[b]Problem Section #4 a) There is a $6 * 6$ grid, each square filled with a grasshopper. After the bell rings, each grasshopper jumps to an adjacent square (A square that shares a side). What is the maximum number of empty squares possible?

Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

The current in a river is flowing steadily at $3$ miles per hour. A motor boat which travels at a constant rate in still water goes downstream $4$ miles and then returns to its starting point. The trip takes one hour, excluding the time spent in turning the boat around. The ratio of the downstream to the upstream rate is $\textbf{(A) }4:3\qquad\textbf{(B) }3:2\qquad\textbf{(C) }5:3\qquad\textbf{(D) }2:1\qquad \textbf{(E) }5:2$

Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$? [i]2021 CCA Math Bonanza Lightning Round #3.1[/i]

A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

Let $x, y, z, a,b, c$ are pairwise different integers from the set $\{1,2,3, 4,5,6\}$. Find the smallest possible value for expression $xyz + abc - ax - by - cz$.

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

Carlos and Yue play the following game: First Carlos writes a $+$ sign or a $-$ sign in front of each of the $50$ numbers $1,2,\cdots,50$. Then, in turns, each one chooses a number from the sequence obtained; Start by choosing Yue. If the absolute value of the sum of the $25$ numbers that Carlos chose is greater than or equal to the absolute value of the sum of the $25$ numbers that Yue chose, Carlos wins. In the other case, Yue wins. Determine which of the two players can develop a strategy that will ensure victory, no matter how well their opponent plays, and describe said strategy.

The numbers $1,2,...,1970$ are written on a board. One is allowed to remove $2$ numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.

Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.

(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$. (b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class? $\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$

On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column? [i]Proposed by Vlad Spătaru[/i]

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 $f(8)$.

Determine all positive integers $n$ such that for every positive devisor $ d $ of $n$, $d+1$ is devisor of $n+1$.

$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$? $ \textbf{(A)} \ 55 \qquad \textbf{(B)} \ 56 \qquad \textbf{(C)} \ 57 \qquad \textbf{(D)} \ 58 \qquad \textbf{(E)} \ 59$

Dima wrote several natural numbers on the blackboard and underlined some of them. Misha wants to erase several numbers (but not all) such that a multiple of three underlined numbers remain and the total amount of the remaining numbers would be divisible by $2013$; but after trying for a while he realizes that it's impossible to do this. What is the largest number of the numbers on the board?

Let $ABCD$ be a parallelogram, let $X$ and $Y$ in the segments $AB$ and $CD$, respectively. The segments $AY$ and $DX$ intersects in $P$ and the segments $BY$ and $DX$ intersects in $Q$, show that the line $PQ$ passes in a fixed point(independent of the positions of the points $X$ and $Y$). I guess that the fixed point is the midpoint of $BD$.

Find all ordered pairs of positive integers $\left(a,b\right)$ such that $$\frac{1}{a}+\frac{a}{b}+\frac{1}{ab}=1.$$

In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $HE$ and $FG$? [asy] defaultpen(linewidth(0.8)); size(200); pair A=(0,8), B=(10,8), C=(10,0), D=origin; pair E=(4,8), F=(10,3), G=(6,0), H=(0,5); pair I=H+4*dir(H--E); pair J=foot(I, F, G); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); draw(I--J); draw(rightanglemark(H,I,J)); draw(rightanglemark(F,J,I)); label("$A$", A, dir((5,4)--A)); label("$B$", B, dir((5,4)--B)); label("$C$", C, dir((5,4)--C)); label("$D$", D, dir((5,4)--D)); label("$E$", E, dir((5,4)--E)); label("$F$", F, dir((5,4)--F)); label("$G$", G, dir((5,4)--G)); label("$H$", H, dir((5,4)--H)); label("$d$", I--J, SW); label("3", H--A, W); label("4", E--A, N); label("6", E--B, N); label("5", F--B, dir(1)); label("3", F--C, dir(1)); label("5", H--D, W); label("4", C--G, S); label("6", D--G, S); [/asy] $ \textbf{(A)}\ 6.8\qquad\textbf{(B)}\ 7.1\qquad\textbf{(C)}\ 7.6\qquad\textbf{(D)}\ 7.8\qquad\textbf{(E)}\ 8.1 $

Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.

Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows: \[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\] Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence. Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.