problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Given an integer $n\geq 2$ , determine the maximum value the sum $x_1+\cdots+x_n$ may achieve, as the $x_i$ run through the positive integers, subject to $x_1\leq x_2\leq \cdots \leq x_n$ and $x_1+\cdots+x_n=x_1 x_2\cdots x_n$ . | S = 2n | 0.875 |
On the planet Mars there are $100$ states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions:
(1) Each block must have at most $50$ states.
(2) Every pair of states must be together in at least one block.
Find the minimum number of blocks that must be formed. | 6 | 0.625 |
Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$ 5^m+63n+49=a^k $$ holds. Find the minimum value of $k$ . | 5 | 0.875 |
$8$ singers take part in a festival. The organiser wants to plan $m$ concerts. For every concert there are $4$ singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises $m$ . | m = 14 | 0.875 |
The diagram below shows a circle with center $F$ . The angles are related with $\angle BFC = 2\angle AFB$ , $\angle CFD = 3\angle AFB$ , $\angle DFE = 4\angle AFB$ , and $\angle EFA = 5\angle AFB$ . Find the degree measure of $\angle BFC$ .
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
dotfactor=4;
draw(unitcircle);
pair A,B,C,D,E,F;
A=dir(90);
B=dir(66);
C=dir(18);
D=dir(282);
E=dir(210);
F=origin;
dot(" $F$ ",F,NW);
dot(" $A$ ",A,dir(90));
dot(" $B$ ",B,dir(66));
dot(" $C$ ",C,dir(18));
dot(" $D$ ",D,dir(306));
dot(" $E$ ",E,dir(210));
draw(F--E^^F--D^^F--C^^F--B^^F--A);
[/asy] | 48^\circ | 0.875 |
Find all prime numbers $p$ , for which the number $p + 1$ is equal to the product of all the prime numbers which are smaller than $p$ . | p = 5 | 0.5 |
As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$ ). If $CE = 14$ , $AB > 14$ , compute the minimal area of $\triangle AEG$ .
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(real x, real y) {
pair P = (x,y);
dot(P,linewidth(3)); return P;
}
int big = 30, small = 14;
filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));
draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);
label(" $A$ ",D2(0,big),NW);
label(" $B$ ",D2(0,0),SW);
label(" $C$ ",D2(big,0),SW);
label(" $D$ ",D2(big,big),N);
label(" $E$ ",D2(big+small,0),SE);
label(" $F$ ",D2(big+small,small),NE);
label(" $G$ ",D2(big,small),NE);
[/asy] | 98 | 0.375 |
How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square? | 1032 | 0.75 |
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$ . The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$ , where $r$ , $s$ , and $t$ are positive integers, and $r+s+t<1000$ . Find $r+s+t$ . | 330 | 0.25 |
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$ ? For example, with $n = 4$ , there are four ways: $4$ , $2 + 2$ , $1 + 1 + 2$ , $1 + 1 + 1 + 1$ . | n | 0.875 |
At the meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each with one another, among the remaining people they do not have a common friend. For each pair individuals $A$ and $B$ are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting? | 100 | 0.75 |
Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$ . Let $ \mathcal Q$ be the locus of the midpoint of $ AB$ . It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$ . | \frac{7}{8} | 0.875 |
Compute $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$ .)
*Proposed by Tristan Shin* | 2019 | 0.5 |
In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$ ? For example, $2+10$ , $10+2$ , and $3+2+2+5$ are three such ways. | 70 | 0.25 |
A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$ , how old is the tortoise? | 77 | 0.875 |
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$ . Find $a_{100}$ . | 53 | 0.125 |
The *subnumbers* of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime.
*Proposed by Lewis Chen* | 9 | 0.625 |
Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$ . What is the maximum possible value for $n$ ?
*Proposed by James Lin* | 5 | 0.75 |
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$ . Find the maximum value of $n$ . | 64 | 0.125 |
How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$ , where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$ .) | 13 | 0.75 |
Given $n \in\mathbb{N}$ , find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$
\[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \] | f(x) = 0 | 0.875 |
Nine weights are placed in a scale with the respective values $1kg,2kg,...,9kg$ . In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one? | 2 | 0.5 |
Let $\ell$ be a line with negative slope passing through the point $(20,16)$ . What is the minimum possible area of a triangle that is bounded by the $x$ -axis, $y$ -axis, and $\ell$ ?
*Proposed by James Lin* | 640 | 0.875 |
A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$ , such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column). | n = 7 | 0.75 |
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational. | W(x) = ax + b | 0.25 |
For each prime $p$ , let $\mathbb S_p = \{1, 2, \dots, p-1\}$ . Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that
\[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \]
for all $n \in \mathbb S_p$ .
*Andrew Wen* | 2 | 0.875 |
A *palindromic table* is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
O & M & O
N & M & N
O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters $O$ and $M$ ? (The table may contain only a single letter.)
*Proposed by Evan Chen* | 16 | 0.375 |
Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$ . Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$ . Find the least $n$ such that $\tfrac{1}{E_n}
> 2017$ . | 13446 | 0.25 |
During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$ , in degrees, of the bottom of the cone when we look at it from the side?
 | 60^\circ | 0.125 |
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$ , where $p$ is an integer. Find $p$ . | 192 | 0.25 |
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$ . | 660 | 0.875 |
Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and $$ \frac{a+b}{a+c}=\frac{b+c}{b+a}. $$ | (1, 1, 1) | 0.5 |
Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$ while their common internal tangent has length $2009$ . Find the product of the radii of $\Omega_1$ and $\Omega_2$ .
*Proposed by David Altizio* | 8052 | 0.875 |
Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=? | 150^\circ | 0.625 |
Find all natural numbers $a>1$ , with the property that every prime divisor of $a^6-1$ divides also at least one of the numbers $a^3-1$ , $a^2-1$ .
*K. Dochev* | a = 2 | 0.75 |
Triangle $ABC$ has side lengths $AB=13, BC=14,$ and $CA=15$ . Let $\Gamma$ denote the circumcircle of $\triangle ABC$ . Let $H$ be the orthocenter of $\triangle ABC$ . Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$ . Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$ . Suppose $DG$ intersects $AC$ at $X$ . Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$ .
*Proposed by Kenan Hasanaliyev (claserken)* | 24 | 0.125 |
Each face of a tetrahedron is a triangle with sides $a, b,$ c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$ . | 8 | 0.75 |
Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation $$ \sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4. $$ | x = 1 | 0.75 |
Find the sum of all positive integers whose largest proper divisor is $55$ . (A proper divisor of $n$ is a divisor that is strictly less than $n$ .)
| 550 | 0.25 |
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his *movie collection*. If every student has watched every movie at most once, at least how many different movie collections can these students have? | 23 | 0.25 |
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$ , $x_2y_1-x_1y_2=5$ , and $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of $y_1^2+5y_2^2$ | 23 | 0.625 |
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits. | 1944 | 0.5 |
Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)
| 48 | 0.25 |
A right rectangular prism has integer side lengths $a$ , $b$ , and $c$ . If $\text{lcm}(a,b)=72$ , $\text{lcm}(a,c)=24$ , and $\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?
*Proposed by Deyuan Li and Andrew Milas* | 3024 | 0.625 |
Let $ m,n\in \mathbb{N}^*$ . Find the least $ n$ for which exists $ m$ , such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$ , $ n \minus{} 1$ of length $ 2$ , $ ...$ , $ 1$ square of length $ n$ . For the found value of $ n$ give the example of covering. | n = 8 | 0.375 |
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$ , Anne and Carl in $5$ . How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back? | \frac{59}{20} | 0.75 |
Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap? | \frac{4\pi}{3} | 0.5 |
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$ . Find all values of $n$ such that $n=d_2^2+d_3^3$ . | 68 | 0.625 |
$4.$ Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the number of galleons between the three of them is a multiple of $7$ . What are all the possible number of galleons that Harry can have? | 6 | 0.875 |
Let $ABC$ be a triangle such that $AB = 7$ , $BC = 8$ , and $CA = 9$ . There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$ . If $XA = \tfrac ab$ , where $a$ and $b$ are coprime positive integers, find $a + b$ .
*Proposed by Alexander Wang* | 61 | 0.75 |
$PS$ is a line segment of length $4$ and $O$ is the midpoint of $PS$ . A semicircular arc is drawn with $PS$ as diameter. Let $X$ be the midpoint of this arc. $Q$ and $R$ are points on the arc $PXS$ such that $QR$ is parallel to $PS$ and the semicircular arc drawn with $QR$ as diameter is tangent to $PS$ . What is the area of the region $QXROQ$ bounded by the two semicircular arcs? | 2\pi - 2 | 0.125 |
Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 3
x^4 - y^4 = 8x - y \end{cases}$ | (2, 1) | 0.75 |
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?
*(5 points)* | 1000 | 0.25 |
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$ . Find $c$ .
*Proposed by Andy Xu* | 1012 | 0.875 |
Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with common difference $\sqrt2$ . Given that the product $abcd = 2021$ , $d$ can be written as $\frac{m+\sqrt{n}}{\sqrt{p}}$ , where $m, n,$ and $p$ are positive integers not divisible by the square of any prime. Find $m + n + p$ . | 100 | 0.875 |
$2021$ people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up $1000$ positions to the left of their original position. | 1021000 | 0.125 |
Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an
empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers
(call them $b$ and $c$ ) such that $b + c = a$ . The player then adds $b\times c$ to her score. She repeats the step
several times until she ends up with all $1$ 's on the whiteboard. Then the game is over, and the final score is
calculated. Let $M, m$ be the maximum and minimum final score that can be possibly obtained respectively.
Find $M-m$ . | 0 | 0.375 |
Given a positive integer $n$ , determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover. | (n+1)^2 | 0.5 |
For a positive integer $n$ , let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$ . | 18 | 0.875 |
Peter and Basil play the following game on a horizontal table $1\times{2019}$ . Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays. | n = 2 | 0.25 |
For any natural number, let $S(n)$ denote sum of digits of $n$ . Find the number of $3$ digit numbers for which $S(S(n)) = 2$ . | 100 | 0.25 |
Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains? | \frac{5}{84} | 0.625 |
In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$ , let $ BC\equal{}1000$ and $ AD\equal{}2008$ . Let $ \angle A\equal{}37^\circ$ , $ \angle D\equal{}53^\circ$ , and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$ , respectively. Find the length $ MN$ . | 504 | 0.5 |
In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);
pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw(" $O$ ", B, dir(180));
draw(" $B$ ", A, dir(45));
draw(" $A$ ", C, dir(45));
draw(" $Q$ ", E, dir(45));
draw(" $P$ ", D, dir(45));
draw(" $R$ ", Z, dir(45));
draw(" $S$ ", F, dir(45));
draw(" $a$ ", Y, dir(210));
draw(" $b$ ", J, dir(100));
[/asy] | 2 : 1 | 0.375 |
Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$ Denote by $P$ the center of $C$ . When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$ . | \frac{2\sqrt{2}}{3} | 0.875 |
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$ , tens digit $y$ , and units digit $z$ , and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$ . How many three-digit numbers $\underline{abc}$ , none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$ ? | 120 | 0.25 |
The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$ . Compute the greatest possible sum of the other two roots.
*2015 CCA Math Bonanza Lightning Round #2.4* | 5 | 0.875 |
If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$ ), two values (for example, $2666$ ), three values (for example, $5215$ ), or four values (for example, $4236$ ). The mean number of values that show is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 887 | 0.75 |
Find the smallest constant M, so that for any real numbers $a_1, a_2, \dots a_{2023} \in [4, 6]$ and $b_1, b_2, \dots b_{2023} \in [9, 12] $ following inequality holds: $$ \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots + a_{2023} b_{2023} \right) $$ *Proposed by Zaza Meliqidze, Georgia* | \frac{3}{2\sqrt{2}} | 0.125 |
Let $x_1, x_2, . . . , x_{2022}$ be nonzero real numbers. Suppose that $x_k + \frac{1}{x_{k+1}} < 0$ for each $1 \leq k \leq 2022$ , where $x_{2023}=x_1$ . Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_n > 0$ . | 1011 | 0.625 |
Given any triangle $ABC$ and any positive integer $n$ , we say that $n$ is a *decomposable* number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$ . Determine the positive integers that are decomposable numbers for every triangle. | n = 1 | 0.125 |
$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours? | 2 | 0.25 |
What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)
Alexandru Mihalcu | 16 | 0.375 |
There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect. | k = 50 | 0.375 |
Two beads, each of mass $m$ , are free to slide on a rigid, vertical hoop of mass $m_h$ . The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio $m/m_h$ such that the hoop always remains in contact with the ground? Neglect friction.
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,0)--(3,0));
draw(circle((1.5,1),1));
filldraw(circle((1.4,1.99499),0.1), gray(.3));
filldraw(circle((1.6,1.99499),0.1), gray(.3));
[/asy] | \frac{3}{2} | 0.125 |
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$ . The lines through $E$ parallel to $AD$ , $DC$ , $CB$ , $BA$ meet $AB$ , $BC$ , $CD$ , $DA$ at $K$ , $L$ , $M$ , $N$ , respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$ | 1 | 0.5 |
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?
*Author: Ray Li* | 2 | 0.625 |
Let $x$ be the largest root of $x^4 - 2009x + 1$ . Find the nearest integer to $\frac{1}{x^3-2009}$ . | -13 | 0.875 |
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference. | 3 | 0.625 |
There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$ , and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group. | \frac{3}{455} | 0.5 |
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number | x = 6 | 0.875 |
Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$ .
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
| 23 | 0.75 |
Consider the $4\times4$ array of $16$ dots, shown below.
[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]
Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1\times1$ squares, four $2\times2$ squares, and one $3\times3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex? | 4 | 0.375 |
In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$ -axis. | \sqrt{3} | 0.875 |
Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$ , there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying
\[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\]
where $y_{n+1}=y_1,y_{n+2}=y_2$ . | n-1 | 0.25 |
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$ . | f(x) = x^2 | 0.875 |
**p1.** $17.5\%$ of what number is $4.5\%$ of $28000$ ?**p2.** Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$ . The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p3.** In the $xy$ -plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$ . Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at?**p4.** What are the last two digits of the sum of the first $2021$ positive integers?**p5.** A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$ th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p6.** How many terms are in the arithmetic sequence $3$ , $11$ , $...$ , $779$ ?**p7.** Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$ ?**p8.** What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$ ?**p9.** Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$ , $AB = 10$ , $BC = 9$ , and the area of $\vartriangle ABC$ is $36$ . Compute the length of $AC$ .
**p10.** If $x + y - xy = 4$ , and $x$ and $y$ are integers, compute the sum of all possible values of $ x + y$ .**p11.** What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap?**p12.** $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$ . Compute the smallest possible value of $N$ .**p13.** Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years?**p14.** Say there is $1$ rabbit on day $1$ . After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$ ?**15.** Ajit draws a picture of a regular $63$ -sided polygon, a regular $91$ -sided polygon, and a regular $105$ -sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have?**p16.** Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p17.** Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$ 's adjacent.**p18.** From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$ , where $a$ and $b$ are positive integers. Compute $a + b$ .**p19.** Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$ . He starts by putting the first marble in bucket $1$ , the second marble in bucket $2$ , the third marble in bucket $3$ , etc. After placing a marble in bucket $9$ , he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag?
**p20.** What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$ ?
PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | 1 | 0.125 |
Given any two positive real numbers $x$ and $y$ , then $x\Diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x\Diamond y$ satisfies the equations $(x\cdot y)\Diamond y=x(y\Diamond y)$ and $(x\Diamond 1)\Diamond x=x\Diamond 1$ for all $x,y>0$ . Given that $1\Diamond 1=1$ , find $19\Diamond 98$ . | 19 | 0.875 |
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$ . Determine the number of elements $A \cup B$ has. | 8 | 0.75 |
Let $ f(n,k)$ be the number of ways of distributing $ k$ candies to $ n$ children so that each child receives at most $ 2$ candies. For example $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$ . Determine the value of $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$ . | 3^{2005} | 0.625 |
Among all fractions (whose numerator and denominator are positive integers) strictly between $\tfrac{6}{17}$ and $\tfrac{9}{25}$ , which one has the smallest denominator? | \frac{5}{14} | 0.875 |
An ordered pair $(n,p)$ is *juicy* if $n^{2} \equiv 1 \pmod{p^{2}}$ and $n \equiv -1 \pmod{p}$ for positive integer $n$ and odd prime $p$ . How many juicy pairs exist such that $n,p \leq 200$ ?
Proposed by Harry Chen (Extile) | 36 | 0.375 |
Let $G$ be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs $(n,k)$ , $n\ge 3$ , such that:
a) For all graph $G$ with $n$ vertex and $k$ edges, it is always possible to perform a series of switching process so that all edges are eventually blue.
b) There exist a graph $G$ with $n$ vertex and $k$ edges and it is possible to perform a series of switching process so that all edges are eventually blue. | (n, k) | 0.125 |
How many 3-term geometric sequences $a$ , $b$ , $c$ are there where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ and $c = 8000$ ? | 39 | 0.25 |
For positive integers $a>b>1$ , define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least $d$ such that for any $a,b$ , the sequence $x_n$ does not contain $d$ consecutive prime numbers.
*V. Senderov* | 3 | 0.75 |
For an integer $n>3$ denote by $n?$ the product of all primes less than $n$ . Solve the equation $n?=2n+16$ .
*V. Senderov* | n = 7 | 0.625 |
Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$ . Find the number of such matrices for which the number of $ 1$ -s in each row and in each column is even. | 2^{(m-1)(n-1)} | 0.5 |
In the $xy$ -coordinate plane, the $x$ -axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$ -axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$ . They are $(126, 0)$ , $(105, 0)$ , and a third point $(d, 0)$ . What is $d$ ? (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.) | 111 | 0.25 |
Let $A$ be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from $A$ differs at most in 1 position from one of the chosen? | 4 | 0.375 |
Let $a_n$ be the closest to $\sqrt n$ integer.
Find the sum $1/a_1 + 1/a_2 + ... + 1/a_{1980}$ . | 88 | 0.5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.