Found problems: 85335
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if
$\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} x < 4$
Prove that for every pair of positive integers $k$ and $n$, there exists integer $x_1$, $x_2$,$...$, $x_k$ with $0 \le x_j \le 2^{k-1}\cdot \sqrt[k]{n}$ for $j = 1$, $2$, $...$, $k$, and such that $$x_1 + x^2_2+ x^3_3+...+ x^k_k= n.$$
Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules:
[list=disc]
[*]if more than one chests are unlocked, it locks one of them, or
[*]if there is only one unlocked chest, it unlocks all the chests.
[/list]
Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.
Let $P$ and $Q$ be points on the plane $ABC$ such that $m(\widehat{BAC})=90^\circ$, $|AB|=1$, $|AC|=\sqrt 2$, $|PB|=1=|QB|$, $|PC|=2=|QC|$, and $|PA|>|QA|$. What is $|PA|/|QA|$?
$ \textbf{(A)}\ \sqrt 2 +\sqrt 3
\qquad\textbf{(B)}\ 5-\sqrt 6
\qquad\textbf{(C)}\ \sqrt 6 -\sqrt 2
\qquad\textbf{(D)}\ \sqrt 6 + 1
\qquad\textbf{(E)}\ \text{None}
$
Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$
$\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$.
Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?
$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.
In $\triangle ABC$, $c-a$ is equal to height on side $AC$. Then, the value of $\sin\frac{C-A}{2}+\cos\frac{C+A}{2}$ is
$\text{(A)}1\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}-1$
$f$ is a continuous real-valued function such that $f(x+y)=f(x)f(y)$ for all real $x$, $y$. If $f(2)=5$, find $f(5)$.
Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
Find the ratio $L/d$.
$ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $
Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set
$$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$
Prove that
$$ \prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$
There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
Show that no one $n$-th root of a rational (for $n$ a positive integer) can be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$.
A frictionless roller coaster ride is given a certain velocity at the start of the ride. At which point in the diagram is the velocity of the cart the greatest? Assume a frictionless surface.
[asy]pair A = (1.7,3.9);
pair B = (3.2,2.7);
pair C = (5,1.2);
pair D = (8,2.7);
size(8cm);
path boundary = (0,0.5)--(8,0.5)--(8,5)--(0,5)--cycle;
path track = (0,3.2)..A..(3,3)..B..(4,1.8)..C..(6,1.5)..(7,2.3)..D;
path sky = (0,5)--track--(8,5)--cycle;
for (int a=0; a<=8; ++a) { draw((a,0)--(a,5), black+1); }
for (int a=0; a<=5; ++a) { draw((0,a)--(8,a), black+1); }
for (int a=-100; a<=100; ++a) { draw((0,a)--(8,a+8)); }
for (int a=-100; a<=100; ++a) { draw((8,a)--(0,a+8)); }
fill(sky,white);
draw(track, black+3);
clip(boundary);
label("$A$", A, dir(120));
label("$B$", B, dir(60));
label("$C$", C, dir(90));
label("$D$", D, dir(135));[/asy]
$ \textbf {(A) } \text {A} \qquad \textbf {(B) } \text {B} \qquad \textbf {(C) } \text {C} \qquad \textbf {(D) } \text {D} \\ \textbf {(E) } \text {There is insufficient information to decide} $
[i]Problem proposed by Kimberly Geddes[/i]
Triangle $\triangle ABC$ has $AB= 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $\triangle BMC$ at points $X$ and $Y$, then $XY^2 = \frac{m}{n}$ for some relatively prime positive integers $m,n$. Find $m+n$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i]
Determine all integers $n>1$ such that every power of $n$ has an odd number of digits.
If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$.
[i]Proposed by Muztaba Syed[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{9}$
$a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$.
[/hide]
On January 15 in the stormy town of Stormville, there is a $50\%$ chance of rain. Every day, the probability of it raining has a $50\%$ chance of being $\frac{2017}{2016}$ times that of the previous day (or $100\%$ if this new quantity is over $100\%$) and a $50\%$ chance of being $\frac{1007}{2016}$ times that of the previous day. What is the probability that it rains on January 20?
[i]2018 CCA Math Bonanza Lightning Round #3.3[/i]
There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board.
$a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board
$b)$ For such $n$, find numbers written on the board at the beginning
Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.
Show that the number $2017^{2016}-2016^{2017}$ is divisible by $5$ .