[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks. [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ x & & & d & e \\ \hline & f & a & c & c \\ + & g & h & i & \\ \hline f & f & f & c & c \\ \end{tabular}$ [b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning. [b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For which natural numbers $n$ can the polynomial $f (x) = x^n + x^{n-1} +...+ x + 1$ as write $f (x) = g (h (x))$, where $g$ and $h$ should be real polynomials of degrees greater than $1$?

Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of \[3x^2+Rx+R=90x\sqrt{x+1}.\] Find the value of $\lfloor x\rfloor$.

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}_0$ by $f(1)=0$ and \[f(n)=\max_j\{ f(j)+f(n-j)+j\}\quad\forall\, n\ge 2 \] Determine $f(2000)$.

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]. [i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.

Let $\mathbb{Q}_{>1}$ be the set of rational numbers greater than $1$. Let $f:\mathbb{Q}_{>1}\to \mathbb{Z}$ be a function that satisfies \[f(q)=\begin{cases} q-3&\textup{ if }q\textup{ is an integer,}\\ \lceil q\rceil-3+f\left(\frac{1}{\lceil q\rceil-q}\right)&\textup{ otherwise.} \end{cases}\] Show that for any $a,b\in\mathbb{Q}_{>1}$ with $\frac{1}{a}+\frac{1}{b}=1$, we have $f(a)+f(b)=-2$. [i]Proposed by usjl[/i]

For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and \[ \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals. (Proposed by Gerhard Woeginger, Austria)

Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation \[\lfloor a_1x\rfloor+\lfloor a_2x\rfloor+\cdots+\lfloor a_nx\rfloor=sx+r\] has exactly $ms$ solutions in $x$, where $s=a_1+a_2+\cdots+a_n+\frac1m$. [i]Linus Tang[/i]

Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$

Find the number of positive integers less than 101 that [i]can not [/i] be written as the difference of two squares of integers.

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?$\phantom{h}$ [asy] unitsize(0.4 cm); pair transx, transy; int i, j; int x, y; transx = (13,0); transy = (0,-9); for (i = 0; i <= 2; ++i) { for (j = 0; j <= 1; ++j) { if (i <= 1 || j <= 0) { for (x = 1; x <= 10; ++x) { draw(shift(i*transx + j*transy)*((x,0)--(x,5)),gray(0.7) + dashed); } for (y = 1; y <= 5; ++y) { draw(shift(i*transx + j*transy)*((0,y)--(10,y)),gray(0.7) + dashed); } draw(shift(i*transx + j*transy)*((0,0)--(11,0)),Arrow(6)); draw(shift(i*transx + j*transy)*((0,0)--(0,6)),Arrow(6)); label("time", (5,-0.5) + i*transx + j*transy); label(rotate(90)*"distance", (-0.5,2.5) + i*transx + j*transy); } }} draw((0,0)--(1.5,2.5)--(7.5,2.5)--(9,5),linewidth(1.5*bp)); draw((0,0)--(10,5),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(2.5,2.5)--(7.5,2.5)--(10,5)),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(2.5,3)--(7,2)--(10,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(2.5,2.5)--(6.5,2.5)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(7.5,2)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(2.5,2)--(7.5,3)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(9,5)),linewidth(1.5*bp)); label("(A)", (-1,6)); label("(B)", (-1,6) + transx); label("(C)", (-1,6) + 2*transx); label("(D)", (-1,6) + transy); label("(E)", (-1,6) + transx + transy); [/asy]

Determine all $a\in\mathbb R$ such that the inequality \[x^4+x^3-2(a+1)x^2-ax+a^2<0\] has at least one real solution $x.$

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]

[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions: $$ax + 2y = 1$$ $$2x + ay = 1$$ [b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$. [b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$. [b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer? [b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces. PS. You should use hide for answers.

Find all real numbers $a, b, c$ such that $x^3 + ax^2 + bx + c$ has three real roots $\alpha, \beta,\gamma$ (not necessarily all distinct) and the equation $x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3$ has roots $\alpha^3, \beta^3,\gamma^3$ .

Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$ I.Voronovich

Given the following system of equations $a_1 + a_2 + a_3 = 1$ $a_2 + a_3 + a_4 = 2$ $a_3 + a_4 + a_5 = 3$ $...$ $a_{12} + a_{13} + a_{14} = 12$ $a_{13} + a_{14} + a_1 = 13$ $a_{14 }+ a_1 + a_2 = 14$ find the value of $a_{14}$.

The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$.

[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$? [b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airplanes, what is the expected number airplanes that will fly straight? [b]p3.[/b] It takes Joshua working alone $24$ minutes to build a birdhouse, and his son working alone takes $16$ minutes to build one. The effective rate at which they work together is the sum of their individual working rates. How long in seconds will it take them to make one birdhouse together? [b]p4.[/b] If Katherine’s school is located exactly $5$ miles southwest of her house, and her soccer tournament is located exactly $12$ miles northwest of her house, how long, in hours, will it take Katherine to bike to her tournament right after school given she bikes at $0.5$ miles per hour? Assume she takes the shortest path possible. [b]p5.[/b] What is the largest possible integer value of $n$ such that $\frac{4n+2022}{n+1}$ is an integer? [b]p6.[/b] A caterpillar wants to go from the park situated at $(8, 5)$ back home, located at $(4, 10)$. He wants to avoid routes through $(6, 7)$ and $(7, 10)$. How many possible routes are there if the caterpillar can move in the north and west directions, one unit at a time? [b]p7.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 2\sqrt{13}$, $BC = 6\sqrt2$. Construct square $BCDE$ such that $\vartriangle ABC$ is not contained in square $BCDE$. Given that $ACDB$ is a trapezoid with parallel bases $\overline{AC}$, $\overline{BD}$, find $AC$. [b]p8.[/b] How many integers $a$ with $1 \le a \le 1000$ satisfy $2^a \equiv 1$ (mod $25$) and $3^a \equiv 1$ (mod $29$)? [b]p9.[/b] Let $\vartriangle ABC$ be a right triangle with right angle at $B$ and $AB < BC$. Construct rectangle $ADEC$ such that $\overline{AC}$,$\overline{DE}$ are opposite sides of the rectangle, and $B$ lies on $\overline{DE}$. Let $\overline{DC}$ intersect $\overline{AB}$ at $M$ and let $\overline{AE}$ intersect $\overline{BC}$ at $N$. Given $CN = 6$, $BN = 4$, find the $m+n$ if $MN^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. [b]p10.[/b] An elimination-style rock-paper-scissors tournament occurs with $16$ players. The $16$ players are all ranked from $1$ to $16$ based on their rock-paper-scissor abilities where $1$ is the best and $16$ is the worst. When a higher ranked player and a lower ranked player play a round, the higher ranked player always beats the lower ranked player and moves on to the next round of the tournament. If the initial order of players are arranged randomly, and the expected value of the rank of the $2$nd place player of the tournament can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$ what is the value of $m+n$? [b]p11.[/b] Estimation (Tiebreaker) Estimate the number of twin primes (pairs of primes that differ by $2$) where both primes in the pair are less than $220022$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ p$ be a positive integer. Define the function $ f: \mathbb{N}\to\mathbb{N}$ by $ f(n)\equal{}a_1^p\plus{}a_2^p\plus{}\cdots\plus{}a_m^p$, where $ a_1, a_2,\ldots, a_m$ are the decimal digits of $ n$ ($ n\equal{}\overline{a_1a_2\ldots a_m}$). Prove that every sequence $ (b_k)^\infty_{k\equal{}0}$ of positive integer that satisfy $ b_{k\plus{}1}\equal{}f(b_k)$ for all $ k\in\mathbb{N}$, has a finite number of distinct terms. $ \mathbb{N}\equal{}\{1,2,3\ldots\}$

Show that among any seven coplanar unit vectors there are at least two of them such that the magnitude of their sum is greater than $ \sqrt 3. $ [i]Ion Tecu[/i] and [i]Teodor Radu[/i]

An equilateral triangle $ABC$ is given, together with a point $P$ inside it. [asy] draw((0,0)--(4,0)--(2,3.464)--(0,0)); draw((1.3, 1.2)--(0,0)); draw((1.3, 1.2)--(2,3.464)); draw((1.3, 1.2)--(4,0)); label("$A$",(0,0),SW); label("$B$",(4,0),SE); label("$C$",(2,3.464),N); label("$P$",(1.3,1.2),S); [/asy] Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.

The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\ \[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\] for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.

Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$