogonal to $\omega_3$, is externally tangent to $\omega_1$ and $\omega_2$. Compute the radius of $\omega_4$. | \frac{92}{61} |
48,130 | Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
[i]Proposed by Evan Chen[/i] | 183 |
48,192 | There are $N$ [permutations](https://artofproblemsolving.com/wiki/index.php/Permutation) $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \leq n < n+m \leq 30$. Find the remainder when $N$ is divided by $1000$. | 440 |
48,195 | A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The answer is $n=2^{101}-1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
The text provided is already in English, so no translation is needed. Here is the text as requested:
A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The answer is $n=2^{101}-1$. | 2^{101}-1 |
48,196 | The [i]Olimpia[/i] country is formed by $n$ islands. The most populated one is called [i]Panacenter[/i], and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions:
a) No pair of islands is joined by more than one bridge.
b) Using the bridges we can reach every island from Panacenter.
c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing.
Determine the number of ways we can build the bridges. | (n-1)! |
48,200 | Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends.
Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.)
[i](Birgit Vera Schmidt)[/i]
[hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person.
In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang.
Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide] | 1011 |
48,207 | Three, as shown, in the triangular prism $A B C A_{1} B_{1} C_{1}$, all nine edges are equal to 1, and $\angle A_{1} A B$ $=\angle A_{1} A C$ $=\angle B A C$. Point $P$ is on the diagonal $A_{1} B$ of the side face $A_{1} A B B_{1}$, with $A_{1} P=\frac{\sqrt{3}}{3}$. Connect $P C_{1}$. Find the degree measure of the angle formed by the skew lines $P C_{1}$ and $A C$. | 30^{\circ} |
48,238 | Three. (50 points) There are 2008 students participating in a large public welfare activity. If two students know each other, then these two students are considered as a cooperative group.
(1) Find the minimum number of cooperative groups $m$, such that no matter how the students know each other, there exist three students who are pairwise in a cooperative group;
(2) If the number of cooperative groups is $\left[\frac{m}{22}\right]$, prove: there exist four students $A$, $B$, $C$, $D$, such that $A$ and $B$, $B$ and $C$, $C$ and $D$, $D$ and $A$ are each in a cooperative group. | 1008017 |
48,244 | Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 298 |
48,285 | Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$. | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.