Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.

Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$.

One ticket to a show costs $ \$20$ at full price. Susan buys 4 tickets using a coupon that gives her a $25\%$ discount. Pam buys 5 tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.

Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.

The absolute value of the difference of the solutions of the equation $x^2 + px + q = 0$, with $p, q \in R$, is equal to $4$. Find the solutions of the equation if it is known that $(q + 1) p^2 + q^2$ takes the minimum value.

Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers: \begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

Let $ABCDA'B'C'D'$ be a rectangular parallelepiped and $M,N, P$ projections of points $A, C$ and $B'$ respectively on the diagonal $BD'$. a) Prove that $BM + BN + BP = BD'$. b) Prove that $3 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2$ if and only if $ABCDA'B'C'D'$ is a cube.

An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.

Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after one initial move is applied to the sequence -- no matter what move -- there is always a way to continue with a finite sequence of moves so as to obtain in the end a constant sequence.

The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram: [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 3; /* # of vertical lines, including AB */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); } [/asy] For each $n \geq 2,$ find the number of existing parallelograms.

The sequence $ \{a_n\}$ is defined by \[ a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2. \]The sequence $ \{b_n\}$ is defined by \[ b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2. \]Find $ \frac {b_{32}}{a_{32}}$.

Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define $S_m=\displaystyle\sum_{\substack{i-j=m \\0 \le i,j \le n}} \dfrac{1}{2^{i+j}}$. Then \[\lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 +... + S_n^2 \right)\] can be expressed in the form $\dfrac{p}{q}$ for relatively prime positive integers $p,q$. Compute $100p+q$. [i]Proposed by Vincent Huang[/i]

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$. Let $H,O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$. Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$, respectively.

[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched to Allstate? [b]p2.[/b] The $7$-digit number $N$ can be written as $\underline{A} \,\, \underline{2} \,\,\underline{0} \,\,\underline{B} \,\,\underline{2} \,\, \underline{1} \,\,\underline{5}$. How many values of $N$ are divisible by $9$? [b]p3.[/b] The solutions to the equation $x^2-18x-115 = 0$ can be represented as $a$ and $b$. What is $a^2+2ab+b^2$? [b]p4.[/b] The exterior angles of a regular polygon measure to $4$ degrees. What is a third of the number of sides of this polygon? [b]p5.[/b] Charlie Brown is having a thanksgiving party. $\bullet$ He wants one turkey, with three different sizes to choose from. $\bullet$ He wants to have two or three vegetable dishes, when he can pick from Mashed Potatoes, Saut´eed Brussels Sprouts, Roasted Butternut Squash, Buttery Green Beans, and Sweet Yams; $\bullet$ He wants two desserts out of Pumpkin Pie, Apple Pie, Carrot Cake, and Cheesecake. How many different combinations of menus are there? [b]p6.[/b] In the diagram below, $\overline{AD} \cong \overline{CD}$ and $\vartriangle DAB$ is a right triangle with $\angle DAB = 90^o$. Given that the radius of the circle is $6$ and $m \angle ADC = 30^o$, if the length of minor arc $AB$ is written as $a\pi$, what is $a$? [img]https://cdn.artofproblemsolving.com/attachments/d/9/ea57032a30c16f4402886af086064261d6828b.png[/img] [b]p7.[/b] This Halloween, Owen and his two friends dressed up as guards from Squid Game. They needed to make three masks, which were black circles with a white equilateral triangle, circle, or square inscribed in their upper halves. Resourcefully, they used black paper circles with a radius of $5$ inches and white tape to create these masks. Ignoring the width of the tape, how much tape did they use? If the length can be expressed $a\sqrt{b}+c\sqrt{d}+ \frac{e}{f} \pi$ such that $b$ and $d$ are not divisible by the square of any prime, and $e$ and $f$ are relatively prime, find $a + b + c + d + e + f$. [img]https://cdn.artofproblemsolving.com/attachments/0/c/bafe3f9939bd5767ba5cf77a51031dd32bbbec.png[/img] [b]p8.[/b] Given $LCM (10^8, 8^{10}, n) = 20^{15}$, where $n$ is a positive integer, find the total number of possible values of $n$. [b]p9.[/b] If one can represent the infinite progression $\frac{1}{11} + \frac{2}{13} + \frac{3}{121} + \frac{4}{169} + \frac{5}{1331} + \frac{6}{2197}+ ...$ as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, what is $a$? [b]p10.[/b] Consider a tiled $3\times 3$ square without a center tile. How many ways are there to color the squares such that no two colored squares are adjacent (vertically or horizontally)? Consider rotations of an configuration to be the same, and consider the no-color configuration to be a coloring. [b]p11.[/b] Let $ABC$ be a triangle with $AB = 4$ and $AC = 7$. Let $AD$ be an angle bisector of triangle $ABC$. Point $M$ is on $AC$ such that $AD$ intersects $BM$ at point $P$, and $AP : PD = 3 : 1$. If the ratio $AM : MC$ can be expressed as $\frac{a}{b}$ such that $a$, $b$ are relatively prime positive integers, find $a + b$. [b]p12.[/b] For a positive integer $n$, define $f(n)$ as the number of positive integers less than or equal to $n$ that are coprime with $n$. For example, $f(9) = 6$ because $9$ does not have any common divisors with $1$, $2$, $4$, $5$, $7$, or $8$. Calculate: $$\sum^{100}_{i=2} \left( 29^{f(i)}\,\,\, mod \,\,i \right).$$ [b]p13.[/b] Let $ABC$ be an equilateral triangle. Let $P$ be a randomly selected point in the incircle of $ABC$. Find $a+b+c+d$ if the probability that $\angle BPC$ is acute can be expressed as $\frac{a\sqrt{b} -c\pi}{d\pi }$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c, d) = 1$ and $b$ is not divisible by the square of any prime. [b]p14.[/b] When the following expression is simplified by expanding then combining like terms, how many terms are in the resulting expression? $$(a + b + c + d)^{100} + (a + b - c - d)^{100}$$ [b]p15.[/b] Jerry has a rectangular box with integral side lengths. If $3$ units are added to each side of the box, the volume of the box is tripled. What is the largest possible volume of this box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Cole is trying to solve the [i]Collatz conjecture[/i]. She decides to make a model with a piece of wood with a hole for every natural number. For every even number there is a rope from $n$ to $\frac{n}{2}$ and for every odd number there is a rope from $n$ to $3n+1$. She wants to bring her model to a convention but in order to do that she needs to cut off the part containing the first $240$ holes. How many ropes did she break? [i]2017 CCA Math Bonanza Individual Round #4[/i]

We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.

In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.

Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$? $ \textbf{(A) }17\qquad\textbf{(B) }18\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21 $

Two perfect squares are [i]friends[/i] if one is obtained from the other adding the digit $1$ at the left. For instance, $1225 = 35^2$ and $225 = 15^2$ are friends. Prove that there are infinite pairs of odd perfect squares that are friends.

Inside an angle $\angle BOC$ there are three disjoint circles: $k_1,k_2$ and $k_3$, which are, each one, tangent to its sides $BO$ and $OC$. Let $r_1, r_2$ and $r_3$, respectively, be the radii of these circles, with $r_1<r_2<r_3$. The circles $k_1$ and $k_3$ are tangent to the side $OB$ at $A$ and $B$, respectively, and $k_2$ is tangent to the side $OC$ at $C$. Let $K=AC\cap k_1,L=AC\cap k_2,M=BC\cap k_2$ and $N=BC\cap k_3$. Besides that, let $P=AM\cap BK,Q=AM\cap BL,R=AN\cap BK$ and $S=AN\cap BL$. If the intersections of $CP,CQ,CR$ and $CS$ with $AB$ are $X,Y,Z$ and $T$, respectively, prove that $XZ = YT$.