Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
600 |
Determine the number of $ 8$ -tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$ , for each $ k \equal{} 1,2,3,4$ , and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$ .
|
1540
| 100 |
601 |
There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$ . The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$ . The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$ , pick up that card, slide the cards in cells $ h \plus{} 1$ , $ h \plus{} 2$ , ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$ . Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]
|
2^n - 1
| 90.625 |
602 |
Restore the acute triangle $ABC$ given the vertex $A$ , the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$ ).
|
ABC
| 0 |
603 |
What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?
|
61
| 35.15625 |
604 |
Five lighthouses are located, in order, at points $A, B, C, D$ , and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$ , the lights from $B, C, D$ , and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written $m +\sqrt{n}$ , where $m$ and $n$ are positive integers. Find $m + n$ .
|
95
| 0 |
605 |
Find the minimum value of $$ \big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big| $$ for real numbers $x$ not multiple of $\frac{\pi}{2}$ .
|
2\sqrt{2} - 1
| 36.71875 |
606 |
Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$ ). Find the least $m$ such that $m-b_m = 1990$ .
|
2^{1990} - 1
| 0 |
607 |
Suppose that $x$ , $y$ , and $z$ are complex numbers of equal magnitude that satisfy
\[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\]
and
\[xyz=\sqrt{3} + i\sqrt{5}.\]
If $x=x_1+ix_2, y=y_1+iy_2,$ and $z=z_1+iz_2$ for real $x_1,x_2,y_1,y_2,z_1$ and $z_2$ then
\[(x_1x_2+y_1y_2+z_1z_2)^2\]
can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$ . Compute $100a+b.$
|
1516
| 0.78125 |
608 |
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?
|
25
| 43.75 |
609 |
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?
|
432
| 47.65625 |
610 |
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$ , respectively, are drawn with center $O$ . Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$ , respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$ , and denote by $P$ the reflection of $B$ across $\ell$ . Compute the expected value of $OP^2$ .
*Proposed by Lewis Chen*
|
10004
| 14.84375 |
611 |
The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$ . Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$ . What’s the least number of zeros $f$ can have in the interval $[0;2014]$ ? Does this change, if $f$ is also continuous?
|
107
| 1.5625 |
612 |
Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.
|
9
| 86.71875 |
613 |
The numbers $1, 2, \dots ,N$ are arranged in a circle where $N \geq 2$ . If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of $N$ ? $
\textbf{a)}\ 18
\qquad\textbf{b)}\ 19
\qquad\textbf{c)}\ 28
\qquad\textbf{d)}\ 29
\qquad\textbf{e)}\ \text{None of above}
$
|
29
| 7.8125 |
614 |
There are $20$ geese numbered $1-20$ standing in a line. The even numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd numbered geese are standing behind them in the order, $1,3,5,\dots ,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
*Author: Ray Li*
|
55
| 20.3125 |
615 |
Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$ . Find the smallest positive value of $k$ .
|
\frac{1}{667}
| 50 |
616 |
Consider a $20$ -sided convex polygon $K$ , with vertices $A_1, A_2,...,A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $(A_1A_2, A_4A_5, A_{11}A_{12})$ is an admissible triple while $(A_1A_2, A_4A_5, A_{19}A_{20})$ is not.
|
520
| 0 |
617 |
In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\angle BMC = 90^\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ .
|
30^\circ
| 6.25 |
618 |
In triangle $ABC$ , $AB = 28$ , $AC = 36$ , and $BC = 32$ . Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$ , and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$ . Find the length of segment $AE$ .
*Ray Li*
|
18
| 2.34375 |
619 |
An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this operation a *step*. If $C$ is the original coloring, let $S(C)$ be the least number of steps required to make all the unit squares black. Find with proof the greatest possible value of $S(C)$ .
|
32
| 72.65625 |
620 |
Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$ , in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n \minus{} 1$ that can be divided by $ d$ .
|
u_n = 2n-1
| 0 |
621 |
Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?
|
280
| 92.1875 |
622 |
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$ , determine $ f(9) .$
|
9
| 89.0625 |
623 |
Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$ . Denote
\[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \]
Initially, some pieces are put on the each point in $Q$ , with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly:
(1) Remove $p$ pieces on $(x, y, z)$ and place a piece on $(x-1, y, z)$ ;
(2) Remove $q$ pieces on $(x, y, z)$ and place a piece on $(x, y-1, z)$ ;
(3) Remove $r$ pieces on $(x, y, z)$ and place a piece on $(x, y, z-1)$ .
Find the smallest positive integer $M$ such that one can always perform a sequence of operations, making a piece placed on $(0,0,0)$ , no matter how the pieces are distributed initially.
|
p^a q^b r^c
| 1.5625 |
624 |
Let $a$ , $b$ , and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$ . Over all possible values of $a$ , $b$ , and $c$ , determine the maximum possible value of $a + 2b + 3c$ .
*Proposed by Andrew Wen*
|
25
| 84.375 |
625 |
$ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ( $ n $ is natural number)
|
n
| 38.28125 |
626 |
We call a path Valid if
i. It only comprises of the following kind of steps:
A. $(x, y) \rightarrow (x + 1, y + 1)$ B. $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis.
Let $M(n)$ = set of all valid paths from $(0,0) $ , to $(2n,0)$ , where $n$ is a natural number.
Consider a Valid path $T \in M(n)$ .
Denote $\phi(T) = \prod_{i=1}^{2n} \mu_i$ ,
where $\mu_i$ =
a) $1$ , if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$ b) $y$ , if the $i^{th} $ step is $(x, y) \rightarrow (x + 1, y - 1)$
Now Let $f(n) =\sum _{T \in M(n)} \phi(T)$ . Evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$
|
0
| 21.09375 |
627 |
Let n be a non-negative integer. Define the *decimal digit product* \(D(n)\) inductively as follows:
- If \(n\) has a single decimal digit, then let \(D(n) = n\).
- Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\).
Let \(P_k(1)\) be the probability that \(D(i) = 1\) where \(i\) is chosen uniformly randomly from the set of integers between 1 and \(k\) (inclusive) whose decimal digit products are not 0.
Compute \(\displaystyle\lim_{k\to\infty} P_k(1)\).
*proposed by the ICMC Problem Committee*
|
0
| 96.09375 |
628 |
Let $x$ , $y$ , and $z$ be real numbers such that $$ 12x - 9y^2 = 7 $$ $$ 6y - 9z^2 = -2 $$ $$ 12z - 9x^2 = 4 $$ Find $6x^2 + 9y^2 + 12z^2$ .
|
9
| 6.25 |
629 |
*Version 1*. Let $n$ be a positive integer, and set $N=2^{n}$ . Determine the smallest real number $a_{n}$ such that, for all real $x$ ,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
*Version 2*. For every positive integer $N$ , determine the smallest real number $b_{N}$ such that, for all real $x$ ,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
|
2^{n-1}
| 0.78125 |
630 |
Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.
|
13
| 81.25 |
631 |
Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?
|
105
| 33.59375 |
632 |
Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$ . For each $j$ , let $w_j$ be one of $z_j$ or $i z_j$ . Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$ .
|
784
| 3.125 |
633 |
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
*Dumitru Bușneag*
|
P(x) = ax + b
| 52.34375 |
634 |
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$ .
(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$ .
(ii) Determine the smallest possible $a$ .
|
3
| 10.9375 |
635 |
The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$ : \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$ .
|
660
| 85.9375 |
636 |
Find $ [\sqrt{19992000}]$ where $ [x]$ is the greatest integer less than or equal to $ x$ .
|
4471
| 100 |
637 |
The inhabitants of a planet speak a language only using the letters $A$ and $O$ . To avoid mistakes, any two words of equal length differ at least on three positions. Show that there are not more than $\frac{2^n}{n+1}$ words with $n$ letters.
|
\frac{2^n}{n+1}
| 98.4375 |
638 |
Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously:
\begin{align*}
x^3+y=z^2
y^3+z=x^2
z^3+x =y^2
\end{align*}
|
(0, 0, 0)
| 43.75 |
639 |
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$ .
|
a = 0
| 65.625 |
640 |
Let $P(x)$ be the polynomial of degree at most $6$ which satisfies $P(k)=k!$ for $k=0,1,2,3,4,5,6$ . Compute the value of $P(7)$ .
|
3186
| 73.4375 |
641 |
A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties: $a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer.
Determine all possible values of $c$ and the associated values of $a$ and $b$ .
|
5
| 20.3125 |
642 |
For $n \in \mathbb{N}$ , let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$ . Consider the set $A=\{n \in \mathbb{N}: P(n)$ is non-zero, square free and $S(n)$ is a proper divisor of $P(n)\}$ .
Find the maximum possible number of digits of the numbers in $A$ .
|
92
| 0 |
643 |
Denote by $S(n)$ the sum of the digits of the positive integer $n$ . Find all the solutions of the equation $n(S(n)-1)=2010.$
|
402
| 97.65625 |
644 |
Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$ , and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$ . Let $M$ the intersection point of the medians in $ABC$ . Let $TM$ intersect $(ATC)$ at $K$ . Find $TM/MK$ .
|
\frac{TM}{MK} = 2
| 0 |
645 |
Let $A$ be the set $\{1,2,\ldots,n\}$ , $n\geq 2$ . Find the least number $n$ for which there exist permutations $\alpha$ , $\beta$ , $\gamma$ , $\delta$ of the set $A$ with the property: \[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \]
*Marcel Chirita*
|
n = 28
| 0 |
646 |
Given triangle $ABC$ , let $D$ , $E$ , $F$ be the midpoints of $BC$ , $AC$ , $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$ , how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?
|
2
| 3.90625 |
647 |
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$ , $p_ {32} = 6$ , $p_ {203} = 6$ . Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$ . Find the largest prime that divides $S $ .
|
103
| 94.53125 |
648 |
Let $ABC$ be a triangle with $\angle C=90^\circ$ , and $A_0$ , $B_0$ , $C_0$ be the mid-points of sides $BC$ , $CA$ , $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$ . Find the angle $C_0C_1C_2$ .
|
30^\circ
| 25.78125 |
649 |
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B$ . The probability that team $A$ finishes with more points than team $B$ is $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
|
831
| 60.15625 |
650 |
A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?
|
\frac{243}{256}
| 0 |
651 |
As shown, $U$ and $C$ are points on the sides of triangle MNH such that $MU = s$ , $UN = 6$ , $NC = 20$ , $CH = s$ , $HM = 25$ . If triangle $UNC$ and quadrilateral $MUCH$ have equal areas, what is $s$ ?
|
s = 4
| 0 |
652 |
Some language has only three letters - $A, B$ and $C$ . A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all $A$ . What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?
|
2^{40}
| 7.03125 |
653 |
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
|
10
| 4.6875 |
654 |
Square $S_1$ is inscribed inside circle $C_1$ , which is inscribed inside square $S_2$ , which is inscribed inside circle $C_2$ , which is inscribed inside square $S_3$ , which is inscribed inside circle $C_3$ , which is inscribed inside square $S_4$ .
[center]<see attached>[/center]
Let $a$ be the side length of $S_4$ , and let $b$ be the side length of $S_1$ . What is $\tfrac{a}{b}$ ?
|
2\sqrt{2}
| 42.96875 |
655 |
A four-digit number $n$ is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by $5$ , the result is $1434$ . For example, $1984$ is *literally 1434* because $1$ mod $5$ is $1$ , $9$ mod $5$ is $4$ , $8$ mod $5$ is $3$ , and $4$ mod $5$ is $4$ . Find the sum of all four-digit positive integers that are *literally 1434*.
*Proposed by Evin Liang*
<details><summary>Solution</summary>*Solution.* $\boxed{67384}$ The possible numbers are $\overline{abcd}$ where $a$ is $1$ or $6$ , $b$ is $4$ or $9$ , $c$ is $3$ or $8$ , and $d$ is $4$ or $9$ . There are $16$ such numbers and the average is $\dfrac{8423}{2}$ , so the total in this case is $\boxed{67384}$ .</details>
|
67384
| 99.21875 |
656 |
Given an integer $n\ge\ 3$ , find the least positive integer $k$ , such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$ .
|
k = 3
| 0 |
657 |
A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks.
|
7
| 97.65625 |
658 |
A *Beaver-number* is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of *Beaver-numbers* differing by exactly $1$ a *Beaver-pair*. The smaller number in a *Beaver-pair* is called an *MIT Beaver*, while the larger number is called a *CIT Beaver*. Find the positive difference between the largest and smallest *CIT Beavers* (over all *Beaver-pairs*).
|
79200
| 83.59375 |
659 |
Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$ , where all $a_k$ equal $0$ or $1$ . After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$ , where all $b_k$ also equal $0$ or $1$ . Then Masha either does nothing or says “Mutabor” and replaces both sequences: her own sequence by $(a_n, a_{n-1}, \ldots , a_1)$ , and Tanya’s sequence by $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$ . Masha’s sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya’s sequence and tell the sequence covered by the napkin. For what $n$ Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word “Mutabor” has been pronounced.
|
n
| 57.8125 |
660 |
Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards?
|
1680
| 91.40625 |
661 |
Let $T_k = \frac{k(k+1)}{2}$ be the $k$ -th triangular number. The in finite series $$ \sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)} $$ has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .
|
451
| 0 |
662 |
Find the least positive integer $n$ ( $n\geq 3$ ), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
|
n = 7
| 0 |
663 |
Let $n$ be positive integer and $S$ = { $0,1,…,n$ }, Define set of point in the plane. $$ A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$ , We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area
|
n+1
| 53.90625 |
664 |
**H**orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$ . Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$ , $BC$ , and $DA$ . If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$ , find the area of $ABCD$ .
|
\frac{225}{2}
| 0.78125 |
665 |
Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
|
298
| 5.46875 |
666 |
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$ , where $m$ and $n$ are positive integers. Find $100m+n$ .
*Proposed by Yannick Yao*
|
407
| 0 |
667 |
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $ . Here $[x]$ denotes the greatest integer less than or equal to $x.$
|
2997
| 85.15625 |
668 |
Compute the number of ways to erase 24 letters from the string ``OMOMO $\cdots$ OMO'' (with length 27), such that the three remaining letters are O, M and O in that order. Note that the order in which they are erased does not matter.
[i]Proposed by Yannick Yao
|
455
| 0 |
669 |
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get?
|
127
| 56.25 |
670 |
$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$ . $B$ must guess the value of $n$ by choosing several subsets of $S$ , then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each.
What is the least value of $k_1 + k_2 + k_3$ such that $B$ has a strategy to correctly guess the value of $n$ no matter what $A$ chooses?
|
28
| 0 |
671 |
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$ , $BC = 10$ , $CD = 11$ , and $DA = 14$ . The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$ , where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$ .
|
446
| 0 |
672 |
Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares.
|
280
| 82.8125 |
673 |
You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$ . Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$ ?
For example, if $a = [-1, 2, 2]$ , then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$ .
*(Proposed by Anton Trygub)*
|
2^{n-1}
| 35.15625 |
674 |
Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)*
|
8 \cdot (8!)^3
| 0 |
675 |
Find the number of pairs $(a,b)$ of natural nunbers such that $b$ is a 3-digit number, $a+1$ divides $b-1$ and $b$ divides $a^{2} + a + 2$ .
|
16
| 97.65625 |
676 |
Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$ , where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$ . The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$ . The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$ . Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where $A$ and $B$ are nonempty decreasing and increasing sets ( $\mid \cdot \mid$ denotes the number of elements of the set).
|
\frac{1}{20^{2012}}
| 16.40625 |
677 |
The Lucas numbers $L_n$ are defined recursively as follows: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}$ for $n\geq2$ . Let $r=0.21347\dots$ , whose digits form the pattern of the Lucas numbers. When the numbers have multiple digits, they will "overlap," so $r=0.2134830\dots$ , **not** $0.213471118\dots$ . Express $r$ as a rational number $\frac{p}{q}$ , where $p$ and $q$ are relatively prime.
|
\frac{19}{89}
| 42.1875 |
678 |
Find the largest possible number of rooks that can be placed on a $3n \times 3n$ chessboard so that each rook is attacked by at most one rook.
|
4n
| 0.78125 |
679 |
Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$ , where $F_n$ is the Fibonacci sequence
( $F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$ )
Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$
|
\frac{1}{2}
| 86.71875 |
680 |
For real $\theta_i$ , $i = 1, 2, \dots, 2011$ , where $\theta_1 = \theta_{2012}$ , find the maximum value of the expression
\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}.
\]
*Proposed by Lewis Chen*
|
1005
| 3.90625 |
681 |
Find the smallest value of $n$ for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds $3^{2007}$ .
|
2000
| 5.46875 |
682 |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$
|
12
| 99.21875 |
683 |
In a group of $n > 20$ people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of $n$ and $k,$ this group has a peculiar property: If any $20$ people are removed from the group, the number of pairs of people that know each other is
at most $\frac{n-k}{n}$ times that of the original group of people.
(a) If $k = 41,$ for what positive integers $n$ could such a group exist?
(b) If $k = 39,$ for what positive integers $n$ could such a group exist?
|
n \ge 381
| 0 |
684 |
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.
|
1
| 92.96875 |
685 |
Define $ P(\tau ) = (\tau + 1)^3$ . If $x + y = 0$ , what is the minimum possible value of $P(x) + P(y)$ ?
|
2
| 99.21875 |
686 |
The eleven members of a cricket team are numbered $1,2,...,11$ . In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?
|
2
| 87.5 |
687 |
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$ .
|
987
| 78.125 |
688 |
A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$ . Let $f$ be a function such that:
$f(n)=0$ , if n is perfect
$f(n)=0$ , if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$
|
0
| 54.6875 |
689 |
For which integers $n>1$ does there exist a rectangle that can be subdivided into $n$ pairwise noncongruent rectangles similar to the original rectangle?
|
n \ge 3
| 0 |
690 |
Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations $x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$ .
Find the maximum possible value of $x + 1000y$ .
|
1991
| 55.46875 |
691 |
Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$ . The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$ . Determine $\angle AMN$ .
|
90^\circ
| 69.53125 |
692 |
Three circles, each of radius 3, are drawn with centers at $(14,92)$ , $(17,76)$ , and $(19,84)$ . A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
|
24
| 94.53125 |
693 |
Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$ . Determine the maximum possible value of degree of $T(x)$
|
p-2
| 30.46875 |
694 |
Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\sqrt n$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Determine $m+n$ .
[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label(" $A$ ",A,S);
label(" $B$ ",B,S);
label(" $C$ ",C,N);
label(" $D$ ",D,dir(B--D));
label(" $E$ ",E,dir(C--E));
[/asy]
|
10
| 3.125 |
695 |
With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$ . How many numbers can be written?
|
57
| 77.34375 |
696 |
An integer $n>0$ is written in decimal system as $\overline{a_ma_{m-1}\ldots a_1}$ . Find all $n$ such that
\[n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)\]
|
18
| 2.34375 |
697 |
In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$ . The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
label(" $A$ ", (0,23), NW);
label(" $B$ ", (23, 23), NE);
label(" $C$ ", (23,0), SE);
label(" $D$ ", (0,0), SW);
draw((0,6)--(23,6));
draw((0,19)--(23,19));
draw((5,0)--(5,23));
draw((12,0)--(12,23));
label("13", (17/2, 21));
label("111",(35/2,25/2));
label("37",(17/2,3));
label("123",(2.5,12.5));[/asy]
*Proposed by Lewis Chen*
|
180
| 0 |
698 |
Determine the range of $w(w + x)(w + y)(w + z)$ , where $x, y, z$ , and $w$ are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]
|
0
| 67.96875 |
699 |
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what is $m+n$ ?
*2016 CCA Math Bonanza Lightning #4.4*
|
1002
| 29.6875 |
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