A [i]permutation[/i] of a finite set $S$ is a one-to-one function from $S$ to $S$. Given a permutation $f$ of the set $\{ 1, 2, \ldots, 100 \}$, define the [i]displacement[/i] of $f$ to be the sum \[ \sum_{i = 1}^{100} \left\lvert f(i) - i \right\rvert . \] How many permutations of $\{ 1, 2, \ldots, 100 \}$ have displacement 4?

A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.

Let $p$ be an odd prime. Determine the positive integers $x$ and $y$ with $x\leq y$ for which the number $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is non-negative and as small as possible.

Let $S$ be the set of fractions of the form $\frac{{\text {lcm}}(A, B)}{A+B}$, where $A$ and $B$ are positive integers and ${\text{lcm}}(A, B)$ is the least common multiple of $A$ and $B$. What is the smallest number exceeding 3 in $S$?

Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \] [i]Robert Szasz[/i]

What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)

The cells of an $n\times n$ table ($n\ge 3$) are painted black and white in the chess-like manner. Per move one can choose any $2\times 2$ square and reverse the color of the cells inside it. Find all $n$ for which one can obtain a table with all cells of the same color after finitely many such moves.

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles? [asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E); [/asy] $\textbf{(A) }3.5\qquad\textbf{(B) }4.0\qquad\textbf{(C) }4.5\qquad\textbf{(D) }5.0\qquad \textbf{(E) }5.5$

You and your friend play a game on a $ 7 \times 7$ grid of buckets. Your friend chooses $5$ “lucky” buckets by marking an “$X$” on the bottom that you cannot see. However, he tells you that they either form a vertical, or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets: either $\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\}$, or $\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, a)\}$, with $1\le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a “lucky” bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is “lucky”?

Let $ (b_0,b_1,b_2,b_3)$ be a permutation of the set $ \{54,72,36,108\}$. Prove that $ x^5\plus{}b_3x^3\plus{}b_2x^2\plus{}b_1x\plus{}b_0$ is irreducible in $ \mathbb Z[x]$.

Find all real numbers $ x$ such that \[ \sqrt{x\plus{}1}\plus{}\sqrt{x\plus{}3} \equal{} \sqrt{2x\minus{}1}\plus{}\sqrt{2x\plus{}1}.\]

Let $ABC$ be a triangle with $\angle B = 90^o$. Given that there exists a point $D$ on $AC$ such that $AD = DC$ and $BD = BC$, compute the value of the ratio $\frac{AB}{BC}$ .

Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number. [i]I. Tonov[/i]

Let $ \Delta ABC$ with incircle $ \Gamma$, let $ D, E$ and $ F$ the tangency points of $ \Gamma$ with sides $ BC, AC$ and $ AB$, respectively and let $ P$ the intersection point of $ AD$ with $ \Gamma$. $ a)$ Prove that $ BC, EF$ and the straight line tangent to $ \Gamma$ for $ P$ concur at a point $ A'$. $ b)$ Define $ B'$ and $ C'$ in an anologous way than $ A'$. Prove that $ A'\minus{}B'\minus{}C'$

Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$. Let $D, E, F$ be the reflections of $H$ over $BC, CA, AB$, respectively, and let $A', B', C'$ be the reflections of $A, B, C$ over $BC, CA, AB$, respectively. Let $S$ be the circumcenter of triangle $A'B'C'$, and let $H'$ be the orthocenter of triangle $DEF$. Define $J$ as the center of the circle passing through the three projections of $H$ onto the lines $B'C', C'A', A'B'$. Prove that $HJ$ is parallel to $H'S$.

Let be a number $ a\in \left[ 1,\infty \right) $ and a function $ f\in\mathcal{C}^2(-a,a) . $ Show that the sequence $$ \left( \sum_{k=1}^n f\left( \frac{k}{n^2} \right) \right)_{n\ge 1} $$ is convergent, and determine its limit.

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$ and $\angle B = 30^{\circ}$. The perpendicular at the midpoint $M$ of $BC$ meets the bisector $BK$ of the angle $B$ at the point $E$. The perpendicular bisector of $EK$ meets $AB$ at $D$. Prove that $KD$ is perpendicular to $DE$. [i]Proposed by Greece[/i]

There is an infinite set of points $S$ on the plane, and any $1\times 1$ square contains a finite number of points from the set $S$. Prove that there are two different points $A$ and $B$ from $S$ such that for any other point $X$ from $S$ the following inequalities hold: $$|XA|, |XB| \ge 0.999|AB|.$$

A criminal is at point $X$, and three policemen at points $A, B$ and $C$ block him up, i.e. the point $X$ lies inside the triangle $ABC$. Each evening one of the policemen is replaced in the following way: a new policeman takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points $A, B$ and $C$ (it is known that at any moment $X$ does not lie on a side of the triangle)? by V.Protasov

Let $S = {1, 2, \cdots, 100}.$ $X$ is a subset of $S$ such that no two distinct elements in $X$ multiply to an element in $X.$ Find the maximum number of elements of $X$. [i]2022 CCA Math Bonanza Individual Round #3[/i]

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

Let $x\in\left(-\frac{1}{2},0\right)$, $\alpha_1=\cos(\sin x\pi),\alpha_2=\sin(\cos x\pi),\alpha_1=\cos (x+1)\pi$, then $\text{(A)}\alpha_3<\alpha_2<\alpha_1\qquad\text{(B)}\alpha_1<\alpha_3<\alpha_2\qquad\text{(C)}\alpha_3<\alpha_1<\alpha_2\qquad\text{(D)}\alpha_2<\alpha_3<\alpha_1$

In an equilateral triangle $ ABC $, point $ O $ is chosen and perpendiculars $ OM $, $ ON $, $ OP $ are dropped to the sides $ BC $, $ CA $, $ AB $, respectively. Prove that the sum of the segments $ AP $, $ BM $, $ CN $ does not depend on the position of point $ O $.

[b]p1.[/b] Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls $1$, $2$, $3$, $4$, $5$, and then $6$ in that order is $p$. The probability that he rolls $2$, $2$, $4$, $4$, $6$, and then $6$ in that order is $q$. What is $p - q$? [b]p2.[/b] What is the smallest positive integer $x$ such that $x \equiv 2017$ (mod $2016$) and $x \equiv 2016$ (mod $2017$) ? [b]p3.[/b] The vertices of triangle $ABC$ lie on a circle with center $O$. Suppose the measure of angle $ACB$ is $45^o$. If $|AB| = 10$, then what is the distance between $O$ and the line $AB$? [b]p4.[/b] A “word“ is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$? [b]p5.[/b] How many distinct complex roots does the polynomial $p(x) = x^{12} - x^8 - x^4 + 1$ have? [b]p6.[/b] Notice that $1 = \frac12 + \frac13 + \frac16$ , that is, $1$ can be expressed as the sum of the three fractions $\frac12 $, $\frac13$ , and $\frac16$ , where each fraction is in the form $\frac{1}{n}$, with each $n$ different. Give a $6$-tuple of distinct positive integers $(a, b, c, d, e, f)$ where $a < b < c < d < e < f$ such that $\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1$ and explain how you arrived at your $6$-tuple. Multiple answers will be accepted. [b]p7.[/b] You have a Monopoly board, an $11 \times 11$ square grid with the $9 \times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard $6$-sided dice. Let $S$ be the set of squares on the board such that if you are initially on a square in $S$, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in $S$ as your starting position. What is the probability that you land on Go? [b]p8.[/b] Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps? [b]p9.[/b] Does there exist a pair of positive integers $(x, y)$, where $x < y$, such that $x^2 + y^2 = 1009^3$? If so, give a pair $(x, y)$ and explain how you found that pair. If not, explain why. [b]p10.[/b] Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s_As_Bs_C$. [b]p11.[/b] Julia and Dan want to divide up $256$ dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until $4$ rejections have been made; once $4$ rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round? [b]p12.[/b] A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of $3$ are $\{1, 1, 1\}$ and $\{1, 2\}$. $\{1, 1, 3, 4\}$ is NOT a perfect partition of $9$ because the sum $4$ can be achieved in two different ways: $4$ and $1 + 3$. How many integers $1 \le N \le 40$ each have exactly one perfect partition? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].