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Let $\alpha\in\mathbb R\backslash \{ 0 \}$ and suppose that $F$ and $G$ are linear maps (operators) from $\mathbb R^n$ into $\mathbb R^n$ satisfying $F\circ G - G\circ F=\alpha F$ .
a) Show that for all $k\in\mathbb N$ one has $F^k\circ G-G\circ F^k=\alpha kF^k$ .
b) Show that there exists $k\geq 1$ such that $F^k=0$ . | F^k = 0 | 0.625 |
From the positive integers, $m,m+1,\dots,m+n$ , only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$ . Find the maximum value of $n$ . | 15 | 0.125 |
There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays. | 50 | 0.875 |
Crisp All, a basketball player, is *dropping dimes* and nickels on a number line. Crisp drops a dime on every positive multiple of $10$ , and a nickel on every multiple of $5$ that is not a multiple of $10$ . Crisp then starts at $0$ . Every second, he has a $\frac{2}{3}$ chance of jumping from his current location $x$ to $x+3$ , and a $\frac{1}{3}$ chance of jumping from his current location $x$ to $x+7$ . When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp *stops on a dime*? | \frac{20}{31} | 0.25 |
How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) | 36 | 0.25 |
A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$ , and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$ . Find the smallest $x$ for which $f(x)=f(2001)$ . | 429 | 0.5 |
Consider the polynomials
\[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\]
\[g(p) = p^3 + 2p + m.\]
Find all integral values of $m$ for which $f$ is divisible by $g$ . | m = 3 | 0.5 |
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$ .
*2017 CCA Math Bonanza Lightning Round #2.4* | 13 | 0.875 |
Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$ .
*Proposed by Harun Khan* | 15 | 0.5 |
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$ , $\text{lcm}(x, z) = 900$ , and $\text{lcm}(y, z) = n$ , where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property. | 9 | 0.25 |
Tony plays a game in which he takes $40$ nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself $5$ points for each nickel that lands in the jar, and takes away $2$ points from his score for each nickel that hits the ground. After Tony is done tossing all $40$ nickels, he computes $88$ as his score. Find the greatest number of nickels he could have successfully tossed into the jar. | 24 | 0.875 |
Call the Graph the set which composed of several vertices $P_1,\ \cdots P_2$ and several edges $($ segments $)$ connecting two points among these vertices. Now let $G$ be a graph with 9 vertices and satisfies the following condition.
Condition: Even if we select any five points from the vertices in $G,$ there exist at least two edges whose endpoints are included in the set of 5 points.
What is the minimum possible numbers of edges satisfying the condition? | 9 | 0.25 |
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$ ? | 8 | 0.75 |
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$ . For instance, $ f(2) \equal{} 1$ , $ f(5) \equal{} 2$ . For how many $ 1 < m < 2008$ is $ f(m)$ odd? | 242 | 0.75 |
A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$ . Find $100r^2$ .
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draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy] | 4 | 0.875 |
Let $a \star b = ab + a + b$ for all integers $a$ and $b$ . Evaluate $1 \star ( 2 \star ( 3 \star (4 \star \ldots ( 99 \star 100 ) \ldots )))$ . | 101! - 1 | 0.75 |
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin) | 1990 | 0.875 |
Two (not necessarily different) numbers are chosen independently and at random from $\{1, 2, 3, \dots, 10\}$ . On average, what is the product of the two integers? (Compute the expected product. That is, if you do this over and over again, what will the product of the integers be on average?) | 30.25 | 0.875 |
The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$ | 118 | 0.5 |
Point $P$ is $\sqrt3$ units away from plane $A$ . Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$ ? | 8\pi | 0.875 |
Find all positive integers $n$ such that the number $n^5+79$ has all the same digits when it is written in decimal represantation. | n = 2 | 0.5 |
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted? | 382 | 0.75 |
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. $5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice? | 36 | 0.625 |
For a natural number $n \ge 3$ , we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$ -gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$ ?
| 41 | 0.125 |
Let $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of $S$ ? | 6561 | 0.625 |
Determine all real values of $x$ for which
\[\frac{1}{\sqrt{x} + \sqrt{x - 2}} + \frac{1}{\sqrt{x} + \sqrt{x + 2}} = \frac{1}{4}.\] | \frac{257}{16} | 0.75 |
There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property:
We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph). | k = n | 0.25 |
For any positive integer $n$ , let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$ , where $[x]$ is the largest integer that is equal or less than $x$ . Determine the value of $a_{2015}$ . | 2015 | 0.5 |
Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$ , at least one of $x+y$ , $x+z$ , or $y+z$ is also contained in $S$ . Find the largest possible number of elements that $S$ could have. | 7 | 0.25 |
The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers? | 4 | 0.625 |
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3. | 669 | 0.625 |
Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$ ' and whose sides is colored by the three colors respectively. | n | 0.25 |
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$ . | 72 | 0.25 |
Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by $$ F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}. $$ Show that $F$ is injective and find the set of values of $F$ . | (\ln(2), \infty) | 0.25 |
The vertices of a regular $2012$ -gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$ , then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$ . How is the tenth vertex labeled?
*Proposed by A. Golovanov* | A_{28} | 0.375 |
Given a (fixed) positive integer $N$ , solve the functional equation
\[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\]
*(Dan Schwarz)* | f(a) = 0 | 0.75 |
There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$ , there is either a one way road from $A$ to $B$ , or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$ . Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country. | \frac{3}{2} | 0.125 |
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime. | 105 | 0.875 |
Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and
a) Product of all divisor of $sm, sn$ are equal.
b) Number of divisors of $sm,sn$ are equal. | (m, n) | 0.375 |
Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of distinct digits that could be used in the access code, calculate the number of possible combinations when the digits are known but their order and frequency are not known. For example, if there are smudges on $3$ and $9,$ two possible codes are $393939$ and $993999.$ Which number of distinct digits in the access code offers the most combinations? | 5 | 0.5 |
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$ ? | 1006 | 0.125 |
In the following diagram (not to scale), $A$ , $B$ , $C$ , $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$ . Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$ . Find $\angle OPQ$ in degrees.
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[/asy] | 30^\circ | 0.625 |
Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$ , and let $M = \left( \sqrt{3}+1 \right)x + 2y$ . Determine the maximum possible value of $M^2$ .
*Proposed by Varun Mohan* | 16104 | 0.75 |
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$ . What is the quotient when $a+b$ is divided by $ab$ ? | 1509 | 0.625 |
The function $g$ , with domain and real numbers, fulfills the following: $\bullet$ $g (x) \le x$ , for all real $x$ $\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$ Find $g (1990)$ . | 1990 | 0.625 |
Let $p$ be an odd prime of the form $p=4n+1$ . [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$ . [*] Calculate the value $n^{n}$ $\pmod{p}$ . [/list] | 1 | 0.875 |
Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$ , $3$ , $5$ , $7$ . | 9 | 0.375 |
Find the sum of all positive integers $n$ where the mean and median of $\{20, 42, 69, n\}$ are both integers.
*Proposed by bissue* | 45 | 0.625 |
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$ ? | 4 | 0.625 |
Let $G$ be the centroid of triangle $ABC$ with $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from $G$ to the three sides of the triangle.
Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.
*2019 CCA Math Bonanza Individual Round #11* | \frac{2348}{195} | 0.5 |
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
*Proposed by Brandon Wang* | 2019 | 0.875 |
For how many ordered pairs of positive integers $(x, y)$ is the least common multiple of $x$ and $y$ equal to $1{,}003{,}003{,}001$ ? | 343 | 0.125 |
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, y^3 \equal{} 12y\plus{}3z\minus{}2, z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
*Razvan Gelca.* | (2, 4, 6) | 0.75 |
$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:
Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell.
At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?
*Proposed by Shayan Dashmiz* | m^2 + mn + n^2 | 0.625 |
Let $ABCDE$ be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$ . Find angle $\angle DAE$ . | 30^\circ | 0.75 |
Find all triples $(p,q,n)$ that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where $p,q$ are odd primes and $n$ is an positive integer. | (3, 3, n) | 0.75 |
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$ , $OG = 1$ and $OG \parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.) | 12 | 0.5 |
An $n \times n$ complex matrix $A$ is called \emph{t-normal} if $AA^t = A^t A$ where $A^t$ is the transpose of $A$ . For each $n$ ,
determine the maximum dimension of a linear space of complex $n
\times n$ matrices consisting of t-normal matrices.
Proposed by Shachar Carmeli, Weizmann Institute of Science
| \frac{n(n+1)}{2} | 0.875 |
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$ . | 384 | 0.125 |
Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$ . | 216 | 0.5 |
Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$ , where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$ . Find $n$ .
*Proposed by **AOPS12142015*** | 56 | 0.75 |
A rectangular prism has three distinct faces of area $24$ , $30$ , and $32$ . The diagonals of each distinct face of the prism form sides of a triangle. What is the triangle’s area? | 25 | 0.625 |
In each cell of the table $4 \times 4$ , in which the lines are labeled with numbers $1,2,3,4$ , and columns with letters $a,b,c,d$ , one number is written: $0$ or $1$ . Such a table is called *valid* if there are exactly two units in each of its rows and in each column. Determine the number of *valid* tables. | 90 | 0.375 |
Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$ .
*Proposed by Kyle Lee* | 875 | 0.75 |
There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a *friend circle* to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles. | 864 | 0.125 |
A positive integer $N$ has $20$ digits when written in base $9$ and $13$ digits when written in base $27$ . How many digits does $N$ have when written in base $3$ ?
*Proposed by Aaron Lin* | 39 | 0.875 |
How many $6$ -digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit. | 3003 | 0.625 |
Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes. | 2002 | 0.75 |
A natural number $n$ is called *perfect* if it is equal to the sum of all its natural divisors other than $n$ . For example, the number $6$ is perfect because $6 = 1 + 2 + 3$ . Find all even perfect numbers that can be given as the sum of two cubes positive integers.
| 28 | 0.25 |
Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors. | 2004 | 0.875 |
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$ , compute the smallest possible value of $|P(0)|.$ | 25 | 0.75 |
Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle. Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent. | 30^\circ | 0.625 |
Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$ | (0, 1, 4) | 0.625 |
Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.
Find the minimum value of $n$ .
*Leng Gangsong* | 11 | 0.25 |
12. What is the sum of all possible $\left(\begin{array}{l}i j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$ , and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 1\end{array}\right)$ and $\left(\begin{array}{c}10 9\end{array}\right)$ . | 27633 | 0.25 |
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$ . Suppose that $AZ-AX=6$ , $BX-BZ=9$ , $AY=12$ , and $BY=5$ . Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$ , where $O$ is the midpoint of $AB$ .
*Proposed by Evan Chen* | 23 | 0.25 |
Convex pentagon $ABCDE$ has side lengths $AB=5$ , $BC=CD=DE=6$ , and $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$ .
| 60 | 0.625 |
[IMO 2007 HKTST 1](http://www.mathlinks.ro/Forum/viewtopic.php?t=107262)
Problem 1
Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$ . Find the maximum value of $3p+2q+r-4|s|$ . | -2\sqrt{2} | 0.875 |
In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies. | 2015 | 0.375 |
For each natural number $n\geq 2$ , solve the following system of equations in the integers $x_1, x_2, ..., x_n$ : $$ (n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n $$ where $$ S=x_1^2+x_2^2+\dots+x_n^2. $$ | x_i = 1 | 0.125 |
Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain?
(An *arithmetic progression* is a set of the form $\{a,a+d,\dots,a+kd\}$ , where $a,d,k$ are positive integers, and $k\geqslant 2$ ; thus an arithmetic progression has at least three elements, and successive elements have difference $d$ , called the *common difference* of the arithmetic progression.) | n^2 | 0.875 |
While taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the College Board's limiting rules, you realize that there are actually $32$ ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could mark $AB$ , or $AC$ , or $ABD$ , or even $ABCDE$ , or nothing at all!
You begin to wonder how many ways you could mark off the 10 questions you haven't yet answered. To increase the challenge, you wonder how many ways you could mark off the rest of your answer sheet without ever marking the same letter twice in a row. (For example, if $ABD$ is marked for one question, $AC$ cannot be marked for the next one because $A$ would be marked twice in a row.) If the number of ways to do this can be expressed in the form $2^m p^n$ , where $m,n > 1$ are integers and $p$ is a prime, compute $100m+n+p$ .
*Proposed by Alexander Dai* | 2013 | 0.25 |
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to
the other one. For each kid $A$ , among the kids to whom $A$ has sent a message, exactly $25 \% $ have sent a message to $A$ . How many possible two-digit values of $n$ are there?
*Proposed by Bulgaria* | 26 | 0.25 |
Let $a_0 = 1$ and define the sequence $\{a_n\}$ by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If $a_{2017}$ can be expressed in the form $a+b\sqrt{c}$ in simplest radical form, compute $a+b+c$ .
*2016 CCA Math Bonanza Lightning #3.2* | 4 | 0.625 |
Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .
(A *common external tangent* is a tangent line to two circles such that the circles are on the same side of the line, while a *common internal tangent* is a tangent line to two circles such that the circles are on opposite sides of the line).
*Proposed by Connor Gordon)* | 24 | 0.75 |
Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares *forbidden*, meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game.
What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.) | 2017 | 0.5 |
Evaluate $$ \frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6} $$ | 108 | 0.75 |
There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$ | 126 | 0.875 |
Let $m$ and $n$ be positive integers where $m$ has $d$ digits in base ten and $d\leq n$ . Find the sum of all the digits (in base ten) of the product $(10^n-1)m$ . | 9n | 0.125 |
A $3$ by $3$ determinant has three entries equal to $2$ , three entries equal to $5$ , and three entries equal to $8$ . Find the maximum possible value of the determinant. | 405 | 0.875 |
Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$ .) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$ | 2^k | 0.75 |
Denote by $P(n)$ the greatest prime divisor of $n$ . Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\] | n = 3 | 0.625 |
Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$ . Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\] | (n-1)^2 | 0.875 |
Yura put $2001$ coins of $1$ , $2$ or $3$ kopeykas in a row. It turned out that between any two $1$ -kopeyka coins there is at least one coin; between any two $2$ -kopeykas coins there are at least two coins; and between any two $3$ -kopeykas coins there are at least $3$ coins. How many $3$ -koyepkas coins could Yura put? | 501 | 0.625 |
Given a grid rectangle of size $2010 \times 1340$ . A grid point is called *fair* if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle?
 | 669 | 0.25 |
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\]
*2019 CCA Math Bonanza Team Round #4* | 7 | 0.25 |
As a reward for working for NIMO, Evan divides $100$ indivisible marbles among three of his volunteers: David, Justin, and Michael. (Of course, each volunteer must get at least one marble!) However, Evan knows that, in the middle of the night, Lewis will select a positive integer $n > 1$ and, for each volunteer, steal exactly $\frac 1n$ of his marbles (if possible, i.e. if $n$ divides the number of marbles). In how many ways can Evan distribute the $100$ marbles so that Lewis is unable to steal marbles from every volunteer, regardless of which $n$ he selects?
*Proposed by Jack Cornish* | 3540 | 0.875 |
The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$ , contains the number $0$ . For every other cell $C$ , we consider a route from $(1, 1)$ to $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$ . For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$ , the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$ , and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell $(2021, 2021)$ ?
| 5 | 0.5 |
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$ . | 2 | 0.625 |
A rectangular wooden block has a square top and bottom, its volume is $576$ , and the surface area of its vertical sides is $384$ . Find the sum of the lengths of all twelve of the edges of the block. | 112 | 0.875 |
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