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Let be given an integer $n\ge 2$ and a positive real number $p$ . Find the maximum of
\[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\]
where $x_i$ are non-negative real numbers with sum $p$ . | \frac{p^2}{4} | 0.875 |
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$ ? | 7 | 0.875 |
Let $p=4k+1$ be a prime. $S$ is a set of all possible residues equal or smaller then $2k$ when $\frac{1}{2} \binom{2k}{k} n^k$ is divided by $p$ . Show that \[ \sum_{x \in S} x^2 =p \] | p | 0.5 |
**i.)** Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$ ?
**ii.)** If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of
\[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number $y.$
**iii.)** The function $f$ satisfies the functional equation $f(x) + f(y) = f(x+y) - x \cdot y - 1$ for every pair $x,y$ of real numbers. If $f(1) = 1,$ then find the numbers of integers $n,$ for which $f(n) = n.$ | 2 | 0.875 |
There's a real number written on every field of a $n \times n$ chess board. The sum of all numbers of a "cross" (union of a line and a column) is $\geq a$ . What's the smallest possible sum of all numbers on the board¿ | \frac{n^2 a}{2n-1} | 0.875 |
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$ , there exist 3 numbers $a$ , $b$ , $c$ among them satisfying $abc=m$ . | 11 | 0.25 |
Let $R$ be the region in the Cartesian plane of points $(x,y)$ satisfying $x\geq 0$ , $y\geq 0$ , and $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$ . Determine the area of $R$ . | \frac{9}{2} | 0.625 |
How many binary strings of length $10$ do not contain the substrings $101$ or $010$ ? | 178 | 0.5 |
In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event? | 43890 | 0.125 |
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides? | 30 | 0.5 |
We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron? | 20 | 0.125 |
Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$ .
*Proposed by Alex Li* | 2 | 0.875 |
Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . Points $P$ , $Q$ , and $R$ are chosen on segments $BC$ , $CA$ , and $AB$ , respectively, such that triangles $AQR$ , $BPR$ , $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$ . What is the perimeter of $PQR$ ?
*2021 CCA Math Bonanza Individual Round #2* | 30 | 0.875 |
For how many integers $n$ with $1 \le n \le 2012$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero? | 335 | 0.375 |
Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$ . | 588 | 0.875 |
Kira has $3$ blocks with the letter $A$ , $3$ blocks with the letter $B$ , and $3$ blocks with the letter $C$ . She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have distances $1, 3$ , and $4$ between one another, the blocks with the letter $B$ have distances $2, 4$ , and $6$ between one another, and the blocks with the letter $C$ have distances $2, 4$ , and $6$ between one another. Altogether, we got distances of $1, 2, 3, 4$ , and $6$ ; these are $5$ distinct distances. What is the maximum number of distinct distances that can occur? | 7 | 0.125 |
Let $S$ be a set of $n$ distinct real numbers. Let $A_{S}$ be the set of numbers that occur as averages of two distinct
elements of $S$ . For a given $n \geq 2$ , what is the smallest possible number of elements in $A_{S}$ ? | 2n-3 | 0.75 |
The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$ . | -24 | 0.625 |
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | 105263157894736842 | 0.375 |
Let $ABF$ be a right-angled triangle with $\angle AFB = 90$ , a square $ABCD$ is externally to the triangle. If $FA = 6$ , $FB = 8$ and $E$ is the circumcenter of the square $ABCD$ , determine the value of $EF$ | 7 \sqrt{2} | 0.75 |
The integers $1, 2, \dots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$ , $345$ , $128$ , $19$ , and $9.5$ in some order. Compute $n$ .
*Proposed by Evan Chen* | 2014 | 0.25 |
A $39$ -tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers $a$ , $b$ (these maxima aren't necessarily achieved using the same tuple of real numbers). Find $a + b$ .
*Proposed by Evan Chang* | 37 | 0.125 |
Marisa has a collection of $2^8-1=255$ distinct nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^8-2=254$ times until there is only one set left in the collection. What is the expected size of this set?
| \frac{1024}{255} | 0.875 |
Let $ABC$ be a right-angled triangle ( $\angle C = 90^\circ$ ) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$ , respectively, meet at point $F$ . Find the ratio $S_{ABF}:S_{ABC}$ .
Note: $S_{\alpha}$ means the area of $\alpha$ . | \frac{4}{3} | 0.125 |
Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$ ? | 120 | 0.875 |
For any positive integer $n$ , and $i=1,2$ , let $f_{i}(n)$ denote the number of divisors of $n$ of the form $3 k+i$ (including $1$ and $n$ ). Define, for any positive integer $n$ , $$ f(n)=f_{1}(n)-f_{2}(n) $$ Find the value of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$ . | f(5^{2022}) = 1 | 0.125 |
A right circular cone has base radius $ r$ and height $ h$ . The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$ , where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$ . | 14 | 0.875 |
Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$ . Find $b - a$ . | 223 | 0.75 |
On Binary Island, residents communicate using special paper. Each piece of paper is a $1 \times n$ row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has $10$ random consecutive squares each of which is randomly colored red or green.
Malmer and Weven would like to develop a scheme that allows them to send messages of length $2016$ between one another. They would like to be able to send any message of length $2016$ , and they want their scheme to work with perfect accuracy. What is the smallest value of $n$ for which they can develop such a strategy?
*Note that when sending a message, one can see which 10 squares are colored and what colors they are. One also knows on which square the message begins, and on which square the message ends.*
| 2026 | 0.125 |
Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$ , where $m$ and $n$ are nonnegative integers. If $1776 $ is one of the numbers that is not expressible, find $a + b$ . | 133 | 0.75 |
A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? | 13 | 0.875 |
The side lengths of a scalene triangle are roots of the polynomial $$ x^3-20x^2+131x-281.3. $$ Find the square of the area of the triangle. | 287 | 0.625 |
Equilateral triangle $ABC$ has side length $6$ . Circles with centers at $A$ , $B$ , and $C$ are drawn such that their respective radii $r_A$ , $r_B$ , and $r_C$ form an arithmetic sequence with $r_A<r_B<r_C$ . If the shortest distance between circles $A$ and $B$ is $3.5$ , and the shortest distance between circles $A$ and $C$ is $3$ , then what is the area of the shaded region? Express your answer in terms of pi.
[asy]
size(8cm);
draw((0,0)--(6,0)--6*dir(60)--cycle);
draw(circle((0,0),1));
draw(circle(6*dir(60),1.5));
draw(circle((6,0),2));
filldraw((0,0)--arc((0,0),1,0,60)--cycle, grey);
filldraw(6*dir(60)--arc(6*dir(60),1.5,240,300)--cycle, grey);
filldraw((6,0)--arc((6,0),2,120,180)--cycle, grey);
label(" $A$ ",(0,0),SW);
label(" $B$ ",6*dir(60),N);
label(" $C$ ",(6,0),SE);
[/asy] | \frac{29\pi}{24} | 0.875 |
Let $ABCD$ be a trapezoid such that $|AC|=8$ , $|BD|=6$ , and $AD \parallel BC$ . Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$ , respectively. If $|PS|=5$ , find the area of the trapezoid $ABCD$ . | 24 | 0.875 |
It is known that, for all positive integers $k,$
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$ | 112 | 0.75 |
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
*Proposed by Evan Chen* | 4 | 0.875 |
You have a circular necklace with $10$ beads on it, all of which are initially unpainted. You randomly select $5$ of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted? | \frac{17}{42} | 0.125 |
Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$ .
Titu Andreescu, Mathematics Department, College of Texas, USA | (2, 3, 5) | 0.25 |
A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$ . Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer. | 17 | 0.625 |
Toner Drum and Celery Hilton are both running for president. A total of $129$ million people cast their vote in a random order, with exactly $63$ million and $66$ million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page?
*2017 CCA Math Bonanza Individual Round #13* | \frac{1}{43} | 0.375 |
Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an
integer. Find the smallest $k$ for which the set A can be partitioned into two subsets
having the same number of elements, the same sum of elements, the same sum
of the squares of elements, and the same sum of the cubes of elements. | k = 4 | 0.375 |
Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$ , chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$ , $b$ , and $c$ . As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach? | \frac{11}{18} | 0.125 |
Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$ . Determine all $n<2001$ with the property that $d_9-d_8=22$ . | n = 1995 | 0.25 |
A rectangular box with side lengths $1$ , $2$ , and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes.
*2020 CCA Math Bonanza Lightning Round #2.2* | 258 | 0.625 |
For a given integer $n\ge 2$ , let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$ . Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$ . | 3n | 0.75 |
Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$ ). What is the maximal possible number of different integers on the blackboard? | 2 | 0.875 |
Find all real values of the parameter $a$ for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these values of $a$ . | a = 33 | 0.875 |
Find all positive integers $n$ such that it is possible to split the numbers from $1$ to $2n$ in two groups $(a_1,a_2,..,a_n)$ , $(b_1,b_2,...,b_n)$ in such a way that $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$ .
*Proposed by Alef Pineda* | n = 1 | 0.25 |
Let $S$ be the set of integers of the form $2^x+2^y+2^z$ , where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$ th smallest element of $S$ . | 577 | 0.375 |
Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$ , \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$ .
*Pakawut Jiradilok* | f(n!) \ge n! | 0.75 |
Hello all. Post your solutions below.**Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.**
4/1/31. A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and
no one else has mangos. The friends split the mangos according to the following rules:
• sharing: to share, a friend passes two mangos to the left and one mango to the right.
• eating: the mangos must also be eaten and enjoyed. However, no friend wants to be
selfish and eat too many mangos. Every time a person eats a mango, they must also
pass another mango to the right.
A person may only share if they have at least three mangos, and they may only eat if they
have at least two mangos. The friends continue sharing and eating, until so many mangos
have been eaten that no one is able to share or eat anymore.
Show that there are exactly eight people stuck with mangos, which can no longer be
shared or eaten. | 8 | 0.375 |
Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5. | 1350 | 0.125 |
Let $\phi(n,m), m \neq 1$ , be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$ , where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality:
\[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\]
for every positive integer $n.$ | m | 0.375 |
Consider three points $P(0,-\sqrt{2}),\ Q(0,\ \sqrt{2}),\ A(a,\ \sqrt{a^2+1})\ (0\leq a\leq 1)$ .
(1) Show that the difference of two line segments $PA-AQ$ is constant regardless of $a$ , then find the value.
(2) Let $B$ be the point of intersection between the half-line passing through $A$ with the end point $Q$ and the parabola $y=\frac{\sqrt{2}}{8}x^2$ , and let $C$ be the point of intersection between the perpendicular line drawn from the point $B$ to the line $y=2$ .
Show that the sum of the line segments $PA+AB+BC$ is constant regardless of $a$ , then find the value. | 4 + \sqrt{2} | 0.875 |
Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$ . Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$ . Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$ . | 30^\circ | 0.625 |
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length. | n = 13 | 0.375 |
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$ . Find the remainder of $A$ divided by $3$ . | 0 | 0.375 |
Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$ , for any $n\ge 1$ . Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit. | 1 | 0.875 |
Find the maximum value of real number $k$ such that
\[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\]
holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$ . | k = 4 | 0.875 |
An integer is between $0$ and $999999$ (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that the sum will be $19$ ? | \frac{7623}{250000} | 0.5 |
$\triangle ABC$ has area $240$ . Points $X, Y, Z$ lie on sides $AB$ , $BC$ , and $CA$ , respectively. Given that $\frac{AX}{BX} = 3$ , $\frac{BY}{CY} = 4$ , and $\frac{CZ}{AZ} = 5$ , find the area of $\triangle XYZ$ .
[asy]
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defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label(" $A$ ",A,N);
label(" $B$ ",B,S);
label(" $C$ ",C,E);
label(" $X$ ",X,W);
label(" $Y$ ",Y,S);
label(" $Z$ ",Z,NE);[/asy] | 122 | 0.625 |
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)? | 20 | 0.5 |
Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$ . Find the maximum possible value of $AB$ . | 69 | 0.125 |
A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices. | 333 | 0.75 |
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$ , we color red the foot of the perpendicular from $C$ to $\ell$ . The set of red points enclose a bounded region of area $\mathcal{A}$ . Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$ ).
*Proposed by Yang Liu* | 157 | 0.25 |
Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that $e$ can take. | 27.5 | 0.875 |
Let $ \overline{AB}$ be a diameter of circle $ \omega$ . Extend $ \overline{AB}$ through $ A$ to $ C$ . Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$ . Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$ . Suppose $ AB \equal{} 18$ , and let $ m$ denote the maximum possible length of segment $ BP$ . Find $ m^{2}$ . | 432 | 0.125 |
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person. | 216 | 0.5 |
In a room, each person is an painter and/or a musician. $2$ percent of the painters are musicians, and $5$ percent of the musicians are painters. Only one person is both an painter and a musician. How many people are in the room?
*Proposed by Evan Chang* | 69 | 0.875 |
Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x\neq y$ , we have $xy\notin A$ . Determine the maximum possible size of $A$ . | 999001 | 0.5 |
Consider all $6$ -digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$ -digit numbers that are divisible by $7$ . | 70 | 0.375 |
The sum of the (decimal) digits of a natural number $n$ equals $100$ , and the sum of digits of $44n$ equals $800$ . Determine the sum of digits of $3n$ . | 300 | 0.75 |
$a$ , $b$ , $c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$ | \frac{17}{3} | 0.75 |
Define $p(n)$ to be th product of all non-zero digits of $n$ . For instance $p(5)=5$ , $p(27)=14$ , $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\] | 103 | 0.125 |
For all positive integers $n$ , let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$ . Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$ . | 11431 | 0.125 |
Find all positive integers that can be represented as $$ \frac{abc+ab+a}{abc+bc+c} $$ for some positive integers $a, b, c$ .
*Proposed by Oleksii Masalitin* | 1 \text{ and } 2 | 0.875 |
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$ . Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of $p_{2}+2p_{1}-3p_{0}.$ | 18 | 0.375 |
Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$ . The square $BDEF$ is inscribed in $\triangle ABC$ , such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$ , respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the square $BDEF$ , such that the radius of $\omega_1$ and $\omega_3$ are both equal to $b$ and the radius of $\omega_2$ and $\omega_4$ are both equal to $a$ . The circle $\omega_1$ is tangent to $ED$ , the circle $\omega_3$ is tangent to $BF$ , $\omega_2$ is tangent to $EF$ and $\omega_4$ is tangent to $BD$ , each one of these circles are tangent to the two closest circles and the circles $\omega_1$ and $\omega_3$ are tangents. Determine the ratio $\frac{c}{a}$ . | 2 | 0.5 |
Find the maximal number of edges a connected graph $G$ with $n$ vertices may have, so that after deleting an arbitrary cycle, $G$ is not connected anymore. | 2n-3 | 0.25 |
The cost of five water bottles is \ $13, rounded to the nearest dollar, and the cost of six water bottles is \$ 16, also rounded to the nearest dollar. If all water bottles cost the same integer number of cents, compute the number of possible values for the cost of a water bottle.
*Proposed by Eugene Chen* | 11 | 0.875 |
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$ .
Determine $f(2014)$ . | 2014 | 0.75 |
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$ . | 7 | 0.75 |
Find the largest $n$ such that the last nonzero digit of $n!$ is $1$ .
| 1 | 0.5 |
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$ . When $a$ , $b$ , $c$ run over all the real numbers, find the smallest possible value of $F$ . | \frac{1}{4} | 0.375 |
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
*(A. Golovanov)* | 1 | 0.125 |
Let $p$ be an odd prime number. Let $g$ be a primitive root of unity modulo $p$ . Find all the values of $p$ such that the sets $A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}$ and $B=\left\{g^m:1\le m\le\frac{p-1}2\right\}$ are equal modulo $p$ . | p = 3 | 0.75 |
Patrick tosses four four-sided dice, each numbered $1$ through $4$ . What's the probability their product is a multiple of four? | \frac{13}{16} | 0.75 |
Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$ . Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$ , he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$ ). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .
*Proposed by Ray Li* | 213 | 0.375 |
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $\$ 1, \ $ 2,\$ 5 $, and $ \ $ 10$ bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $\$ 1{,}000{,}000 $, and whoever makes the last move wins the $ \ $1{,}000{,}000$ in the pile (if the pile is worth more than $\$ 1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?
*Proposed by Yannick Yao* | 1,000,000 | 0.875 |
Anna and Boris move simultaneously towards each other, from points A and B respectively. Their speeds are constant, but not necessarily equal. Had Anna started 30 minutes earlier, they would have met 2 kilometers nearer to B. Had Boris started 30 minutes earlier instead, they would have met some distance nearer to A. Can this distance be uniquely determined?
*(3 points)* | 2 \text{ km} | 0.75 |
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$ ? | 10071 | 0.75 |
Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$ . Find the leading coefficient of $g(\alpha)$ . | \frac{1}{16} | 0.5 |
Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$ 's column and strictly less than the average of $c$ 's row.
*Proposed by Holden Mui* | (n-1)^2 | 0.125 |
In a triangle $ABC, L$ and $K$ are the points of intersections of the angle bisectors of $\angle ABC$ and $\angle BAC$ with the segments $AC$ and $BC$ , respectively. The segment $KL$ is angle bisector of $\angle AKC$ , determine $\angle BAC$ . | 120^\circ | 0.125 |
A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
| 438 | 0.375 |
What is the sum of all distinct values of $x$ that satisfy $x^4-x^3-7x^2+13x-6=0$ ?
*2018 CCA Math Bonanza Lightning Round #1.4* | 0 | 0.75 |
Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $k$ for which it is possible that Emilio can do $k$ conveniently thought steps in such a way that with the received information, he can determine the area of each one of the $n^2$ smaller rectangles.
| 2n - 1 | 0.75 |
Consider a standard ( $8$ -by- $8$ ) chessboard. Bishops are only allowed to attack pieces that are along the same diagonal as them (but cannot attack along a row or column). If a piece can attack another piece, we say that the pieces threaten each other. How many bishops can you place a chessboard without any of them threatening each other? | 14 | 0.375 |
For all positive integers $n > 1$ , let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$ , compute
\[\frac{f(N)}{f(f(f(N)))}.\] | 25 | 0.875 |
In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$ . Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$ . Determine $AB$ | \sqrt{10} | 0.875 |
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