Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
400 |
Call a $ 3$ -digit number *geometric* if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
|
840
| 97.65625 |
401 |
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine $ n$ .
|
2002
| 53.90625 |
402 |
In triangle $ABC$ , find the smallest possible value of $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$
|
\frac{8\sqrt{3}}{9}
| 78.90625 |
403 |
Alice is counting up by fives, starting with the number $3$ . Meanwhile, Bob is counting down by fours, starting with the number $2021$ . How many numbers between $3$ and $2021$ , inclusive, are counted by both Alice and Bob?
|
101
| 70.3125 |
404 |
Given that nonzero reals $a,b,c,d$ satisfy $a^b=c^d$ and $\frac{a}{2c}=\frac{b}{d}=2$ , compute $\frac{1}{c}$ .
*2021 CCA Math Bonanza Lightning Round #2.2*
|
16
| 84.375 |
405 |
What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$ ?
*2016 CCA Math Bonanza Lightning #1.1*
|
7
| 96.09375 |
406 |
Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$ . Which one of the numbers $x^y$ , $y^x$ is bigger ?
|
x^y = y^x
| 86.71875 |
407 |
$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$ . Find $\lim_{n \to \infty} \sqrt[n]{a_n}$ .
|
k
| 95.3125 |
408 |
If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$ , where $m$ and $n$ are positive integers, what is the value of $m+n$ ?
|
2011
| 74.21875 |
409 |
Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]
|
(2014, 2014, 2014)
| 63.28125 |
410 |
$x$ is a base- $10$ number such that when the digits of $x$ are interpreted as a base- $20$ number, the resulting number is twice the value as when they are interpreted as a base- $13$ number. Find the sum of all possible values of $x$ .
|
198
| 60.15625 |
411 |
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$ .
*Ray Li*
|
40320
| 46.09375 |
412 |
Let $r$ be a positive integer. Show that if a graph $G$ has no cycles of length at most $2r$ , then it has at most $|V|^{2016}$ cycles of length exactly $2016r$ , where $|V|$ denotes the number of vertices in the graph $G$ .
|
|V|^{2016}
| 97.65625 |
413 |
In a year that has $365$ days, what is the maximum number of "Tuesday the $13$ th" there can be?
Note: The months of April, June, September and November have $30$ days each, February has $28$ and all others have $31$ days.
|
3
| 100 |
414 |
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$ , there exists an unique positive integer $k$ , satisfying $f^k(n)\leq n+k+1$ .
|
f(n) = n + 2
| 2.34375 |
415 |
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
|
3
| 63.28125 |
416 |
Determine all positive integers $n$ such that $$ n\cdot 2^{n-1}+1 $$ is a perfect square.
|
5
| 99.21875 |
417 |
Let $f(x) = 2^x + 3^x$ . For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$ ?
|
11
| 41.40625 |
418 |
The altitudes of the triangle ${ABC}$ meet in the point ${H}$ . You know that ${AB = CH}$ . Determine the value of the angle $\widehat{BCA}$ .
|
45^\circ
| 83.59375 |
419 |
A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied:
(i) All the squares are congruent.
(ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.
How many positive integers $n$ are there with $2018\leq n \leq 3018$ , such that there exists a collection of $n$ squares that is tri-connected?
|
501
| 93.75 |
420 |
Let $T=TNFTPP$ . Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$ , $CD = 5$ , and $BD = 6$ . Determine the area of quadrilateral $ACDB$ .
[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]
[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=10$</details>.
|
44
| 35.9375 |
421 |
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.
|
17
| 50 |
422 |
Let $a_1$ , $a_2$ , $a_3$ , $a_4$ , $a_5$ be real numbers satisfying
\begin{align*}
2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4
2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3
2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2
2a_4+a_5 &= 6 + a_1
\end{align*}
Compute $a_1+a_2+a_3+a_4+a_5$ .
*Proposed by Evan Chen*
|
2
| 3.90625 |
423 |
For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$ . Find the smallest $n$ for which
\[ P(P(P(n))) > 10^{12} \]
|
6
| 93.75 |
424 |
Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$ . Here we make the convention $\sqrt[1]{abc}=abc$ .
|
3
| 41.40625 |
425 |
Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$ ?
|
5
| 97.65625 |
426 |
Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that
- each member in $S$ is a positive integer not exceeding $2002$ ,
- if $a,b\in S$ (not necessarily different), then $ab\not\in S$ .
|
1958
| 4.6875 |
427 |
Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$ .
|
1
| 49.21875 |
428 |
Let $\triangle ABC$ be a triangle with $AB < AC$ . Let the angle bisector of $\angle BAC$ meet $BC$ at $D$ , and let $M$ be the midpoint of $BC$ . Let $P$ be the foot of the perpendicular from $B$ to $AD$ . $Q$ the intersection of $BP$ and $AM$ . Show that : $(DQ) // (AB) $ .
|
DQ \parallel AB
| 73.4375 |
429 |
In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$ ; he notices that the expected value of the score of the resulting grid is equal to $k$ , too! Given that $k > 0.9999$ , find the minimum possible value of $n$ .
*Proposed by Andrew Wu*
|
51
| 21.875 |
430 |
Determine all positive integers $n$ for which the equation
\[ x^n + (2+x)^n + (2-x)^n = 0 \]
has an integer as a solution.
|
n=1
| 50.78125 |
431 |
The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is
\[
\frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots.
\]
Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$ . Find $m+n$ .
|
61
| 50 |
432 |
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
|
1 \text{ and } 27
| 0.78125 |
433 |
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ( $0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1,5$ , but there might be two contestants whose ordered scores are $7,1,2$ and $7,2,1$ ). Find the maximum number of contestants.
|
64
| 18.75 |
434 |
Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$ ?
|
(0, A^2)
| 0.78125 |
435 |
We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.
|
4n^2 - 4n + 2
| 13.28125 |
436 |
There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$ . Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of $k$ are there?
*Ray Li*
|
501
| 46.875 |
437 |
More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.
|
n = 12
| 0 |
438 |
The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
|
25
| 63.28125 |
439 |
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?
|
\frac{1}{4}
| 15.625 |
440 |
A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$ .
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$ .
|
7
| 21.09375 |
441 |
Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$ ?
|
333
| 85.9375 |
442 |
In triangle $ABC$ , angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$ , and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$
|
291
| 4.6875 |
443 |
Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$ , such that each triangle in $T$ has a side that is not a side of any other triangle in $T$ ?
|
n
| 18.75 |
444 |
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?
|
9
| 11.71875 |
445 |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions?
|
36
| 94.53125 |
446 |
Find all primes $p$ such that there exist positive integers $q$ and $r$ such that $p \nmid q$ , $3 \nmid q$ , $p^3 = r^3 - q^2$ .
|
p = 7
| 0 |
447 |
The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$ , where $m$ , $n$ , and $p$ are integers. Find $m+n+p$ .
|
348
| 0.78125 |
448 |
Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$
|
-(x - 1)^2 e^x + C
| 14.84375 |
449 |
Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$ . Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$ . If the lines $AD$ and $OE$ meet at $F$ , find $|AF|/|FD|$ .
|
\frac{1}{2}
| 0 |
450 |
Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.
|
\frac{3}{4}
| 81.25 |
451 |
We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?
*Proposed by Nathan Xiong*
|
16
| 96.09375 |
452 |
Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ .
|
4
| 100 |
453 |
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
|
990
| 94.53125 |
454 |
Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$ . If $\angle ACB = 40^o$ , $AD = 7$ , $CD = 24$ , $CE = 15$ , and $BE = 20$ , what is the measure of $\angle ABC $ in,degrees?
|
70^\circ
| 81.25 |
455 |
On the coordinate plane, let $C$ be a circle centered $P(0,\ 1)$ with radius 1. let $a$ be a real number $a$ satisfying $0<a<1$ . Denote by $Q,\ R$ intersection points of the line $y=a(x+1) $ and $C$ .
(1) Find the area $S(a)$ of $\triangle{PQR}$ .
(2) When $a$ moves in the range of $0<a<1$ , find the value of $a$ for which $S(a)$ is maximized.
*2011 Tokyo University entrance exam/Science, Problem 1*
|
a = 2 - \sqrt{3}
| 0 |
456 |
Let $n$ be a natural number divisible by $3$ . We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ( $m > 1$ ), the number of black squares is not more than white squares. Find the maximum number of black squares.
|
\frac{4n^2}{9}
| 33.59375 |
457 |
Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ *Adrian Zanoschi*
|
(0, 6)
| 38.28125 |
458 |
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$ . If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are $n$ iterations of $L$ . For example,
\[
a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).
\]
As $n$ approaches infinity, what value does $n a_n$ approach?
|
\frac{34}{19}
| 0.78125 |
459 |
$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ( $ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$ . For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$ . Find the smallest positive integer $ t$ , such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$ .
|
t = 2
| 0 |
460 |
We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \[k = \sum_{i=2}^n a_i F_i,\] where $a_i\in\{0,1\}$ for all $i$ , $a_n=1$ , and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ( $F_0=0$ , $F_1=1$ , and $F_i=F_{i-1}+F_{i-2}$ for all $i\ge2$ ). This representation is said to be $\textit{minimal}$ if it has fewer 1’s than any other Fibonacci representation of $k$ . Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.
|
1596
| 60.9375 |
461 |
Find all positive integers $n>1$ such that
\[\tau(n)+\phi(n)=n+1\]
Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$ , and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$ .
*Raja Oktovin, Pekanbaru*
|
n = 4
| 0 |
462 |
A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.
|
\frac{55}{64}
| 23.4375 |
463 |
Given two natural numbers $ w$ and $ n,$ the tower of $ n$ $ w's$ is the natural number $ T_n(w)$ defined by
\[ T_n(w) = w^{w^{\cdots^{w}}},\]
with $ n$ $ w's$ on the right side. More precisely, $ T_1(w) = w$ and $ T_{n+1}(w) = w^{T_n(w)}.$ For example, $ T_3(2) = 2^{2^2} = 16,$ $ T_4(2) = 2^{16} = 65536,$ and $ T_2(3) = 3^3 = 27.$ Find the smallest tower of $ 3's$ that exceeds the tower of $ 1989$ $ 2's.$ In other words, find the smallest value of $ n$ such that $ T_n(3) > T_{1989}(2).$ Justify your answer.
|
1988
| 0 |
464 |
For two sets $A, B$ , define the operation $$ A \otimes B = \{x \mid x=ab+a+b, a \in A, b \in B\}. $$ Set $A=\{0, 2, 4, \cdots, 18\}$ and $B=\{98, 99, 100\}$ . Compute the sum of all the elements in $A \otimes B$ .
*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 7)*
|
29970
| 96.09375 |
465 |
On semicircle, with diameter $|AB|=d$ , are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$
|
8
| 0.78125 |
466 |
A math contest consists of $9$ objective type questions and $6$ fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly $14$ out of $15$ questions. Let $O_1, O_2, \dots , O_9$ be the nine objective questions and $F_1, F_2, \dots , F_6$ be the six fill inthe blanks questions. Let $a_{ij}$ be the number of students who attemoted both questions $O_i$ and $F_j$ . If the sum of all the $a_{ij}$ for $i=1, 2,\dots , 9$ and $j=1, 2,\dots , 6$ is $972$ , then find the number of students who took the test in the school.
|
21
| 0 |
467 |
Find the smallest constant $ C$ such that for all real $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds.
|
\frac{4}{3}
| 3.125 |
468 |
Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.
|
\lambda = 2
| 0 |
469 |
In triangle $ABC$ , $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$ . Find $CE$ given that $AC=35$ , $BC=7$ , and $BE=5$ .
|
10
| 6.25 |
470 |
(**4**) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$ . Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$ ).
|
\frac{3}{8}
| 17.1875 |
471 |
Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .
*Proposed by Andy Xu*
|
42
| 66.40625 |
472 |
The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$ .
|
463
| 84.375 |
473 |
Mariana plays with an $8\times 8$ board with all its squares blank. She says that two houses are *neighbors* if they have a common side or vertex, that is, two houses can be neighbors vertically, horizontally or diagonally. The game consists of filling the $64$ squares on the board, one after the other, each with a number according to the following rule: she always chooses a house blank and fill it with an integer equal to the number of neighboring houses that are still in White. Once this is done, the house is no longer considered blank.
Show that the value of the sum of all $64$ numbers written on the board at the end of the game does not depend in the order of filling. Also, calculate the value of this sum.
Note: A house is not neighbor to itself.
<details><summary>original wording</summary>Mariana brinca com um tabuleiro 8 x 8 com todas as suas casas em branco. Ela diz que duas
casas s˜ao vizinhas se elas possu´ırem um lado ou um v´ertice em comum, ou seja, duas casas podem ser vizinhas
verticalmente, horizontalmente ou diagonalmente. A brincadeira consiste em preencher as 64 casas do tabuleiro,
uma ap´os a outra, cada uma com um n´umero de acordo com a seguinte regra: ela escolhe sempre uma casa
em branco e a preenche com o n´umero inteiro igual `a quantidade de casas vizinhas desta que ainda estejam em
branco. Feito isso, a casa n˜ao ´e mais considerada em branco.
Demonstre que o valor da soma de todos os 64 n´umeros escritos no tabuleiro ao final da brincadeira n˜ao depende
da ordem do preenchimento. Al´em disso, calcule o valor dessa soma.
Observa¸c˜ao: Uma casa n˜ao ´e vizinha a si mesma</details>
|
210
| 4.6875 |
474 |
Find the smallest number $k$ , such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively.
*Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov*
|
\frac{4}{3}
| 0 |
475 |
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$
|
121
| 0 |
476 |
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$ , $a^2=m_2^2+n_2^2$ , and $m_1-n_1=m_2-n_2.$
|
261
| 0 |
477 |
Find all sets of integers $n\geq 2$ and positive integers $(a_1, a_2, \dots, a_n)$ that satisfy all of the following conditions:
- $a_1 < a_2 < \cdots < a_n$
- $a_n$ is a prime number.
- For any integer $k$ between $1$ and $n$ , $a_k$ divides $a_1+a_2+\cdots+a_n$ .
|
(1, 2, 3)
| 10.15625 |
478 |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in\mathbb{R}$ \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]*Proposed by Dániel Dobák, Budapest*
|
f(x) = x^2 + 1
| 3.90625 |
479 |
What is the maximal number of solutions can the equation have $$ \max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 $$ where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$ .
|
2
| 1.5625 |
480 |
A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$ , what is left is just $1-2z$ . Determine, with proof, $b_{32}$ .
|
2^{27} - 2^{11}
| 0 |
481 |
In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.
|
52
| 45.3125 |
482 |
َA natural number $n$ is given. Let $f(x,y)$ be a polynomial of degree less than $n$ such that for any positive integers $x,y\leq n, x+y \leq n+1$ the equality $f(x,y)=\frac{x}{y}$ holds. Find $f(0,0)$ .
|
\frac{1}{n}
| 0 |
483 |
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$ .
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]
|
0
| 41.40625 |
484 |
Find a necessary and sufficient condition on the positive integer $n$ that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root.
|
n = 1
| 0 |
485 |
Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$ and centroid $G$ . The circumcircle $k_1$ of triangle $AGC$ and the circumcircle $k_2$ of triangle $BGC$ intersect $AB$ at $P$ and $Q$ , respectively. The perpendiculars from $P$ and $Q$ respectively to $AC$ and $BC$ intersect $k_1$ and $k_2$ at $X$ and $Y$ . Determine the value of $\frac{CX \cdot CY}{AB^2}$ .
|
\frac{4}{9}
| 0.78125 |
486 |
Together, Kenneth and Ellen pick a real number $a$ . Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$ ) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?
|
3
| 4.6875 |
487 |
Let $f: N \to N$ satisfy $n=\sum_{d|n} f(d), \forall n \in N$ . Then sum of all possible values of $f(100)$ is?
|
40
| 51.5625 |
488 |
Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $ , we have $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of $ a_{2013} $ .
|
2016
| 13.28125 |
489 |
How many distinct four letter arrangements can be formed by rearranging the letters found in the word **FLUFFY**? For example, FLYF and ULFY are two possible arrangements.
|
72
| 87.5 |
490 |
What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another?
|
50
| 93.75 |
491 |
Find all integer triples $(a,b,c)$ with $a>0>b>c$ whose sum equal $0$ such that the number $$ N=2017-a^3b-b^3c-c^3a $$ is a perfect square of an integer.
|
(36, -12, -24)
| 75 |
492 |
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$ , $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$ .
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$ . In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of $f(k)$ . Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
*Proposed by YaWNeeT*
|
187
| 14.84375 |
493 |
Evan has $10$ cards numbered $1$ through $10$ . He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$ , the remainder is $1$ . Find the maximum number of cards he could have chose.
*Proposed by Evan Chang*
|
6
| 61.71875 |
494 |
Real numbers $a$ , $b$ , $c$ which are differ from $1$ satisfies the following conditions;
(1) $abc =1$ (2) $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$ .
|
-\frac{3}{2}
| 14.0625 |
495 |
Cynthia and Lynnelle are collaborating on a problem set. Over a $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.
|
\frac{4}{9}
| 18.75 |
496 |
Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that
\[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \]
holds for all complex numbers $x, y$ .
*Proposed by Evan Chen*
|
18
| 4.6875 |
497 |
In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth $1$ , $5$ , $25$ , and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
*Proposed by Aaron Lin*
|
82
| 86.71875 |
498 |
A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$ ; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?
|
k
| 62.5 |
499 |
Let $a$ be the sum of the numbers: $99 \times 0.9$ $999 \times 0.9$ $9999 \times 0.9$ $\vdots$ $999\cdots 9 \times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$ .
Find the sum of the digits in the number $a$ .
|
891
| 17.96875 |
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