Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
4,200 |
Define a $good~word$ as a sequence of letters that consists only of the letters $A,$ $B,$ and $C$ $-$ some of these letters may not appear in the sequence $-$ and in which $A$ is never immediately followed by $B,$ $B$ is never immediately followed by $C,$ and $C$ is never immediately followed by $A.$ How many seven-letter good words are there?
|
192
| 96.875 |
4,201 |
Let $c$ be a fixed real number. Show that a root of the equation
\[x(x+1)(x+2)\cdots(x+2009)=c\]
can have multiplicity at most $2$ . Determine the number of values of $c$ for which the equation has a root of multiplicity $2$ .
|
1005
| 2.34375 |
4,202 |
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$ , $b$ , $c$ reals such that any quadratic trinomial obtained by a permutation of $f$ 's coefficients has an integer root (including $f$ itself).
Show that $f(1)=0$ .
|
f(1) = 0
| 6.25 |
4,203 |
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$ . Find $ab$ .
|
\frac{128}{3}
| 9.375 |
4,204 |
The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$ . Find the distance from $P$ to the third side.
|
5\sqrt{3} - 4
| 92.1875 |
4,205 |
Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point.
|
2 + 2\sqrt{2}
| 25.78125 |
4,206 |
A company that sells keychains has to pay $\mathdollar500$ in maintenance fees each day and then it pays each work $\mathdollar15$ an hour. Each worker makes $5$ keychains per hour, which are sold at $\mathdollar3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an $8$ -hour workday?
|
126
| 83.59375 |
4,207 |
Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability $1/2$ and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die $5$ times and gets a $1, 2, 3, 4$ and $5$ in some unspecified
order. Compute the probability that his next roll is a $6$ .
|
\frac{3}{26}
| 7.03125 |
4,208 |
Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}2342\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}3636\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}4223\hline 65\end{tabular}\,.\]
|
3240
| 86.71875 |
4,209 |
Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$ .
|
1899
| 1.5625 |
4,210 |
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.
|
-2
| 12.5 |
4,211 |
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$ .
(b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$ . Show that $ A^{A} \leq R^{B}$ .
|
A^A \leq R^B
| 89.0625 |
4,212 |
Let $n$ be an integer with $n\geq 2.$ Over all real polynomials $p(x)$ of degree $n,$ what is the largest possible number of negative coefficients of $p(x)^2?$
|
2n-2
| 0.78125 |
4,213 |
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \]
|
(1, 11, 3)
| 84.375 |
4,214 |
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$ ?
|
420
| 93.75 |
4,215 |
A massless string is wrapped around a frictionless pulley of mass $M$ . The string is pulled down with a force of 50 N, so that the pulley rotates due to the pull. Consider a point $P$ on the rim of the pulley, which is a solid cylinder. The point has a constant linear (tangential) acceleration component equal to the acceleration of gravity on Earth, which is where this experiment is being held. What is the weight of the cylindrical pulley, in Newtons?
*(Proposed by Ahaan Rungta)*
<details><summary>Note</summary>This problem was not fully correct. Within friction, the pulley cannot rotate. So we responded:
<blockquote>Excellent observation! This is very true. To submit, I'd say just submit as if it were rotating and ignore friction. In some effects such as these, I'm pretty sure it turns out that friction doesn't change the answer much anyway, but, yes, just submit as if it were rotating and you are just ignoring friction. </blockquote>So do this problem imagining that the pulley does rotate somehow.</details>
|
100 \, \text{N}
| 0 |
4,216 |
Determine all pairs $(p, q)$ of prime numbers such that $p^p + q^q + 1$ is divisible by $pq.$
|
(2, 5)
| 15.625 |
4,217 |
Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$ . Show that $p<2n$ .
|
p < 2n
| 97.65625 |
4,218 |
Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that $1$ , $2$ , $3$ , $4$ , $5$ , and $6$ dots are on separate faces. In a fit of mischief, James places his $21$ dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct con figuration. Regardless of the confi guration of dots, both dice are unweighted and have equal probability of showing each face after being rolled.
Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability.
|
\frac{17}{32}
| 0 |
4,219 |
We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ *nice* if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element *nice* subset while keeping it nice, find the minimum value of $n$ .
|
436
| 0.78125 |
4,220 |
Three points $X, Y,Z$ are on a striaght line such that $XY = 10$ and $XZ = 3$ . What is the product of all possible values of $YZ$ ?
|
91
| 85.9375 |
4,221 |
Let $a,b,c,x,y,$ and $z$ be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If $xy+yz+xz=67$ and $x+y+z=2010$ , find the value of $xyz$ .
|
-5892
| 0 |
4,222 |
Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.
|
14:24
| 75.78125 |
4,223 |
Find the real number $a$ such that the integral $$ \int_a^{a+8}e^{-x}e^{-x^2}dx $$ attain its maximum.
|
a = -4.5
| 0 |
4,224 |
The Fibonacci sequence is defined as $f_1=f_2=1$ , $f_{n+2}=f_{n+1}+f_n$ ( $n\in\mathbb{N}$ ). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$ . Show that $b\ge f_{n+1}$ .
|
b \geq f_{n+1}
| 17.1875 |
4,225 |
Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$ . For example, this gives $f(8)=1$ , $f(9)=1$ , $f(10)=2$ , $f(11)=4$ , and $f(12)=5$ . Determine the value of $f(10^{100})$ .
|
10^{101} + 1
| 4.6875 |
4,226 |
Three distinct integers are chosen uniformly at random from the set $$ \{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}. $$ Compute the probability that their arithmetic mean is an integer.
|
\frac{7}{20}
| 92.1875 |
4,227 |
What is the $22\text{nd}$ positive integer $n$ such that $22^n$ ends in a $2$ ? (when written in base $10$ ).
|
85
| 100 |
4,228 |
Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$ .
|
36^\circ
| 92.1875 |
4,229 |
Parallelogram $ABCD$ is such that angle $B < 90$ and $AB<BC$ . Points E and F are on the circumference of $\omega$ inscribing triangle ABC, such that tangents to $\omega$ in those points pass through D. If $\angle EDA= \angle{FDC}$ , find $\angle{ABC}$ .
|
60^\circ
| 89.0625 |
4,230 |
Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:
- for every pair of students there is exactly one problem that was solved by both students;
- for every pair of problems there is exactly one student who solved both of them;
- one specific problem was solved by Laura and exactly $n$ other students.
Determine the number of students in Laura's class.
|
n^2 + n + 1
| 77.34375 |
4,231 |
The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$ . Compute the greatest possible value of $D$ .
|
24
| 78.125 |
4,232 |
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$ , with $ x, y$ being integers.
|
5
| 68.75 |
4,233 |
Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$ . The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$ . Find $x$ .
|
\frac{32}{27}
| 89.84375 |
4,234 |
Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$ , for every natural number $ n$ .
|
f(n) = 1 - n
| 35.15625 |
4,235 |
Sets $A, B$ , and $C$ satisfy $|A| = 92$ , $|B| = 35$ , $|C| = 63$ , $|A\cap B| = 16$ , $|A\cap C| = 51$ , $|B\cap C| = 19$ . Compute the number of possible values of $ |A \cap B \cap C|$ .
|
10
| 4.6875 |
4,236 |
Let $a_0 = \frac{1}{2}$ and $a_n$ be defined inductively by
\[a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{, $n \ge 1$ .} \]
[list=a]
[*] Show that for $n = 0,1,2, \ldots,$
\[a_n = \cos(\theta_n) \text{ for some $0 < \theta_n < \frac{\pi}{2}$ , }\]
and determine $\theta_n$ .
[*] Using (a) or otherwise, calculate
\[ \lim_{n \to \infty} 4^n (1 - a_n).\]
[/list]
|
\frac{\pi^2}{18}
| 100 |
4,237 |
<u>Round 9</u>**p25.** Let $S$ be the set of the first $2017$ positive integers. Find the number of elements $n \in S$ such that $\sum^n_{i=1} \left\lfloor \frac{n}{i} \right\rfloor$ is even.**p26.** Let $\{x_n\}_{n \ge 0}$ be a sequence with $x_0 = 0$ , $x_1 = \frac{1}{20}$ , $x_2 = \frac{1}{17}$ , $x_3 = \frac{1}{10}$ , and $x_n = \frac12 ((x_{n-2} +x_{n-4})$ for $n\ge 4$ . Compute $$ \left\lfloor \frac{1}{x_{2017!} -x_{2017!-1}} \right\rfloor. $$ **p27.** Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$ , find $\angle CDE + \angle EAB$ , in degrees.
<u>Round 10</u>**p28.** Let $S = \{1,2,4, ... ,2^{2016},2^{2017}\}$ . For each $0 \le i \le 2017$ , let $x_i$ be chosen uniformly at random from the subset of $S$ consisting of the divisors of $2^i$ . What is the expected number of distinct values in the set $\{x_0,x_1,x_2,... ,x_{2016},x_{2017}\}$ ?**p29.** For positive real numbers $a$ and $b$ , the points $(a, 0)$ , $(20,17)$ and $(0,b)$ are collinear. Find the minimum possible value of $a+b$ .**p30.** Find the sum of the distinct prime factors of $2^{36}-1$ .
<u>Round 11</u>**p31.** There exist two angle bisectors of the lines $y = 20x$ and $y = 17x$ with slopes $m_1$ and $m_2$ . Find the unordered pair $(m_1,m_2)$ .**p32.** Triangle 4ABC has sidelengths $AB = 13$ , $BC = 14$ , $C A =15$ and orthocenter $H$ . Let $\Omega_1$ be the circle through $B$ and $H$ , tangent to $BC$ , and let $\Omega_2$ be the circle through $C$ and $H$ , tangent to $BC$ . Finally, let $R \ne H$ denote the second intersection of $\Omega_1$ and $\Omega_2$ . Find the length $AR$ .**p33.** For a positive integer $n$ , let $S_n = \{1,2,3, ...,n\}$ be the set of positive integers less than or equal to $n$ . Additionally, let $$ f (n) = |\{x \in S_n : x^{2017}\equiv x \,\, (mod \,\, n)\}|. $$ Find $f (2016)- f (2015)+ f (2014)- f (2013)$ .
<u>Round 12</u>**p34.** Estimate the value of $\sum^{2017}_{n=1} \phi (n)$ , where $\phi (n)$ is the number of numbers less than or equal $n$ that are relatively prime to n. If your estimate is $E$ and the correct answer is $A$ , your score for this problem will be max $\max \left(0,\lfloor 15 - 75 \frac{|A-E|}{A} \rceil \right).$ **p35.** An up-down permutation of order $n$ is a permutation $\sigma$ of $(1,2,3, ..., n)$ such that $\sigma(i ) <\sigma (i +1)$ if and only if $i$ is odd. Denote by $P_n$ the number of up-down permutations of order $n$ . Estimate the value of $P_{20} +P_{17}$ . If your estimate is $E$ and the correct answer is $A$ , your score for this problem will be $\max \left(0, 16 -\lceil \max \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ **p36.** For positive integers $n$ , superfactorial of $n$ , denoted $n\$ $, is defined as the product of the first $ n $ factorials. In other words, we have $ n\ $ = \prod^n_{i=1}(i !)$ . Estimate the number of digits in the product $(20\$ )\cdot (17\ $)$ . If your estimate is $E$ and the correct answer is $A$ , your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [here ](https://artofproblemsolving.com/community/c3h3158491p28715220) and 5-8 [here](https://artofproblemsolving.com/community/c3h3158514p28715373). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).
|
163^\circ
| 0 |
4,238 |
Find the area of the region bounded by a function $y=-x^4+16x^3-78x^2+50x-2$ and the tangent line which is tangent to the curve at exactly two distinct points.
Proposed by Kunihiko Chikaya
|
\frac{1296}{5}
| 0 |
4,239 |
Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$ , then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.
|
a_i = c
| 0 |
4,240 |
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
*Author: Ray Li*
|
256
| 19.53125 |
4,241 |
For each positive integer $n$ , let $s(n)$ be the sum of the squares of the digits of $n$ . For example, $s(15)=1^2+5^2=26$ . Determine all integers $n\geq 1$ such that $s(n)=n$ .
|
1
| 97.65625 |
4,242 |
How many ordered triples $(a, b, c)$ of odd positive integers satisfy $a + b + c = 25?$
|
78
| 99.21875 |
4,243 |
The graph of the equation $x^y =y^x$ in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.
|
(e, e)
| 70.3125 |
4,244 |
Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75\%$ . If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25\%$ . He decides not to make it rain today. Find the smallest positive integer $n$ such that the probability that Lil Wayne *makes it rain* $n$ days from today is greater than $49.9\%$ .
|
n = 9
| 0 |
4,245 |
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$ -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$ .
|
4
| 30.46875 |
4,246 |
What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$ ?
|
60
| 100 |
4,247 |
Let $a$ , $b$ , $c$ be positive integers and $p$ be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b)\equiv 8\pmod{p} \] for each nonnegative integer $n$ . Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$ , and $k$ the remainder when $m^p$ is divided by $p^4$ . Find the maximum possible value of $k$ .
*Proposed by Justin Stevens and Evan Chen*
|
399
| 2.34375 |
4,248 |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$ f(x+y)\leq f(x^2+y) $$ for all $x,y$ .
|
f(x) = c
| 51.5625 |
4,249 |
Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$ . Find the integer part of:
$ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$
|
1
| 89.84375 |
4,250 |
A square of side $1$ is covered with $m^2$ rectangles.
Show that there is a rectangle with perimeter at least $\frac{4}{m}$ .
|
\frac{4}{m}
| 98.4375 |
4,251 |
Given $1962$ -digit number. It is divisible by $9$ . Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$ . Let the sum of the digits of $y$ be $z$ . Find $z$ .
|
9
| 100 |
4,252 |
Consider $S=\{1, 2, 3, \cdots, 6n\}$ , $n>1$ . Find the largest $k$ such that the following statement is true: every subset $A$ of $S$ with $4n$ elements has at least $k$ pairs $(a,b)$ , $a<b$ and $b$ is divisible by $a$ .
|
k = n
| 0 |
4,253 |
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
*Proposed by Aaron Lin*
|
1006
| 22.65625 |
4,254 |
Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.
|
n = 1
| 0 |
4,255 |
Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$ . Find the derivative of $f(x)-f(4x)$ at $x=1$ .
|
19
| 86.71875 |
4,256 |
Let $n$ be some positive integer and $a_1 , a_2 , \dots , a_n$ be real numbers. Denote $$ S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2}, $$ where $a_{n+1} = a_1$ and $a_{n+2} = a_2.$ 1. Show that $S_0 - S_1 \geq 0$ .
2. Show that $3$ is the minimum value of $C$ such that for any $n$ and $a_1 , a_2 , \dots , a_n,$ there holds $C(S_0 - S_1) \geq S_1 - S_2$ .
|
C = 3
| 0 |
4,257 |
Let $S$ be a subset of the natural numbers such that $0\in S$ , and for all $n\in\mathbb N$ , if $n$ is in $S$ , then both $2n+1$ and $3n+2$ are in $S$ . What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$ ?
|
47
| 97.65625 |
4,258 |
In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.
|
2014
| 42.96875 |
4,259 |
Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$ . Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$ .
|
942
| 77.34375 |
4,260 |
Given seven points in the plane, some of them are connected by segments such that:
**(i)** among any three of the given points, two are connected by a segment;
**(ii)** the number of segments is minimal.
How many segments does a figure satisfying **(i)** and **(ii)** have? Give an example of such a figure.
|
9
| 54.6875 |
4,261 |
Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.
|
n = 4
| 0 |
4,262 |
Let $n$ be a positive integer with the following property: $2^n-1$ divides a number of the form $m^2+81$ , where $m$ is a positive integer. Find all possible $n$ .
|
n = 2^k
| 0 |
4,263 |
Jae and Yoon are playing SunCraft. The probability that Jae wins the $n$ -th game is $\frac{1}{n+2}.$ What is the probability that Yoon wins the first six games, assuming there are no ties?
|
\frac{1}{4}
| 92.1875 |
4,264 |
Real numbers $x$ and $y$ satisfy
\begin{align*}
x^2 + y^2 &= 2023
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of $|x-y|$ .
*Proposed by Howard Halim*
|
13\sqrt{13}
| 3.90625 |
4,265 |
The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $P(\min(X,Y)=k)$ in terms of $p_1=P(X=k)$ , $p_2=P(Y=k)$ and $p_3=P(\max(X,Y)=k)$ .
|
p_1 + p_2 - p_3
| 55.46875 |
4,266 |
**8.** Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). **(N. 9)**
|
a = 2
| 2.34375 |
4,267 |
A square $ABCD$ has side length $ 1$ . A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$ , $BC$ , $CD$ , $DA$ when reflected on sides $AB$ , $B$ C, $CD$ , $DA$ , respectively. The area of square $PQRS$ is $a+b\sqrt2$ , where $a$ and $ b$ are integers. Find the value of $a+b$ .
|
a + b = 3 - 2 = 1
| 0 |
4,268 |
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$ . Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$ , respectively, with $P$ lying in between $B$ and $Q$ . If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?
|
799
| 7.8125 |
4,269 |
Find the smallest positive integer $n$ for which $315^2-n^2$ evenly divides $315^3-n^3$ .
*Proposed by Kyle Lee*
|
90
| 35.15625 |
4,270 |
Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$ , the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$ , and the resulting line is reflected in $l_2$ . Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$ . Given that $l$ is the line $y=\frac{19}{92}x$ , find the smallest positive integer $m$ for which $R^{(m)}(l)=l$ .
|
945
| 92.96875 |
4,271 |
Let $N$ be a given positive integer. Consider a permutation of $1,2,3,\cdots,N$ , denoted as $p_1,p_2,\cdots,p_N$ . For a section $p_l, p_{l+1},\cdots, p_r$ , we call it "extreme" if $p_l$ and $p_r$ are the maximum and minimum value of that section. We say a permutation $p_1,p_2,\cdots,p_N$ is "super balanced" if there isn't an "extreme" section with a length at least $3$ . For example, $1,4,2,3$ is "super balanced", but $3,1,2,4$ isn't. Please answer the following questions:
1. How many "super balanced" permutations are there?
2. For each integer $M\leq N$ . How many "super balanced" permutations are there such that $p_1=M$ ?
*Proposed by ltf0501*
|
\binom{N-1}{M-1}
| 0 |
4,272 |
Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$ .
Evaluate
\[\int_0^1 \sin \alpha x\sin \beta x\ dx\]
|
0
| 50.78125 |
4,273 |
A right cylinder is given with a height of $20$ and a circular base of radius $5$ . A vertical planar cut is made into this base of radius $5$ . A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the pieces is $100\sqrt2$ . Then the volume of the larger piece can be written as $a + b\pi$ , where $a, b$ are positive integers. Find $a + b$ .
|
625
| 15.625 |
4,274 |
Consider $\triangle \natural\flat\sharp$ . Let $\flat\sharp$ , $\sharp\natural$ and $\natural\flat$ be the answers to problems $4$ , $5$ , and $6$ , respectively. If the incircle of $\triangle \natural\flat\sharp$ touches $\natural\flat$ at $\odot$ , find $\flat\odot$ .
*Proposed by Evan Chen*
|
2.5
| 3.125 |
4,275 |
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$
|
22
| 0 |
4,276 |
The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
[i]Proposed by Clive Chan
|
1008
| 62.5 |
4,277 |
A *9-cube* is a nine-dimensional hypercube (and hence has $2^9$ vertices, for example). How many five-dimensional faces does it have?
(An $n$ dimensional hypercube is defined to have vertices at each of the points $(a_1,a_2,\cdots ,a_n)$ with $a_i\in \{0,1\}$ for $1\le i\le n$ )
*Proposed by Evan Chen*
|
2016
| 50.78125 |
4,278 |
A triangle has sides of lengths 5, 6, 7. What is 60 times the square of the radius of the inscribed circle?
|
160
| 33.59375 |
4,279 |
Let $x$ , $y$ , $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$ . Find the greatest possible value of $x+y+z$ .
|
20
| 50.78125 |
4,280 |
Let $\triangle ABC$ be a triangle with circumcenter $O$ satisfying $AB=13$ , $BC = 15$ , and $AC = 14$ . Suppose there is a point $P$ such that $PB \perp BC$ and $PA \perp AB$ . Let $X$ be a point on $AC$ such that $BX \perp OP$ . What is the ratio $AX/XC$ ?
*Proposed by Thomas Lam*
|
\frac{169}{225}
| 0 |
4,281 |
For reals $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$ , let $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$ for each $k$ . If $S_2 = 777$ , compute the least possible value of $S_3$ .
*Proposed by Evan Chen*
|
999
| 5.46875 |
4,282 |
Find all integer sequences of the form $ x_i, 1 \le i \le 1997$ , that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$ .
|
x_i = 0
| 0 |
4,283 |
Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$ .
*2020 CCA Math Bonanza Individual Round #8*
|
5
| 40.625 |
4,284 |
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 $\triangle$ .)
*Proposed by Robin Park*
|
22200
| 0.78125 |
4,285 |
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ .
|
45^\circ
| 10.9375 |
4,286 |
In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C?$
|
20
| 97.65625 |
4,287 |
If there is only $1$ complex solution to the equation $8x^3 + 12x^2 + kx + 1 = 0$ , what is $k$ ?
|
6
| 14.0625 |
4,288 |
Three palaces, each rotating on a duck leg, make a full round in $30$ , $50$ , and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?
|
525
| 81.25 |
4,289 |
If $a$ , $6$ , and $b$ , in that order, form an arithmetic sequence, compute $a+b$ .
|
12
| 92.96875 |
4,290 |
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$ ? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
|
5
| 85.15625 |
4,291 |
The measures, in degrees, of the angles , $\alpha, \beta$ and $\theta$ are greater than $0$ less than $60$ . Find the value of $\theta$ knowing, also, that $\alpha + \beta = 2\theta$ and that $$ \sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ). $$
|
30^\circ
| 85.15625 |
4,292 |
A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + b}{c \pi}$ for some positive integers $a, b, c$ with $a$ and $c$ relatively prime. Find $a+b+c$ .
|
34
| 0 |
4,293 |
We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$ ?
|
119
| 24.21875 |
4,294 |
Let $f$ be a function defined for the non-negative integers, such that:
a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$ .
b) $f(n+1)=f(n)-1$ otherwise.
i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$ .
ii) Find $f(2^{1990})$ .
|
f(2^{1990}) = -1
| 0 |
4,295 |
Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$ .
|
(a, b)
| 82.03125 |
4,296 |
Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that: $f(1997)=1998$ $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$
|
t = -1
| 0 |
4,297 |
Source: 2017 Canadian Open Math Challenge, Problem A2
-----
An equilateral triangle has sides of length $4$ cm. At each vertex, a circle with radius $2$ cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$ . Determine $a$ .
[center][asy]
size(2.5cm);
draw(circle((0,2sqrt(3)/3),1));
draw(circle((1,-sqrt(3)/3),1));
draw(circle((-1,-sqrt(3)/3),1));
draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle);
fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray);
draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle);
fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray);
draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle);
fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray);
[/asy][/center]
|
2
| 64.84375 |
4,298 |
Suppose $x$ is a random real number between $1$ and $4$ , and $y$ is a random real number between $1$ and $9$ . If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$ .
*Proposed by Lewis Chen*
|
1112
| 43.75 |
4,299 |
For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$ . Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$ .
|
(n-3)2^n + n + 3
| 0 |
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