Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$. An estimate $E$ earns $\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.2[/i]

Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $\angle KPL$.

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

When the speed of a rear-drive car is increasing on a horizontal road, the direction of the frictional force on the tires is: $ \textbf {(A) } \text {backward on the front tires and forward on the rear tires.} $ $ \textbf {(B) } \text {forward on the front tires and backward on the rear tires.} $ $ \textbf {(C) } \text {forward on all tires.} $ $ \textbf {(D) } \text {backward on all tires.} $ $ \textbf {(E) } \text {zero.} $

The rectangle in the figure has dimensions $16$ x $20$ and is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles?

Let $ A \equal{} (a_{ik})$ be an $ n\times n$ matrix with nonnegative elements such that $ \sum_{k \equal{} 1}^n a_{ik} \equal{} 1$ for $ i \equal{} 1,...,n$. Show that, for every eigenvalue $ \lambda$ of $ A$, either $ |\lambda| < 1$ or there exists a positive integer $ k$ such that $ \lambda^k \equal{} 1$

Two positive integers $r$ and $k$ are given as is an infinite sequence of positive integers $a_1 \le a_2 \le a_3 \le ..$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a positive integer $t$ such that $\frac{t}{a_t}= k$.

Let $x; y; m; n$ be integers greater than $1$ such that

A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is: $\textbf{(A)}\ \text{Least when the point is the center of gravity of the triangle}\qquad\\ \textbf{(B)}\ \text{Greater than the altitude of the triangle} \qquad\\ \textbf{(C)}\ \text{Equal to the altitude of the triangle}\qquad\\ \textbf{(D)}\ \text{One-half the sum of the sides of the triangle} \qquad\\ \textbf{(E)}\ \text{Greatest when the point is the center of gravity}$

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens. Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that $\Pi$ is a square.

[b]p1.[/b] There are $2000$ cans of paint. Show that at least one of the following two statements must be true. There are at least $45$ cans of the same color. There are at least $45$ cans all of different colors. [b]p2.[/b] The measures of the $3$ angles of one triangle are all different from each other but are the same as the measures of the $3$ angles of a second triangle. The lengths of $2$ sides of the first triangle are different from each other but are the same as the lengths of $2$ sides of the second triangle. Must the length of the remaining side of the first triangle be the same as the length of the remaining side of the second triangle? If yes, prove it. If not, provide an example. [b]p3.[/b] Consider the sequence $a_1=1$, $a_2=2$, $a_3=5/2$, ... satisfying $a_{n+1}=a_n+(a_n)^{-1}$ for $n>1$. Show that $a_{10000}>141$. [b]p4.[/b] Prove that no matter how $250$ points are placed in a disk of radius $1$, there is a disk of radius $1/10$ that contains at least $3$ of the points. [b]p5.[/b] Prove that: Given any $11$ integers (not necessarily distinct), one can select $6$ of them so that their sum is divisible by $6$. Given any $71$ integers (not necessarily distinct), one can select $36$ of them so that their sum is divisible by $36$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.

Let $ABC$ be an acute triangle. Let $I$ be a point inside the triangle and let $D$ be a point on the line $AB$. The line through $D$ which is parallel to $AI$ intersects the line $AC$ at the point $E$, and the line through $D$ parallel to $BI$ intersects the line $BC$ in point $F$. prove that $$\frac{EF \cdot CI}{2} \ge area (\vartriangle ABC) $$

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$. (a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$. (b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$. (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.

Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$? $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 165 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 215 $

In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

We consider an equilateral triangle with its circumscribed circle, of center $O$, and radius $4$cm. We rotate the triangle $90º$ around $O$. Compute the common area that was covered by the previous position of the triangle and is also covered by the new one.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \] [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.

How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6? $\textbf{(A)}\text{ 1}\qquad\textbf{(B)}\text{ 2}\qquad\textbf{(C)}\text{ 3}\qquad\textbf{(D)}\text{ 5}\qquad\textbf{(E)}\text{ 6}$

Which of the following is the set of all perfect squares that can be written as sum of three odd composite numbers? $\textbf{a)}\ \{(2k + 1)^2 : k \geq  0\}$ $\textbf{b)}\ \{(4k + 3)^2 : k \geq  1\}$ $\textbf{c)}\ \{(2k + 1)^2 : k \geq  3\}$ $\textbf{d)}\ \{(4k + 1)^2 : k \geq 2\}$ $\textbf{e)}\ \text{None of above}$

Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0.$$