Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.

Evaluate \[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Let $ f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n $, where $i = \sqrt{-1}$. What is $f(2022)$ $ \textbf{(A)}\ -2 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \sqrt{3} \qquad \textbf{(E)}\ 2$

[b]p1.[/b] A dog on a $10$ meter long leash is tied to a $10$ meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence? [b]p2.[/b] Suppose that the equation $$\begin{tabular}{cccccc} &\underline{C} &\underline{H} &\underline{M}& \underline{M}& \underline{C}\\ +& &\underline{H}& \underline{M}& \underline{M} & \underline{T}\\ \hline &\underline{P} &\underline{U} &\underline{M} &\underline{A} &\underline{C}\\ \end{tabular}$$ holds true, where each letter represents a single nonnegative digit, and distinct letters represent different digits (so that $\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ and $ \underline{P}\, \underline{U}\, \underline{M}\, \underline{A}\, \underline{C}$ are both five digit positive integers, and the number $\underline{H }\, \underline{M}\, \underline{M}\, \underline{T}$ is a four digit positive integer). What is the largest possible value of the five digit positive integer$\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ ? [b]p3.[/b] Square $ABCD$ has side length $4$, and $E$ is a point on segment $BC$ such that $CE = 1$. Let $C_1$ be the circle tangent to segments $AB$, $BE$, and $EA$, and $C_2$ be the circle tangent to segments $CD$, $DA$, and $AE$. What is the sum of the radii of circles $C_1$ and $C_2$? [b]p4.[/b] A finite set $S$ of points in the plane is called tri-separable if for every subset $A \subseteq S$ of the points in the given set, we can find a triangle $T$ such that (i) every point of $A$ is inside $T$ , and (ii) every point of $S$ that is not in $A$ is outside$ T$ . What is the smallest positive integer $n$ such that no set of $n$ distinct points is tri-separable? [b]p5.[/b] The unit $100$-dimensional hypercube $H$ is the set of points $(x_1, x_2,..., x_{100})$ in $R^{100}$ such that $x_i \in \{0, 1\}$ for $i = 1$, $2$, $...$, $100$. We say that the center of $H$ is the point $$\left( \frac12,\frac12, ..., \frac12 \right)$$ in $R^{100}$, all of whose coordinates are equal to $1/2$. For any point $P \in R^{100}$ and positive real number $r$, the hypersphere centered at $P$ with radius $r$ is defined to be the set of all points in $R^{100}$ that are a distance $r$ away from $P$. Suppose we place hyperspheres of radius $1/2$ at each of the vertices of the $100$-dimensional unit hypercube $H$. What is the smallest real number $R$, such that a hypersphere of radius $R$ placed at the center of $H$ will intersect the hyperspheres at the corners of $H$? [b]p6.[/b] Greg has a $9\times 9$ grid of unit squares. In each square of the grid, he writes down a single nonzero digit. Let $N$ be the number of ways Greg can write down these digits, so that each of the nine nine-digit numbers formed by the rows of the grid (reading the digits in a row left to right) and each of the nine nine-digit numbers formed by the columns (reading the digits in a column top to bottom) are multiples of $3$. What is the number of positive integer divisors of $N$? [b]p7.[/b] Find the largest positive integer $n$ for which there exists positive integers $x$, $y$, and $z$ satisfying $$n \cdot gcd(x, y, z) = gcd(x + 2y, y + 2z, z + 2x).$$ [b]p8.[/b] Suppose $ABCDEFGH$ is a cube of side length $1$, one of whose faces is the unit square $ABCD$. Point $X$ is the center of square $ABCD$, and $P$ and $Q$ are two other points allowed to range on the surface of cube $ABCDEFHG$. Find the largest possible volume of tetrahedron $AXPQ$. [b]p9.[/b] Deep writes down the numbers $1, 2, 3, ... , 8$ on a blackboard. Each minute after writing down the numbers, he uniformly at random picks some number $m$ written on the blackboard, erases that number from the blackboard, and increases the values of all the other numbers on the blackboard by $m$. After seven minutes, Deep is left with only one number on the black board. What is the expected value of the number Deep ends up with after seven minutes? [b]p10.[/b] Find the number of ordered tuples $(x_1, x_2, x_3, x_4, x_5)$ of positive integers such that $x_k \le 6$ for each index $k = 1$, $2$, $... $,$ 5$, and the sum $$x_1 + x_2 +... + x_5$$ is $1$ more than an integer multiple of $7$. [b]p11.[/b] The equation $$\left( x- \sqrt[3]{13}\right)\left( x- \sqrt[3]{53}\right)\left( x- \sqrt[3]{103}\right)=\frac13$$ has three distinct real solutions $r$, $s$, and $t$ for $x$. Calculate the value of $$r^3 + s^3 + t^3.$$ [b]p12.[/b] Suppose $a$, $b$, and $c$ are real numbers such that $$\frac{ac}{a + b}+\frac{ba}{b + c}+\frac{cb}{c + a}= -9$$ and $$\frac{bc}{a + b}+\frac{ca}{b+c}+\frac{ab}{c + a}= 10.$$ Compute the value of $$\frac{b}{a + b}+\frac{c}{b + c}+\frac{a}{c + a}.$$ [b]p13.[/b] The complex numbers $w$ and $z$ satisfy the equations $|w| = 5$, $|z| = 13$, and $$52w - 20z = 3(4 + 7i).$$ Find the value of the product $wz$. [b]p14.[/b] For $i = 1, 2, 3, 4$, we choose a real number $x_i$ uniformly at random from the closed interval $[0, i]$. What is the probability that $x_1 < x_2 < x_3 < x_4$ ? [b]p15.[/b] The terms of the infinite sequence of rational numbers $a_0$, $a_1$, $a_2$, $...$ satisfy the equation $$a_{n+1} + a_{n-2} = a_na_{n-1}$$ for all integers $n\ge 2$. Moreover, the values of the initial terms of the sequence are $a_0 =\frac52$, $a_1 = 2$ and} $a_2 =\frac52.$ Call a nonnegative integer $m$ lucky if when we write $a_m =\frac{p}{q}$ for some relatively prime positive integers $p$ and $q$, the integer $p + q$ is divisible by $13$. What is the $101^{st}$ smallest lucky number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the Fibonacci sequence $1$, $1$, $2$, $3$, $5$, $8$, $13$, ... What are the last three digits (from left to right) of the $2020$th term?

A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.

$p(x)$ is a real polynomial that for each $x\geq 0$, $p(x)\geq 0$. Prove that there are real polynomials $A(x),B(x)$ that $p(x)=A(x)^{2}+xB(x)^{2}$

Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

If $x<0$, then $\left|x-\sqrt{(x-1)^2}\right|$ equals $\textbf{(A) }1\qquad\textbf{(B) }1-2x\qquad\textbf{(C) }-2x-1\qquad\textbf{(D) }1+2x\qquad \textbf{(E) }2x-1$

Let $D,E,F$ denote the tangent points of the incircle of $ABC$ with sides $BC,AC,AB$ respectively. Let $M$ be the midpoint of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D,M,F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D,M,E$ and the segment $AC$. Prove that the circle passing through $A,K,L$ touches the line $BC$

The sequence $a_{n}$ is defined by $a_{1}\geq 2$ and the recurrence formula \[a_{n+1}=a_{n}\sqrt{\frac{a_{n}^3+2}{2(a_{n}^3+1)}}\] for $n\geq 1$. Prove that for every integer $n$, the inequality $a_{n}>\sqrt{\frac{3}{n}}$ holds.

Ahmad and Salem play the following game. Ahmad writes two integers (not necessarily different) on a board. Salem writes their sum and product. Ahmad does the same thing: he writes the sum and product of the two numbers which Salem has just written. They continue in this manner, not stopping unless the two players write the same two numbers one after the other (for then they are stuck!). The order of the two numbers which each player writes is not important. Thus if Ahmad starts by writing $3$ and $-2$, the first five moves (or steps) are as shown: (a) Step 1 (Ahmad) $3$ and $-2$ (b) Step 2 (Salem) $1$ and $-6$ (c) Step 3 (Ahmad) $-5$ and $-6$ (d) Step 4 (Salem) $-11$ and $30$ (e) Step 5 (Ahmad) $19$ and $-330$ (i) Describe all pairs of numbers that Ahmad could write, and ensure that Salem must write the same numbers, and so the game stops at step 2. (ii) What pair of integers should Ahmad write so that the game finishes at step 4? (iii) Describe all pairs of integers which Ahmad could write at step 1, so that the game will finish after finitely many steps. (iv) Ahmad and Salem decide to change the game. The first player writes three numbers on the board, $u, v$ and $w$. The second player then writes the three numbers $u + v + w,uv + vw + wu$ and $uvw$, and they proceed as before, taking turns, and using this new rule describing how to work out the next three numbers. If Ahmad goes first, determine all collections of three numbers which he can write down, ensuring that Salem has to write the same three numbers at the next step.

A binary sequence is constructed as follows. If the sum of the digits of the positive integer $k$ is even, the $k$-th term of the sequence is $0$. Otherwise, it is $1$. Prove that this sequence is not periodic.

Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$ Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$

Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$: $$p \mid f(n)f(p-1)!+n^{f(p)}.$$ [i]Proposed by Dorlir Ahmeti[/i]

In an acute triangle $ABC$, an arbitrary point $P$ is chosen on the altitude $AH$. The points $E$ and $F$ are the midpoints of $AC$ and $AB$, respectively. The perpendiculars from $E$ on $CP$ and from $F$ on $BP$ intersect at the point $K$. Show that $KB = KC$.

In some club, each member is on two commissions. Furthermore, it is known that two any commissions always have exactly one member in common. Knowing there are five commissions. How many members does the club have?

A particle moves in $3$-space according to the equations: $$ \frac{dx}{dt} =yz,\; \frac{dy}{dt} =xz,\; \frac{dz}{dt}= xy.$$ Show that: (a) If two of $x(0), y(0), z(0)$ equal $0,$ then the particle never moves. (b) If $x(0)=y(0)=1, z(0)=0,$ then the solution is $$ x(t)= \sec t ,\; y(t) =\sec t ,\; z(t)= \tan t;$$ whereas if $x(0)=y(0)=1, z(0)=-1,$ then $$ x(t) =\frac{1}{t+1} ,\; y(t)=\frac{1}{t+1}, z(t)=- \frac{1}{t+1}.$$ (c) If at least two of the values $x(0), y(0), z(0)$ are different from zero, then either the particle moves to infinity at some finite time in the future, or it came from infinity at some finite time in the past (a point $(x, y, z)$ in $3$-space "moves to infinity" if its distance from the origin approaches infinity).

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .$ [i]Proposed by kunny[/i]

The headteacher wants to hire a certain number of new teachers in addition to existing teachers. If he hired an additional $10$ teachers, the number of school students would be reduced number per teacher by $5$. However, if the headmaster hired $20$ new teachers, the number of students per teacher would be reduced by $8$. How many students and how many there are teachers in this school? [img]https://cdn.artofproblemsolving.com/attachments/2/8/c0157ff43fd3d92138c87556a0fca2414e8a3f.png[/img]

Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that \[f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.\] a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$. b) Prove that $\mathcal{F}$ has exactly two elements. [i]Nelu Chichirim[/i]

Let $(a_{n})$ be the sequence defined by $a_{0}=1,a_{n+1}=\sum_{k=0}^{n}\dfrac{a_k}{n-k+2}$. Find the limit \[\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\dfrac{a_{k}}{2^{k}},\] if it exists.