Unnamed: 0
int64 0
56.9k
| problem
stringlengths 16
7.44k
| ground_truth
stringlengths 1
942
| solved_percentage
float64 0
100
|
|---|---|---|---|
1,600 |
Let $ABC$ be an acute triangle with orthocenter $H$ . Let $M$ , $N$ and $R$ be the midpoints of $AB$ , $BC$ an $AH$ , respectively. If $A\hat{B}C=70^\large\circ$ , compute $M\hat{N}R$ .
|
20^\circ
| 61.71875 |
1,601 |
Given a positive integer $n$ , suppose that $P(x,y)$ is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$ } \] What is the minimum degree of $P$ ?
*Proposed by Loke Zhi Kin*
|
2n
| 74.21875 |
1,602 |
Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$ , then we have
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
$n$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \hline
$p(n)$ & $1$ & $2$ & $5$ & $15$ & $52$ & $203$ & $877$ & $4140$ & $21147$ & $115975$ \hline
\end{tabular}
Define a *splitting* of the $n$ distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has $8$ candies, what is the sum of the number of groups over all splittings he can use?
*2020 CCA Math Bonanza Lightning Round #3.4*
|
17007
| 30.46875 |
1,603 |
A circle with diameter $23$ is cut by a chord $AC$ . Two different circles can be inscribed between the large circle and $AC$ . Find the sum of the two radii.
|
\frac{23}{2}
| 10.15625 |
1,604 |
Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$ . Find the angle $\angle MXK $ .
|
135^\circ
| 0 |
1,605 |
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$ .
Find the minimum of $x^2+y^2+z^2+t^2$ .
*proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi*
|
2
| 60.15625 |
1,606 |
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$ . Show that the number $11$ is not an element of this sequence.
|
11
| 88.28125 |
1,607 |
A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions: $\textbf{(i)}$ it contains no digits of $9$ ; $\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$ , the outcome is a square.
Find all squarish nonnegative integers. $\textit{(Vlad Robu)}$
|
0, 3, 8, 15
| 0 |
1,608 |
A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there?
*Proposed by Evan Chen*
|
8
| 32.03125 |
1,609 |
A one-person game with two possible outcomes is played as follows:
After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$ . The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$
|
a = 332, b = 7
| 5.46875 |
1,610 |
**9.** Let $w=f(x)$ be regular in $ \left | z \right |\leq 1$ . For $0\leq r \leq 1$ , denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$ . Show that if the maximal length of the chords of $c_{1}$ is $1$ , then for every $r$ such that $0\leq r \leq 1$ , the maximal length of the chords of c, is not greater than $r$ . **(F. 1)**
|
r
| 0 |
1,611 |
We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection?
|
120^\circ
| 35.15625 |
1,612 |
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$
|
480
| 92.96875 |
1,613 |
Consider the sets $A_1=\{1\}$ , $A_2=\{2,3,4\}$ , $A_3=\{5,6,7,8,9\}$ , etc. Let $b_n$ be the arithmetic mean of the smallest and the greatest element in $A_n$ . Show that the number $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$ is a prime integer.
|
1999
| 33.59375 |
1,614 |
Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common.
Also, $g$ is called a $C$ -gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$ . Determine the largest $C$ -gap $g$ .
|
9
| 6.25 |
1,615 |
Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$ . Find $w_1+w_2$ .
|
25 + 2 = 27
| 0 |
1,616 |
Find the sum of first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - 1$ .
|
30
| 91.40625 |
1,617 |
Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$ . Let $M$ be any point on $l,$ interior to the rectangle.
Find the necessary and sufficient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.
|
a = b
| 32.03125 |
1,618 |
Integers from 1 to 100 are placed in a row in some order. Let us call a number *large-right*, if it is greater than each number to the right of it; let us call a number *large-left*, is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and exactly $k$ large-left numbers. Find the maximal possible value of $k$ .
|
k = 50
| 0 |
1,619 |
Consider the set $E = \{5, 6, 7, 8, 9\}$ . For any partition ${A, B}$ of $E$ , with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$ . Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$ .
|
17
| 88.28125 |
1,620 |
On the bisector $AL$ of triangle $ABC$ a point $K$ is chosen such that $\angle BKL=\angle KBL=30^{\circ}$ . Lines $AB$ and $CK$ intersect at point $M$ , lines $AC$ and $BK$ intersect at point $N$ . FInd the measure of angle $\angle AMN$ *Proposed by D. Shiryaev, S. Berlov*
|
60^\circ
| 17.96875 |
1,621 |
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $$ x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2 $$ is divisible by $3$ .
|
80
| 100 |
1,622 |
Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$ . Show that: $$ \displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044} $$ When the equality holds?
|
8044
| 97.65625 |
1,623 |
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$ . How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$ -axis is an integer?
[asy]
import graph;
size(300);
defaultpen(linewidth(0.8)+fontsize(10));
real k=1.5;
real endp=sqrt(k);
real f(real x) {
return k-x^2;
}
path parabola=graph(f,-endp,endp)--cycle;
filldraw(parabola, lightgray);
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));
label("Region I", (0,2*k/5));
label("Box II", (51/64*endp,13/16*k));
label("area(I) = $\frac23$ \,area(II)",(5/3*endp,k/2));
[/asy]
|
14
| 87.5 |
1,624 |
Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$ . Find the minimum possible value of $\beta$ .
|
32
| 50 |
1,625 |
There are several different positive integers written on the blackboard, and the sum of any two different numbers should be should be a prime power. At this time, find the maximum possible number of integers written on the blackboard. A prime power is an integer expressed in the form $p^n$ using a prime number $p$ and a non-negative integer number $n$ .
|
4
| 13.28125 |
1,626 |
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_ib_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$ for all $i \ge 1$ . If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$ , then what is $b_{6}$ ?
*Proposed by Andrew Wu*
|
257
| 20.3125 |
1,627 |
Let $ABC$ be a triangle in which ( ${BL}$ is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$ , ${AH}$ is an altitude of $\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$ is the midpoint of the side ${AB}$ . It is known that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\vartriangle ABC$ .
|
60^\circ
| 0.78125 |
1,628 |
Let $P$ be a $10$ -degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$ -degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$ . If $P(0) = Q(1) = 2$ , then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively prime integers $a, b$ . Find $a + b$ .
|
19
| 2.34375 |
1,629 |
Bob writes a random string of $5$ letters, where each letter is either $A, B, C,$ or $D$ . The letter in each position is independently chosen, and each of the letters $A, B, C, D$ is chosen with equal probability. Given that there are at least two $A's$ in the string, find the probability that there are at least three $A's$ in the string.
|
\frac{53}{188}
| 53.90625 |
1,630 |
The polynomial of seven variables $$ Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2) $$ is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients: $$ Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2. $$ Find all possible values of $P_1(1,1,\ldots,1)$ .
*(A. Yuran)*
|
3
| 15.625 |
1,631 |
A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$
Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$
|
c = 5
| 0 |
1,632 |
**Q13.** Determine the greatest value of the sum $M=11xy+3x+2012yz$ , where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$
|
503000000
| 2.34375 |
1,633 |
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$ .
|
41
| 100 |
1,634 |
Find the minimum area of the part bounded by the parabola $ y\equal{}a^3x^2\minus{}a^4x\ (a>0)$ and the line $ y\equal{}x$ .
|
\frac{4}{3}
| 61.71875 |
1,635 |
Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds: $\overline{ab}=3 \cdot \overline{cd} + 1$ .
|
2809
| 95.3125 |
1,636 |
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$ . Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$ .
|
665
| 2.34375 |
1,637 |
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$ .
|
p = q
| 0 |
1,638 |
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$ , $b - d$ , and $c + d$ .
|
(1, 2, 0, 1)
| 4.6875 |
1,639 |
Let $x_1=1/20$ , $x_2=1/13$ , and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$ . Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$ .
|
23
| 0.78125 |
1,640 |
Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela will do it. If 2 or more of the dice are a multiple of 3 neither one of them will do it. How many dice should be rolled so that the risk is the same for both Claudia and Adela?
|
x = 2
| 0 |
1,641 |
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$ , for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$ . Determine $x_{1997}$ .
|
23913
| 98.4375 |
1,642 |
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$ . What is the value of $ a_5$ ?
|
9
| 82.8125 |
1,643 |
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$ , $B(0,12)$ , $C(16,0)$ , $A'(24,18)$ , $B'(36,18)$ , and $C'(24,2)$ . A rotation of $m$ degrees clockwise around the point $(x,y)$ , where $0<m<180$ , will transform $\triangle ABC$ to $\triangle A'B'C'$ . Find $m+x+y$ .
|
108
| 18.75 |
1,644 |
Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$ , the number of divisors of $kp+1$ between $k$ and $p$ exclusive is $a_k$ . Find the value of $a_1+a_2+\ldots + a_{p-1}$ .
|
p-2
| 7.03125 |
1,645 |
The rectangle in the figure has dimensions $16$ x $20$ and is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles?
|
24
| 59.375 |
1,646 |
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
|
215
| 83.59375 |
1,647 |
Find (in terms of $n \ge 1$ ) the number of terms with odd coefficients after expanding the product:
\[\prod_{1 \le i < j \le n} (x_i + x_j)\]
e.g., for $n = 3$ the expanded product is given by $x_1^2 x_2 + x_1^2 x_3 + x_2^2 x_3 + x_2^2 x_1 + x_3^2 x_1 + x_3^2 x_2 + 2x_1 x_2 x_3$ and so the answer would be $6$ .
|
n!
| 0 |
1,648 |
Define mutually externally tangent circles $\omega_1$ , $\omega_2$ , and $\omega_3$ . Let $\omega_1$ and $\omega_2$ be tangent at $P$ . The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$ . Let $O$ be the center of $\omega_3$ . If $QP = 420$ and $QO = 427$ , find the radius of $\omega_3$ .
*Proposed by Tanishq Pauskar and Mahith Gottipati*
|
77
| 28.90625 |
1,649 |
An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$ , there is a language spoken by $A$ and not $B$ , and there is another language spoken by $B$ but not $A$ . At least how many languages must be spoken at the firm?
|
10
| 0 |
1,650 |
Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\cdots,a_n$ , that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum,
\[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\]
Is $ 2001$ .
|
3
| 64.0625 |
1,651 |
A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of $\tfrac{2}{3}$ of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?
|
\frac{20}{81}
| 9.375 |
1,652 |
Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$ . The line $OA$ intersects the segment $EH$ in the point $X$ , $OB$ intersects $EF$ in the point $Y$ , $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W$ . If the $(EFGH) = 1$ . Find: $(ABCD) \times (XYZW)$ Note $(P)$ denote the area of the polygon $P$ .
|
1
| 13.28125 |
1,653 |
Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$ .
|
n = 7
| 0 |
1,654 |
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$ , $P_{0}P_{3}P_{6}$ , $P_{0}P_{6}P_{7}$ , $P_{0}P_{7}P_{8}$ , $P_{1}P_{2}P_{3}$ , $P_{3}P_{4}P_{6}$ , $P_{4}P_{5}P_{6}$ . In how many ways can these triangles be labeled with the names $\triangle_{1}$ , $\triangle_{2}$ , $\triangle_{3}$ , $\triangle_{4}$ , $\triangle_{5}$ , $\triangle_{6}$ , $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$ ? Justify your answer.
|
1
| 53.125 |
1,655 |
How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?
|
128
| 24.21875 |
1,656 |
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, *Neymar* chooses a divisor $d>1$ of $n$ , erases $n$ , and writes $n+d$ . If the initial number on the board is $2022$ , what is the largest composite number that *Neymar* will never be able to write on the blackboard?
|
2033
| 0 |
1,657 |
How many integers can be expressed in the form: $\pm 1 \pm 2 \pm 3 \pm 4 \pm \cdots \pm 2018$ ?
|
2037172
| 31.25 |
1,658 |
Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$ . Suppose further that
\[ |x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|. \]
What is the smallest possible value of $n$ ?
|
20
| 75 |
1,659 |
Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than $1.$ Show that $K$ lies in a circle of radius $\frac{1}{\sqrt{3}}.$
|
\frac{1}{\sqrt{3}}
| 96.875 |
1,660 |
Let $a$ and $b$ be integer solutions to $17a+6b=13$ . What is the smallest possible positive value for $a-b$ ?
|
17
| 74.21875 |
1,661 |
Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits, $\overline{AB}$ , such that $A$ and $B$ are both factors of $\overline{AB}$ ."
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"
Cat says, "That's cool, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!"
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
*Proposed by Andrew Wu*
|
24
| 52.34375 |
1,662 |
Let $M$ be a set of six distinct positive integers whose sum is $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
|
84
| 0 |
1,663 |
Let $a,b,c,d,e,f$ be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors $x-x_i$ , with $x_i>0$ for $i=1,2,\ldots,8$ . Determine all possible values of $f$ .
|
\frac{1}{256}
| 95.3125 |
1,664 |
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$ , where $n$ is either $2012$ or $2013$ .
|
338
| 16.40625 |
1,665 |
Consider an acute triangle $ABC$ of area $S$ . Let $CD \perp AB$ ( $D \in AB$ ), $DM \perp AC$ ( $M \in AC$ ) and $DN \perp BC$ ( $N \in BC$ ). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$ , respectively $MND$ . Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$ .
|
S
| 35.9375 |
1,666 |
A circle with area $40$ is tangent to a circle with area $10$ . Let R be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$ . Find $n$ .
[asy]
defaultpen(linewidth(0.7)); size(120);
real R = sqrt(40/pi), r = sqrt(10/pi);
draw(circle((0,0), R)); draw(circle((R+r,0), r));
draw((-R,-R)--(-R,R)--(R+2*r,R)--(R+2*r,-R)--cycle);[/asy]
|
240
| 57.8125 |
1,667 |
Leticia has a $9\times 9$ board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:
- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.
Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.
|
486
| 31.25 |
1,668 |
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.
|
48
| 98.4375 |
1,669 |
A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer $N$ such that, regardless of the students' sitting positions, at the end there is a set $S$ consisting of $N$ students satisfying the following property: every two of them have never been sitting together.
|
13
| 98.4375 |
1,670 |
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$ .
*Yannick Yao*
|
2015
| 40.625 |
1,671 |
A prime number $p$ is a **moderate** number if for every $2$ positive integers $k > 1$ and $m$ , there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If $q$ is the smallest **moderate** number, then determine the smallest prime $r$ which is not moderate and $q < r$ .
|
7
| 19.53125 |
1,672 |
Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$ , such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube.
|
10
| 96.09375 |
1,673 |
Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.
|
14
| 93.75 |
1,674 |
Find $n$ such that $n - 76$ and $n + 76$ are both cubes of positive integers.
|
140
| 81.25 |
1,675 |
4. Let $v$ and $w$ be two randomly chosen roots of the equation $z^{1997} -1 = 0$ (all roots are equiprobable). Find the probability that $\sqrt{2+\sqrt{3}}\le |u+w|$
|
\frac{333}{1997}
| 0 |
1,676 |
We call a number *perfect* if the sum of its positive integer divisors(including $1$ and $n$ ) equals $2n$ . Determine all *perfect* numbers $n$ for which $n-1$ and $n+1$ are prime numbers.
|
6
| 99.21875 |
1,677 |
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$ . The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$ .
|
108^\circ
| 72.65625 |
1,678 |
Given two positive integers $m,n$ , we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$ -*slippery* if it has the following properties:
i) $f$ is continuous;
ii) $f(0) = 0$ , $f(m) = n$ ;
iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$ , then $t_2-t_1 \in \{0,m\}$ .
Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$ -slippery.
|
\gcd(m,n) = 1
| 0 |
1,679 |
For how many integers $a$ with $|a| \leq 2005$ , does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions?
|
90
| 46.875 |
1,680 |
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.
|
5
| 26.5625 |
1,681 |
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
|
126
| 92.96875 |
1,682 |
Find the positive integer $n$ such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.
|
15
| 93.75 |
1,683 |
Let $A$ , $B$ , $C$ , $D$ be four points on a circle in that order. Also, $AB=3$ , $BC=5$ , $CD=6$ , and $DA=4$ . Let diagonals $AC$ and $BD$ intersect at $P$ . Compute $\frac{AP}{CP}$ .
|
\frac{2}{5}
| 75 |
1,684 |
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$ , where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
|
3432
| 31.25 |
1,685 |
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$ .
|
23
| 29.6875 |
1,686 |
Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.
|
209
| 89.84375 |
1,687 |
Let $\alpha$ be a root of $x^6-x-1$ , and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$ . It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$ . Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$ .
|
727
| 19.53125 |
1,688 |
Let $a$ and $b$ be positive integers such that $(a^3 - a^2 + 1)(b^3 - b^2 + 2) = 2020$ . Find $10a + b$ .
|
53
| 78.90625 |
1,689 |
There are $5$ accents in French, each applicable to only specific letters as follows:
- The cédille: ç
- The accent aigu: é
- The accent circonflexe: â, ê, î, ô, û
- The accent grave: à, è, ù
- The accent tréma: ë, ö, ü
Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$ .
|
4
| 0.78125 |
1,690 |
Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard such that they do NOT attack each other?
|
1848
| 79.6875 |
1,691 |
**Q14.** Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$ . Suppose that $IH$ is perpendicular to $BC$ ( $H$ belongs to $BC$ ). If $HB=5 \text{cm}, \; HC=8 \text{cm}$ , compute the area of $\triangle ABC$ .
|
40
| 0.78125 |
1,692 |
In every row of a grid $100 \times n$ is written a permutation of the numbers $1,2 \ldots, 100$ . In one move you can choose a row and swap two non-adjacent numbers with difference $1$ . Find the largest possible $n$ , such that at any moment, no matter the operations made, no two rows may have the same permutations.
|
2^{99}
| 4.6875 |
1,693 |
A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner.
[i]Proposed by Nathan Ramesh
|
11
| 100 |
1,694 |
For positive integers $m$ and $n$ , find the smalles possible value of $|2011^m-45^n|$ .
*(Swiss Mathematical Olympiad, Final round, problem 3)*
|
14
| 89.0625 |
1,695 |
Let $a$ and $b$ be positive integers such that $(2a+b)(2b+a)=4752$ . Find the value of $ab$ .
*Proposed by James Lin*
|
520
| 2.34375 |
1,696 |
Let $A=\{1,2,3,\ldots,40\}$ . Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$ .
|
4
| 29.6875 |
1,697 |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$ , $y$ , and $z$ . Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$ , $17$ , $21$ , and $23$ . Compute $x^2+y^2+z^2$ .
|
371
| 50 |
1,698 |
In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer $n$ for which any tennis tournament with $n$ people has a group of four tennis players that is ordered.
*Ray Li*
|
8
| 76.5625 |
1,699 |
The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$ . An *operation* consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.
|
255
| 96.875 |
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