Unnamed: 0
int64 0
56.9k
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stringlengths 16
7.44k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
1,500 |
The sequence $a_1,a_2,\dots,a_{13}$ is a geometric sequence with $a_1=a$ and common ratio $r$ , where $a$ and $r$ are positive integers. Given that $$ \log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015, $$ find the number of possible ordered pairs $(a,r)$ .
|
26^3
| 0 |
1,501 |
Determine the maximal size of a set of positive integers with the following properties: $1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$ . $2.$ No digit occurs more than once in the same integer. $3.$ The digits in each integer are in increasing order. $4.$ Any two integers have at least one digit in common (possibly at different positions). $5.$ There is no digit which appears in all the integers.
|
32
| 62.5 |
1,502 |
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$ , find the minimum value of $p$ .
|
5
| 93.75 |
1,503 |
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.
|
\frac{1}{2013}
| 39.0625 |
1,504 |
A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$ , but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$ .
Find how many Ecuadorian numbers exist.
|
168
| 96.09375 |
1,505 |
Determine the number of positive integers $n$ satisfying:
- $n<10^6$
- $n$ is divisible by 7
- $n$ does not contain any of the digits 2,3,4,5,6,7,8.
|
104
| 89.0625 |
1,506 |
Find an integral solution of the equation
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \]
(Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to $u$ .)
|
1176
| 10.15625 |
1,507 |
A country has $ 1998$ airports connected by some direct flights. For any three airports, some two are not connected by a direct flight. What is the maximum number of direct flights that can be offered?
|
998001
| 98.4375 |
1,508 |
Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ?
|
\frac{1006}{4025}
| 38.28125 |
1,509 |
Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
|
n = 3
| 100 |
1,510 |
Let $p\ge 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
|
3p^2 - 3p + 1
| 0 |
1,511 |
Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$
|
120
| 96.875 |
1,512 |
Given a parameterized curve $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$ .
Find the area bounded by the curve $ C$ , the $ x$ axis and two lines $ x\equal{}\pm 1$ .
|
\frac{5\sqrt{5}}{2}
| 0 |
1,513 |
Suppose that $n$ is a positive integer and let \[d_{1}<d_{2}<d_{3}<d_{4}\] be the four smallest positive integer divisors of $n$ . Find all integers $n$ such that \[n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.\]
|
n = 130
| 0 |
1,514 |
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
|
1344
| 0 |
1,515 |
The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$ . What is the fewest number of digits he could have erased?
*Ray Li*
|
2
| 92.1875 |
1,516 |
How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
|
750
| 97.65625 |
1,517 |
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \dots,20$ on its sides). He conceals the results but tells you that at least half the rolls are $20$ . Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$ ?
|
\frac{1}{58}
| 15.625 |
1,518 |
We call a set “sum free” if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of $\{ 1, 2, \ldots , 2n - 1 \}$ .
|
n
| 100 |
1,519 |
Solve the system of equation for $(x,y) \in \mathbb{R}$ $$ \left\{\begin{matrix}
\sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5
3x^2+4xy=24
\end{matrix}\right. $$
Explain your answer
|
(2, 1.5)
| 0.78125 |
1,520 |
Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$ .
|
20
| 99.21875 |
1,521 |
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$ ). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
*Proposed by David Altizio*
|
24
| 66.40625 |
1,522 |
We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?
|
16
| 62.5 |
1,523 |
A snowman is built on a level plane by placing a ball radius $6$ on top of a ball radius $8$ on top of a ball radius $10$ as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$ .
[asy]
size(150);
draw(circle((0,0),24));
draw(ellipse((0,0),24,9));
draw(circle((0,-56),32));
draw(ellipse((0,-56),32,12));
draw(circle((0,-128),40));
draw(ellipse((0,-128),40,15));
[/asy]
|
61
| 0.78125 |
1,524 |
Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$ , such that $ (2 \minus{} \sin 2x)\sin (x \plus{} \frac {\pi}{4}) \equal{} 1$ .
|
x = \frac{\pi}{4}
| 0 |
1,525 |
The mayor of a city wishes to establish a transport system with at least one bus line, in which:
- each line passes exactly three stops,
- every two different lines have exactly one stop in common,
- for each two different bus stops there is exactly one line that passes through both.
Determine the number of bus stops in the city.
|
s = 7
| 0 |
1,526 |
In an $m \times n$ grid, each square is either filled or not filled. For each square, its *value* is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$ \frac{f(m,n)}{mn} \le C $$ holds for any positive integers $m,n$ *CSJL*
|
C = 2
| 0 |
1,527 |
Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\tfrac12$ the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?
|
\frac{13}{29}
| 72.65625 |
1,528 |
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$ . We also say in this case that $Q$ is circumscribed to $P$ . Given a triangle $T$ , let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .
|
2
| 21.875 |
1,529 |
For a constant $c$ , a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\lim_{n\to\infty} a_n$ .
|
0
| 88.28125 |
1,530 |
Let n be a set of integers. $S(n)$ is defined as the sum of the elements of n. $T=\{1,2,3,4,5,6,7,8,9\}$ and A and B are subsets of T such that A $\cup$ $B=T$ and A $\cap$ $B=\varnothing$ . The probability that $S(A)\geq4S(B)$ can be expressed as $\frac{p}{q}$ . Compute $p+q$ .
*2022 CCA Math Bonanza Team Round #8*
|
545
| 56.25 |
1,531 |
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).
|
373
| 85.9375 |
1,532 |
The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares.
|
25
| 19.53125 |
1,533 |
Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$ . Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$ .
|
99
| 67.1875 |
1,534 |
Find all four-digit natural numbers $\overline{xyzw}$ with the property that their sum plus the sum of their digits equals $2003$ .
|
1978
| 80.46875 |
1,535 |
Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$ - $y$ plane.
Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$ -axis.
|
\frac{128\pi}{105}
| 0 |
1,536 |
A positive integer $n$ is called $\textit{un-two}$ if there does not exist an ordered triple of integers $(a,b,c)$ such that exactly two of $$ \dfrac{7a+b}{n},\;\dfrac{7b+c}{n},\;\dfrac{7c+a}{n} $$ are integers. Find the sum of all un-two positive integers.
*Proposed by **stayhomedomath***
|
660
| 4.6875 |
1,537 |
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.
|
10
| 94.53125 |
1,538 |
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 $\mathcal{P}$ .
(*Dan Schwarz*)
|
2^{n-1}
| 3.125 |
1,539 |
A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$ . Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.
|
n = 3
| 0 |
1,540 |
Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$ , and $Y$ the foot of the perpendicular from $C$ to $AB$ . If $AX = 20$ and $CY = 22$ , find the area of the parallelogram.
|
880
| 33.59375 |
1,541 |
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .
*Proposed by Michael Tang*
|
423
| 31.25 |
1,542 |
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$ .
*Proposed by Aaron Lin*
|
2083
| 0.78125 |
1,543 |
A parabola has focus $F$ and vertex $V$ , where $VF = 1$ 0. Let $AB$ be a chord of length $100$ that passes through $F$ . Determine the area of $\vartriangle VAB$ .
|
100\sqrt{10}
| 10.15625 |
1,544 |
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$ .
|
10
| 0.78125 |
1,545 |
Given a square $ABCD$ . Let $P\in{AB},\ Q\in{BC},\ R\in{CD}\ S\in{DA}$ and $PR\Vert BC,\ SQ\Vert AB$ and let $Z=PR\cap SQ$ . If $BP=7,\ BQ=6,\ DZ=5$ , then find the side length of the square.
|
10
| 17.96875 |
1,546 |
Let $a$ be a positive real number, $n$ a positive integer, and define the *power tower* $a\uparrow n$ recursively with $a\uparrow 1=a$ , and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$ . For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has $155$ digits. For each positive integer $k$ , let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$ . Which is larger: $x_{42}$ or $x_{43}$ ?
|
x_{42}
| 28.125 |
1,547 |
In rectangle $ABCD$ , point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$ . The perpendicular bisector of $MP$ intersects side $DA$ at point $X$ . Given that $AB = 33$ and $BC = 56$ , find the least possible value of $MX$ .
*Proposed by Michael Tang*
|
33
| 35.15625 |
1,548 |
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .
.
|
769
| 24.21875 |
1,549 |
Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying
\[
\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
\]
|
18
| 6.25 |
1,550 |
Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$ . What is the minimum possible value of $a+b$ ?
|
34
| 91.40625 |
1,551 |
Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?
|
(m, n)
| 2.34375 |
1,552 |
On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$ . Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.
|
43
| 89.0625 |
1,553 |
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
|
8
| 75.78125 |
1,554 |
A positive integer $n$ is called *mythical* if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.
*Proposed by Evan Chen*
|
135
| 9.375 |
1,555 |
If \[ \begin {eqnarray*} x + y + z + w = 20 y + 2z - 3w = 28 x - 2y + z = 36 -7x - y + 5z + 3w = 84 \]then what is $(x,y,z,w)$ ?
|
(4, -6, 20, 2)
| 91.40625 |
1,556 |
Let $ABC$ be an arbitrary triangle. A regular $n$ -gon is constructed outward on the three sides of $\triangle ABC$ . Find all $n$ such that the triangle formed by the three centres of the $n$ -gons is equilateral.
|
n = 3
| 0 |
1,557 |
Let $ n \geq 3$ be an odd integer. Determine the maximum value of
\[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\]
where $ x_{i}$ are positive real numbers from the interval $ [0,1]$ .
|
n - 2 + \sqrt{2}
| 0 |
1,558 |
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$ . Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$ . Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$ , respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$ .
*Proposed by Andrew Wu*
|
31
| 3.90625 |
1,559 |
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.
|
1009
| 11.71875 |
1,560 |
We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion $$ a_n = \bigg \lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \bigg \rfloor $$
for all $n \ge 3$ . Find the value of $a_{2022}$ .
|
674
| 95.3125 |
1,561 |
Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. Find $m+n$ .
*Proposed by **AOPS12142015***
|
9
| 75.78125 |
1,562 |
Ria writes down the numbers $1,2,\cdots, 101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen?
|
68
| 83.59375 |
1,563 |
Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . What is $m+n$ ?
*2021 CCA Math Bonanza Team Round #9*
|
455
| 0 |
1,564 |
Anton and Britta play a game with the set $M=\left \{ 1,2,\dots,n-1 \right \}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$ , and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_1, x_2$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_1, y_2$ from $B$ . Britta wins if $(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$ otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.
|
n
| 78.125 |
1,565 |
Let $ABC$ be a triangle whose angles measure $A$ , $B$ , $C$ , respectively. Suppose $\tan A$ , $\tan B$ , $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$ , find the number of possible integer values for $\tan B$ . (The values of $\tan A$ and $\tan C$ need not be integers.)
*Proposed by Justin Stevens*
|
11
| 4.6875 |
1,566 |
in a right-angled triangle $ABC$ with $\angle C=90$ , $a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$ ; $a,c$ respectively,with radii $r,t$ .find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.
|
\sqrt{2} + 1
| 2.34375 |
1,567 |
Dave has a pile of fair standard six-sided dice. In round one, Dave selects eight of the dice and rolls them. He calculates the sum of the numbers face up on those dice to get $r_1$ . In round two, Dave selects $r_1$ dice and rolls them. He calculates the sum of the numbers face up on those dice to get $r_2$ . In round three, Dave selects $r_2$ dice and rolls them. He calculates the sum of the numbers face up on those dice to get $r_3$ . Find the expected value of $r_3$ .
|
343
| 99.21875 |
1,568 |
On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?
|
126
| 25.78125 |
1,569 |
Let $p$ be a polynomial with integer coefficients such that $p(15)=6$ , $p(22)=1196$ , and $p(35)=26$ . Find an integer $n$ such that $p(n)=n+82$ .
|
28
| 8.59375 |
1,570 |
Positive integers a, b, c, d, and e satisfy the equations $$ (a + 1)(3bc + 1) = d + 3e + 1 $$ $$ (b + 1)(3ca + 1) = 3d + e + 13 $$ $$ (c + 1)(3ab + 1) = 4(26-d- e) - 1 $$ Find $d^2+e^2$ .
|
146
| 56.25 |
1,571 |
How many distinct positive integers can be expressed in the form $ABCD - DCBA$ , where $ABCD$ and $DCBA$ are 4-digit positive integers? (Here $A$ , $B$ , $C$ and $D$ are digits, possibly equal.)
Clarification: $A$ and $D$ can't be zero (because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit integer).
|
161
| 21.09375 |
1,572 |
Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$ . Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]
|
2
| 83.59375 |
1,573 |
Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and $$ \frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1}) $$ $\forall n=1,2,...$ Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$
|
|u_n| \leq 1
| 89.84375 |
1,574 |
The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
*2018 CCA Math Bonanza Lightning Round #3.1*
|
264
| 60.15625 |
1,575 |
Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.
|
390
| 1.5625 |
1,576 |
Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$ . The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$ ?
|
1:2
| 24.21875 |
1,577 |
A four-digit number has the following properties:
(a) It is a perfect square;
(b) Its first two digits are equal
(c) Its last two digits are equal.
Find all such four-digit numbers.
|
7744
| 94.53125 |
1,578 |
Triangle $ABC$ has sidelengths $AB=1$ , $BC=\sqrt{3}$ , and $AC=2$ . Points $D,E$ , and $F$ are chosen on $AB, BC$ , and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, b) = 1$ , find $a + b$ . (Here $[DEF]$ denotes the area of triangle $DEF$ .)
*Proposed by Vismay Sharan*
|
67
| 28.90625 |
1,579 |
Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1 $ . Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions:
**I.** $m\leq x\leq n$ for all $x\in S$ ;
**II.** the product of all elements in $S$ is the square of an integer.
|
n = m + p
| 0 |
1,580 |
Solve the following system of equations for real $x,y$ and $z$ :
\begin{eqnarray*}
x &=& \sqrt{2y+3}
y &=& \sqrt{2z+3}
z &=& \sqrt{2x+3}.
\end{eqnarray*}
|
x = y = z = 3
| 0 |
1,581 |
Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$ ?
*2018 CCA Math Bonanza Tiebreaker Round #3*
|
607
| 9.375 |
1,582 |
Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$ , and $(0, 3, 0)$ , and whose apex is the point $(1, 1, 3)$ . Let $Q$ be a square pyramid whose base is the same as the base of $P$ , and whose apex is the point $(2, 2, 3)$ . Find the volume of the intersection of the interiors of $P$ and $Q$ .
|
\frac{27}{4}
| 0 |
1,583 |
Let the positive integer $n$ have at least for positive divisors and $0<d_1<d_2<d_3<d_4$ be its least positive divisors. Find all positive integers $n$ such that:
\[ n=d_1^2+d_2^2+d_3^2+d_4^2. \]
|
130
| 87.5 |
1,584 |
Given an integer number $n \geq 3$ , consider $n$ distinct points on a circle, labelled $1$ through $n$ .
Determine the maximum number of closed chords $[ij]$ , $i \neq j$ , having pairwise non-empty intersections.
*János Pach*
|
n
| 3.90625 |
1,585 |
A group of $n$ people play a board game with the following rules:
1) In each round of the game exactly $3$ people play
2) The game ends after exactly $n$ rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of $n$
|
7
| 85.15625 |
1,586 |
Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are non-zero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.)
|
6655
| 3.90625 |
1,587 |
Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.
|
313
| 0 |
1,588 |
$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$ , $az+cx=b$ , $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$
|
\frac{1}{2}
| 53.125 |
1,589 |
Let there be a regular polygon of $n$ sides with center $O$ . Determine the highest possible number of vertices $k$ $(k \geq 3)$ , which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$ .
|
1010
| 78.125 |
1,590 |
Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$ . Find $$ \sum_{(a,b,c)\in S} abc. $$
|
\binom{2015}{5}
| 0 |
1,591 |
Let $ T$ be the set of all positive integer divisors of $ 2004^{100}$ . What is the largest possible number of elements of a subset $ S$ of $ T$ such that no element in $ S$ divides any other element in $ S$ ?
|
101^2
| 0.78125 |
1,592 |
Twelve $1$ 's and ten $-1$ 's are written on a chalkboard. You select 10 of the numbers and compute their product, then add up these products for every way of choosing 10 numbers from the 22 that are written on the chalkboard. What sum do you get?
|
-42
| 0.78125 |
1,593 |
Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions: $\bullet$ $r$ is a positive integer with exactly $8$ positive divisors. $\bullet$ $p$ and $q$ are prime numbers.
|
(2, 5, 609)
| 82.03125 |
1,594 |
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$ . $$ xy+yz = 30 $$ $$ yz+zx = 36 $$ $$ zx+xy = 42 $$ *Proposed by Nathan Xiong*
|
13
| 89.0625 |
1,595 |
Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\]
|
0
| 66.40625 |
1,596 |
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
|
231361
| 21.09375 |
1,597 |
Suppose that $0^\circ < A < 90^\circ$ and $0^\circ < B < 90^\circ$ and \[\left(4+\tan^2 A\right)\left(5+\tan^2 B\right) = \sqrt{320}\tan A\tan B\] Determine all possible values of $\cos A\sin B$ .
|
\frac{\sqrt{6}}{6}
| 17.1875 |
1,598 |
A number is called *purple* if it can be expressed in the form $\frac{1}{2^a 5^b}$ for positive integers $a > b$ . The sum of all purple numbers can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$ . Compute $100a + b$ .
*Proposed by Eugene Chen*
|
109
| 89.0625 |
1,599 |
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$ , the sum of the digits of $n$ is $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 19 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 23$
|
15
| 35.15625 |
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