Fourty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat? $\textbf{(A) }27\qquad\textbf{(B) }37\qquad\textbf{(C) }47\qquad\textbf{(D) }57\qquad\textbf{(E) }67$

Let $f(a)$ be the largest integer less than or equal to the fourth root of " $a$". Calculate $$f(1)+f(2)+...+f(2005).$$

Let $G$ be a finite abelian group and $f :\mathbb{Z}^+ \to G$ a completely multiplicative function (i.e. $f(mn) = f(m)f(n)$ for any positive integers $m, n$). Prove that there are infinitely many positive integers $k$ such that $f(k) = f(k + 1).$

If $ |x| + x + y = 10$ and $x + |y| - y = 12$, find $x + y$. $ \textbf{(A)}\ -2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{18}{5}\qquad\textbf{(D)}\ \frac{22}{3}\qquad\textbf{(E)}\ 22 $

$N$ straight lines are drawn on a plane. The $N$ lines can be partitioned into set of lines such that if a line $l$ belongs to a partition set then all lines parallel to $l$ make up the rest of that set. For each $n>=1$,let $a_n$ denote the number of partition sets of size $n$. Now that $N$ lines intersect at certain points on the plane. For each $n>=2$ let $b_n$ denote the number of points that are intersection of exactly $n$ lines. Show that $\sum_{n>= 2}(a_n+b_n)$$\binom{n}{2}$ $=$ $\binom{N}{2}$

Let $f (x) = x^2 + px + q$, where $p, q$ are integers. Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$.

Find the pairs of functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ with $ f $ continuous, $ g $ differentiable and satisfying: $$ -\sin g(x) + \int \cos f(x)dx =\cos g(x) +\int \sin f(x)dx $$

Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.

Find all values of $a$ such that the two polynomials \[x^2+ax-1\qquad\text{and}\qquad x^2-x+a\] share at least 1 root. [i]2018 CCA Math Bonanza Individual Round #7[/i]

Suppose $h$, $i$, $o$ are real numbers that satisfy the products $hi = 12$, $ooh = 18$, and $hohoho = 27$. Find the value of the product $ohio$.

Find the locus of points $P$ in an equilateral triangle $ABC$ for which the square root of the distance of $P$ to one of the sides is equal to the sum of the square root of the distance of $P$ to the two other sides.

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

What conditions must the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation $$ x^3 + ax^2 + bx + c = 0$$ has three distinct real roots forming a geometric progression?

Let $ABCD$ be a cyclic quadrilateral. The bisectors of angles $BAD$ and $BCD$ intersect in point $K$ such that $K \in BD$. Let $M$ be the midpoint of $BD$. A line passing through point $C$ and parallel to $AD$ intersects $AM$ in point $P$. Prove that triangle $\triangle DPC$ is isosceles.

On the median $AD$ of the isosceles triangle $ABC$, point $E$ is marked. Point $F$ is the projection of point $E$ on the line $BC$, point $M$ lies on the segment $EF$, points $N$ and $P$ are projections of point $M$ on the lines $AC$ and $AB$, respectively. Prove that the bisectors of the angles $PMN$ and $PEN$ are parallel.

A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\omega$ in $A$ and $Q$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$, at $P$. The point $L$ lies on $\omega$, and is such that $PL$ is tangent to $\omega$ and $L\neq Q$. Show that $K, L$, and $M$ are collinear.

Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have \[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]

Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

Let $n$ be a positive integer. Assume that $n$ numbers are to be chosen from the table $\begin{array}{cccc}0 & 1 & \cdots & n-1\\ n & n+1 & \cdots & 2n-1\\ \vdots & \vdots & \ddots & \vdots\\(n-1)n & (n-1)n+1 & \cdots & n^2-1\end{array} $ with no two of them from the same row or the same column. Find the maximal value of the product of these $n$ numbers.

Let $A,B,C$ be points in that order along a line, such that $AB=20$ and $BC=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY\|BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?

Let $ABC$ be an acute, scalene triangle with circumcenter $O$, circumcircle $(O)$, orthocenter $H$. Line $AH$ meets $(O)$ again at $D \neq A$. Let $E, F$ be the midpoint of segments $AB, AC$ respectively. The line through $H$ and perpendicular to $HF$ meets line $BC$ at $K$. a) Line $DK$ meets $(O)$ again at $Y \neq D$. Prove that the intersection of line $BY$ and the perpendicular bisector of $BK$ lies on the circumcircle of triangle $OFY$. b) The line through $H$ and perpendicular to $HE$ meets line $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z \neq D$. Let $M$ be the intersection of lines $BZ, OE$; $N$ be the intersection of lines $CY, OF$; $P$ be the intersection of lines $BY, CZ$. Let $T$ be the intersection of lines $YZ, MN$ and $d$ be the line through $T$ and perpendicular to $OA$. Prove that $d$ bisects $AP$.

Let $ABC$ be an actue triangle with $AB<AC$. Let $\Gamma$ be its circumcircle, $I$ its incenter and $P$ is a point on $\Gamma$ such that $\angle API=90^{\circ}$. Let $Q$ be a point on $\Gamma$ such that $$QB\cdot\tan \angle B=QC\cdot \tan \angle C$$ Consider a point $R$ such that $PR$ is tangent to $\Gamma$ and $BR=CR$. Prove that the points $A, Q, R$ are colinear.

For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}

In $\triangle ABC$, $AB>AC$, $l$ is tangent line of the circumscribed circle of $\triangle ABC$ that passes $A$. The circle with center $A$ and radius $AC$, intersects segment $AB$ at $D$, and line $l$ at $E, F$ ($F,B$ are on the same side). Prove that lines $DE, DF$ pass the incenter and an excenter of $\triangle ABC$ respectively.

The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3)); pair[] ps={A,B,C,D,E}; draw(A--B--C--cycle); draw(E--D); draw(rightanglemark(E,D,B)); dot(ps); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE); label("$3$",midpoint(A--C),NW); label("$4$",midpoint(C--B),NE); label("$5$",midpoint(A--B),SW);[/asy] $ \textbf{(A)}\ \frac{4}{3} \qquad \textbf{(B)}\ \sqrt{5} \qquad \textbf{(C)}\ \frac{9}{4} \qquad \textbf{(D)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(E)}\ \frac{5}{2} $