[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order? $ \textbf{(A) } 21 \qquad \textbf{(B) } 34 \qquad \textbf{(C) } 51 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 150$

Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$. If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively, prove that \[ OG \leq \sqrt{ R ( R - 2r ) } . \] [i]Greece[/i]

For positive integers $n,$ let $\nu_3 (n)$ denote the largest integer $k$ such that $3^k$ divides $n.$ Find the number of subsets $S$ (possibly containing 0 or 1 elements) of $\{1, 2, \ldots, 81\}$ such that for any distinct $a,b \in S$, $\nu_3 (a-b)$ is even. [i]Author: Alex Zhu[/i] [hide="Clarification"]We only need $\nu_3(a-b)$ to be even for $a>b$. [/hide]

Consider an acute triangle $ABC$ in which $A_1, B_1,$ and $C_1$ are the feet of the altitudes from $A, B,$ and $C,$ respectively, and $H$ is the orthocenter. The perpendiculars from $H$ onto $A_1C_1$ and $A_1B_1$ intersect lines $AB$ and $AC$ at $P$ and $Q,$ respectively. Prove that the line perpendicular to $B_1C_1$ that passes through $A$ also contains the midpoint of the line segment $PQ$.

Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$

In $\triangle ABC$, $AB = 13$, $BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. The length of $HM$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, A=(0,6), B=(-4,0), C=(5,0), M=B+3.6*dir(B--A); draw(B--C--A--B^^M--H--A^^rightanglemark(A,H,C)); label("$A$", A, NE); label("$B$", B, W); label("$C$", C, E); label("$H$", H, S); label("$M$", M, dir(M)); [/asy] $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 7.5\qquad\textbf{(E)}\ 8 $

Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$? $\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$

A $12 \times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number $N$ such that if $N$ rooks are placed on the unit squares, one rook per square, we can always find $7$ rooks such that no two are attacking each other.

The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure [b]cannot[/b] be formed? [asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p);[/asy] [asy] size(350); defaultpen(linewidth(0.6)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)}; pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)}; pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)}; pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; int i; for(int i=0; i<15; i=i+1) { draw(shift(a[i])*p); draw(shift(b[i])*p); draw(shift(c[i])*p); draw(shift(d[i])*p); draw(shift(e[i])*p); } [/asy] \[ \textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\:\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)} \]

Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds. [i]Proposed by Yunhao Fu[/i]

Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to: ${{ \textbf{(A)}\ t^3 \qquad\textbf{(B)}\ t^2+t \qquad\textbf{(C)}\ |t^2+t| \qquad\textbf{(D)}\ t\sqrt{t^2+1} }\qquad\textbf{(E)}\ |t|\sqrt{1+t^2} } $

For every integer $n\geq 1$ prove that $$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$

Consider a $12$-card deck containing all four suits of $2, 3,$ and $4.$ A [i]double[/i] is defined as two cards directly next to each other in the deck, with the same value. Suppose we scan the deck left to right, and whenever we encounter a double, we remove all the cards up to that point (including the double). Let $N$ denote the number of times we have to remove cards. What is the expected value of $N$?

Let $X_1, X_2, \ldots$ be independent random variables of the same distribution such that their joint distribution is discrete and is concentrated on infinitely many different values. Let $a_n$ denote the probability that $X_1,\ldots, X_{n+1}$ are all different on the condition that $X_1,\ldots, X_n$ are all different ($n\ge 1$). Show that (a) $a_n$ is strictly decreasing and tends to $0$ as $n\to \infty$; and (b) for any sequence $1\le f(1)\le f(2) < \ldots$ of positive integers the joint distribution of $X_1, X_2, \ldots$ can be chosen such that $$\limsup_{n\to\infty}\frac{a_{f(n)}}{a_n}=1$$ holds.

Let $ABC$ be an acute triangle with circumcenter $O$, and select $E$ on $\overline{AC}$ and $F$ on $\overline{AB}$ so that $\overline{BE} \perp \overline{AC}$, $\overline{CF} \perp \overline{AB}$. Suppose $\angle EOF - \angle A = 90^{\circ}$ and $\angle AOB - \angle B = 30^{\circ}$. If the maximum possible measure of $\angle C$ is $\tfrac mn \cdot 180^{\circ}$ for some positive integers $m$ and $n$ with $m < n$ and $\gcd(m,n)=1$, compute $m+n$. [i]Proposed by Evan Chen[/i]

Solve the equation $~$ $x^4+2y^4+4z^4+8t^4=16xyzt$ $~$ in the set of integer numbers.

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that \[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\] is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$

We have a bipartite graph $G$ (with parts $X$ and $Y$). We orient each edge arbitrarily. [i]Hessam[/i] chooses a vertex at each turn and reverse the orientation of all edges that $v$ is one of their endpoint. Prove that with these steps we can reach to a graph that for each vertex $v$ in part $X$, $\deg^{+}(v)\geq \deg^{-}(v)$ and for each vertex in part $Y$, $\deg^{+}v\leq \deg^{-}v$

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$. Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$

Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$. Tangent of $\Gamma$ at $D$ intersects lines $BC$, $CA$, $AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects the circle $\omega$ again at $F$. Prove that points $D,E,F$ are collinear. (Saudi Arabia)