Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
700 |
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$ . Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$ .
|
\frac{9}{4}
| 48.4375 |
701 |
The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find $m$ .
|
6490716149
| 79.6875 |
702 |
How many $4-$ digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ?
|
7
| 96.875 |
703 |
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$ ,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$ , for any $f \in \mathcal{F}$ .
b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$ .
|
3
| 32.03125 |
704 |
Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$ . For all points $(a, b)$ , let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$ . Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4, then compute $n$ .
*Proposed by Lewis Chen*
|
2013
| 12.5 |
705 |
Let $1=d_1<d_2<\ldots<d_k=n$ be all natural divisors of a natural number $n$ .
Find all possible values of $k$ if $n=d_2d_3+d_2d_5+d_3d_5$ .
|
k = 9
| 0 |
706 |
In one of the hotels of the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. We know the following about the guests: if two women know each other then their husbands also know each other, and vice versa. There are several restrictions on the usage of saunas. Each sauna can be used by either men only, or women only (but there is no limit on the number of people using a sauna at once, as long as they are of a single gender). Each woman is only willing to share a sauna with women whom she knows, and each man is only willing to share a sauna with men whom he does not know. What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?
|
2018
| 1.5625 |
707 |
Suppose $a,b,c,x,y,z$ are pairwisely different real numbers. How many terms in the following can be $1$ at most: $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$
|
2
| 31.25 |
708 |
Two circles touch in $M$ , and lie inside a rectangle $ABCD$ . One of them touches the sides $AB$ and $AD$ , and the other one touches $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$ .
|
1:1
| 5.46875 |
709 |
Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote the area of $X.$ Find the maximum value, or show that no maximum exists, of $\tfrac{A(R)+A(S)}{A(T)},$ where $T$ ranges over all triangles and $R,S$ over all rectangles as above.
|
\frac{2}{3}
| 7.03125 |
710 |
Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a *bullet*. Suppose some of the bullets are coloured red. For each pair $(i j)$ , with $ 1 \le i < j \le 8$ , let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$ .
|
n = 14
| 0 |
711 |
Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last $30$ digits all equal to $9$ .
|
34
| 45.3125 |
712 |
Let $a$ , $b$ , $c$ be positive reals for which
\begin{align*}
(a+b)(a+c) &= bc + 2
(b+c)(b+a) &= ca + 5
(c+a)(c+b) &= ab + 9
\end{align*}
If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , compute $100m+n$ .
*Proposed by Evan Chen*
|
4532
| 53.125 |
713 |
For any integer $a$ , let $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of $f(a)$ that are prime?
*Proposed by Alexander Wang*
|
22
| 71.875 |
714 |
$k$ marbles are placed onto the cells of a $2024 \times 2024$ grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).
a) Assume that $k=2024$ . Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions.
b) Determine the largest value of $k$ such that for all arrangements of $k$ marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.
|
k = 2023
| 0 |
715 |
Let $ABCD$ be a convex quadrilateral with side lengths satisfying the equality: $$ AB \cdot CD = AD \cdot BC = AC \cdot BD. $$ Determine the sum of the acute angles of quadrilateral $ABCD$ .
*Proposed by Zaza Meliqidze, Georgia*
|
60^\circ
| 0 |
716 |
For a given integer $k \geq 1$ , find all $k$ -tuples of positive integers $(n_1,n_2,...,n_k)$ with $\text{GCD}(n_1,n_2,...,n_k) = 1$ and $n_2|(n_1+1)^{n_1}-1$ , $n_3|(n_2+1)^{n_2}-1$ , ... , $n_1|(n_k+1)^{n_k}-1$ .
|
(1, 1, \ldots, 1)
| 57.03125 |
717 |
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$ , he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-word name}$ “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute $n$ .
|
16
| 27.34375 |
718 |
$a$ and $b$ are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of $2ab.$ *Author: Ray Li*
|
25
| 67.1875 |
719 |
rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$ . what is the maximum number of days rainbow can live in terms of $n$ ?
|
2n - 1
| 90.625 |
720 |
In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from the first switch until he has to turn on a bulb; as soon as he turns a bulb on, he leaves the room. For example the first person goes in, turns the first switch on and leaves. Then the second man goes in, seeing that the first switch on turns it off and then lights the second bulb. Then the third person goes in, finds the first switch off and turns it on and leaves the room. Then the fourth person enters and switches off the first and second bulbs and switches on the third. The process continues in this way. Finally we find out that first 10 bulbs are off and the 11 -th bulb is on. Then how many people were involved in the entire process?
|
1024
| 67.96875 |
721 |
Given that $x\ge0$ , $y\ge0$ , $x+2y\le6$ , and $2x+y\le6$ , compute the maximum possible value of $x+y$ .
|
4
| 85.9375 |
722 |
Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$ .
*Proposed by Lewis Chen*
|
103
| 10.9375 |
723 |
Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$ . The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all $n$ for which he can select the guests in such an order to serve all the guests.
|
n
| 84.375 |
724 |
Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ be integers selected from the set $\{1,2,\dots,100\}$ , uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when $M$ is divided by $64$ ?
|
\frac{63}{2}
| 0.78125 |
725 |
Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$ . Find the number of elements in $S$ that do not divide $\frac{L}{2016}$ .
*Proposed by Yannick Yao*
|
44
| 35.15625 |
726 |
The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$ . Find the sum of the possible values of $d_2$ .
|
42
| 1.5625 |
727 |
Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$ .
(1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$ .
(2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$ . Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$ .
(3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$ .
|
J > \sqrt{2}
| 30.46875 |
728 |
Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ *Vlad Matei*
|
p = 2
| 0 |
729 |
Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$ , then find the product $ab$ .
|
2022
| 56.25 |
730 |
Let $\alpha$ denote $\cos^{-1}(\tfrac 23)$ . The recursive sequence $a_0,a_1,a_2,\ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$ , $$ a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}. $$ Suppose that the series $$ \sum_{k=0}^\infty\dfrac{a_k}{2^k} $$ can be expressed uniquely as $\tfrac{p\sqrt q}r$ , where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$ .
|
23
| 0 |
731 |
Find all triplets $(a, b, c)$ of positive integers, such that $a+bc, b+ac, c+ab$ are primes and all divide $(a^2+1)(b^2+1)(c^2+1)$ .
|
(1, 1, 1)
| 100 |
732 |
The graph of ${(x^2 + y^2 - 1)}^3 = x^2 y^3$ is a heart-shaped curve, shown in the figure below.
[asy]
import graph;
unitsize(10);
real f(real x)
{
return sqrt(cbrt(x^4) - 4 x^2 + 4);
}
real g(real x)
{
return (cbrt(x^2) + f(x))/2;
}
real h(real x)
{
return (cbrt(x^2) - f(x)) / 2;
}
real xmax = 1.139028;
draw(graph(g, -xmax, xmax) -- reverse(graph(h, -xmax, xmax)) -- cycle);
xaxis(" $x$ ", -1.5, 1.5, above = true);
yaxis(" $y$ ", -1.5, 1.5, above = true);
[/asy]
For how many ordered pairs of integers $(x, y)$ is the point $(x, y)$ inside or on this curve?
|
7
| 88.28125 |
733 |
Say a positive integer $n$ is *radioactive* if one of its prime factors is strictly greater than $\sqrt{n}$ . For example, $2012 = 2^2 \cdot 503$ , $2013 = 3 \cdot 11 \cdot 61$ and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime factors less than $30$ ?
*Proposed by Evan Chen*
|
119
| 95.3125 |
734 |
For every positive integer $n$ , define $S_n$ to be the sum
\[
S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n .
\]
As $n$ approaches infinity, what value does $S_n$ approach?
|
1944
| 67.96875 |
735 |
Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$ , find the smallest possible value for $a$ .
|
-8
| 10.9375 |
736 |
Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if $ x_1 > a$ , then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$ , then the sequence is not monotonic.
|
a = 8^{1/8}
| 0 |
737 |
Let $(a_n)\subset (\frac{1}{2},1)$ . Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$ . Is this sequence convergent? If yes find the limit.
|
1
| 91.40625 |
738 |
$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$ . The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$ . Find the maximum value of $p_1+p_n$ .
|
m(n-1)
| 64.84375 |
739 |
Triangle $ABC$ has $AB=BC=10$ and $CA=16$ . The circle $\Omega$ is drawn with diameter $BC$ . $\Omega$ meets $AC$ at points $C$ and $D$ . Find the area of triangle $ABD$ .
|
24
| 25 |
740 |
Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$ . Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$ .
*B. Berceanu*
|
n
| 17.96875 |
741 |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning.
|
\frac{5}{11}
| 6.25 |
742 |
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$ ?
|
31
| 36.71875 |
743 |
Find $ \int_{ - 1}^1 {n\choose k}(1 + x)^{n - k}(1 - x)^k\ dx\ (k = 0,\ 1,\ 2,\ \cdots n)$ .
|
\frac{2^{n+1}}{n+1}
| 24.21875 |
744 |
Find all positive integer bases $b \ge 9$ so that the number
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers $n$ .
*Proposed by Yang Liu*
|
b = 10
| 0 |
745 |
$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)
|
n
| 49.21875 |
746 |
Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$ , how many chords can be drawn before returning to the starting point ?
|
24
| 78.125 |
747 |
In triangle $ABC$ , let $M$ be the midpoint of $BC$ , $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$ . Suppose that $OA = ON = 11$ and $OH = 7.$ Compute $BC^2$ .
|
288
| 14.84375 |
748 |
We say that a group of $k$ boys is $n-acceptable$ if removing any boy from the group one can always find, in the other $k-1$ group, a group of $n$ boys such that everyone knows each other. For each $n$ , find the biggest $k$ such that in any group of $k$ boys that is $n-acceptable$ we must always have a group of $n+1$ boys such that everyone knows each other.
|
k = 2n-1
| 0 |
749 |
We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $ .
|
25
| 54.6875 |
750 |
For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$ , we define
\[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\]
Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$ , $\Omega(P(k))$ is even. Show that $n$ is an even number.
|
n
| 96.09375 |
751 |
Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $ . If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences?
*Proposed by Isabella Grabski*
|
0
| 97.65625 |
752 |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C=\frac{5}{9}(F-32)$ . An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures $T$ with $32 \leq T \leq 1000$ does the original temperature equal the final temperature?
|
539
| 98.4375 |
753 |
Four right triangles, each with the sides $1$ and $2$ , are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
|
\frac{1}{5}
| 5.46875 |
754 |
Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$ . $D$ is a point on $AB$ such that $CD\perp AB$ . If the area of triangle $ABC$ is $84$ , what is the smallest possible value of $$ AC^2+\left(3\cdot CD\right)^2+BC^2? $$ *2016 CCA Math Bonanza Lightning #2.3*
|
1008
| 77.34375 |
755 |
Find the values of $a\in [0,\infty)$ for which there exist continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ , such that $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$ .
|
a = 0
| 0 |
756 |
Let $ABC$ be a right isosceles triangle with right angle at $A$ . Let $E$ and $F$ be points on A $B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$ . Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$ . Calculate the measure of angle $\angle FDC$ .
|
90^\circ
| 0.78125 |
757 |
Suppose $ABCD$ is a rectangle whose diagonals meet at $E$ . The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$ . Compute the number of possible integer values of $n$ .
|
47
| 2.34375 |
758 |
Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased.
*Proposed by Ankan Bhattacharya*
|
37
| 47.65625 |
759 |
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$ , $v_i$ is connected to $v_j$ if and only if $i$ divides $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.
|
10
| 40.625 |
760 |
Consider $4n$ points in the plane, with no three points collinear. Using these points as vertices, we form $\binom{4n}{3}$ triangles. Show that there exists a point $X$ of the plane that belongs to the interior of at least $2n^3$ of these triangles.
|
2n^3
| 87.5 |
761 |
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$ , and no two of which share a common divisor greater than $1$ .
|
8
| 98.4375 |
762 |
Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$ .
|
(2, 2)
| 100 |
763 |
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied.
One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$ . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.
The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$ . Compute $100a + b$ .
*Proposed by Lewis Chen*
|
102
| 23.4375 |
764 |
$p(n) $ is a product of all digits of n.Calculate: $ p(1001) + p(1002) + ... + p(2011) $
|
91125
| 35.9375 |
765 |
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$ ?
|
50
| 82.8125 |
766 |
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?
|
14
| 13.28125 |
767 |
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$ .
*Proposed by A. Golovanov, M. Ivanov, K. Kokhas*
|
4
| 0 |
768 |
Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?
|
2013
| 17.96875 |
769 |
Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the $2014$ th smallest perimeter of all such right triangles.
|
24168
| 85.9375 |
770 |
At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$ . Find $m+n$ .
|
39
| 3.90625 |
771 |
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
- $I$ contains no ridiculous numbers, and
- There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$ , and no integer square greater than 1 divides $c$ . What is $a + b + c + d$ ?
|
9
| 3.125 |
772 |
Let $ABCD$ be a square of side length $4$ . Points $E$ and $F$ are chosen on sides $BC$ and $DA$ , respectively, such that $EF = 5$ . Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$ .
*Proposed by Andrew Wu*
|
16
| 15.625 |
773 |
Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$ . Determine the least value the sum
\[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers.
*Fedor Petrov*
|
n
| 99.21875 |
774 |
The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$ .
|
722
| 89.0625 |
775 |
In the triangle $ABC$ , $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 = |DB| \cdot |DC|$ . Determine all possible values of the perimeter of the triangle $ABC$ .
|
\sqrt{2} + 1
| 6.25 |
776 |
For real constant numbers $ a,\ b,\ c,\ d,$ consider the function $ f(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ such that $ f( \minus{} 1) \equal{} 0,\ f(1) \equal{} 0,\ f(x)\geq 1 \minus{} |x|$ for $ |x|\leq 1.$
Find $ f(x)$ for which $ \int_{ \minus{} 1}^1 \{f'(x) \minus{} x\}^2\ dx$ is minimized.
|
f(x) = -x^2 + 1
| 7.03125 |
777 |
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
|
3
| 60.15625 |
778 |
In a massive school which has $m$ students, and each student took at least one subject. Let $p$ be an odd prime. Given that:
(i) each student took at most $p+1$ subjects.
(ii) each subject is taken by at most $p$ students.
(iii) any pair of students has at least $1$ subject in common.
Find the maximum possible value of $m$ .
|
p^2
| 5.46875 |
779 |
In a rectangular $57\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?
[i]Proposed by James Lin
|
457
| 0 |
780 |
Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$
|
1
| 78.90625 |
781 |
Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$
|
1
| 68.75 |
782 |
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ .
Find the smallest k that:
$S(F) \leq k.P(F)^2$
|
\frac{1}{16}
| 94.53125 |
783 |
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\triangle '$ . Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$ , and denote its area by $\triangle ''$ . Given that $\triangle ' = 30$ and $\triangle '' = 20$ , find $\triangle$ .
|
45
| 9.375 |
784 |
Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$ -letter strings containing only the letters $A$ and $B$ . Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right.
What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$ ?
|
2^{k-1}
| 8.59375 |
785 |
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$ . Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$ .
|
1
| 27.34375 |
786 |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong*
|
37
| 8.59375 |
787 |
What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$ ?
|
1729
| 96.09375 |
788 |
Find the number of $12$ -digit "words" that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if neighboring digits must differ by exactly $2$ .
|
882
| 100 |
789 |
Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$ , the line $x=a$ and the $x$ -axis around the $x$ -axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$ , the line $y=\frac{a}{a+k}$ and the $y$ -axis around the $y$ -axis.
Find the ratio $\frac{V_2}{V_1}.$
|
k
| 1.5625 |
790 |
Find all triples $(a, b, c)$ of positive integers for which $$ \begin{cases} a + bc=2010 b + ca = 250\end{cases} $$
|
(3, 223, 9)
| 80.46875 |
791 |
Let $a_1$ , $a_2$ , $\ldots\,$ , $a_{2019}$ be a sequence of real numbers. For every five indices $i$ , $j$ , $k$ , $\ell$ , and $m$ from 1 through 2019, at least two of the numbers $a_i$ , $a_j$ , $a_k$ , $a_\ell$ , and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence?
|
8
| 35.15625 |
792 |
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.
|
12
| 53.125 |
793 |
In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$ , but if you spend $35$ minutes on the essay you somehow do not earn any points.
It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?
*Proposed by Evan Chen*
|
75
| 21.09375 |
794 |
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber b^4+c^2a^2=16b \nonumber c^4+a^2b^2=16c \nonumber \end{eqnarray}
|
(2, 2, 2)
| 67.96875 |
795 |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer?
|
12
| 20.3125 |
796 |
Define $\triangle ABC$ with incenter $I$ and $AB=5$ , $BC=12$ , $CA=13$ . A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$ 's area divided by $\pi.$
|
\frac{16}{3}
| 0 |
797 |
Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square.
|
179
| 97.65625 |
798 |
Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real
numbers x, for which this value is achieved.
|
\sin(1) + 1
| 49.21875 |
799 |
Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$ .
|
4037
| 0 |
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