[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of line $AB$ with the line $CD$, and $F$ is the intersection of line $BC$ with the line $AD$. Let $P$ and $Q$ be the foots of the perpendicular of $E$ to the lines $AD$ and $BC$ respectively, and let $R$ and $S$ be the foots of the perpendicular of $F$ to the lines $AB$ and $CD$, respectively.The point $T$ is the intersection of the line $ER$ with the line $FS$. a) Show that, there exists a circle that passes in the points $E, F, P, Q, R$ and $S$. b)Show that, the circumcircle of triangle $RST$ is tangent with the circumcircle of triangle $QRB$.

Let $A, B$ be integer $n\times n$-matrices, where $det (A) = 1$ and $det (B) \ne 0$. Show that there is a $k \in N$, for which $BA^kB^{-1}$ is a matrix with integer entries.

There are a) $26$; b) $30$ identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not?? [i]Proposed by A. Gribalko[/i]

Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by $$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$ Show that $F$ is injective and find the set of values of $F$.

A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$

A cube side $5$ is made up of unit cubes. Two small cubes are [i]adjacent [/i] if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

A Knight always tells the truth. A Knave always lies. A Normal may either lie or tell the truth. You are allowed to ask questions that can be answered with ''yes" or ''no", such as ''is this person a Normal?" (a) There are three people in front of you. One is a Knight, another one is a Knave, and the third one is a Normal. They all know the identities of one another. How can you too learn the identity of each? (1) (b) There are four people in front of you. One is a Knight, another one is a Knave, and the other two are Normals. They all know the identities of one another. Prove that the Normals may agree in advance to answer your questions in such a way that you will not be able to learn the identity of any of the four people. (3)

Find the perimeter of a triangle whose altitudes are $3,4,$ and $6$. $ \textbf{(A)}\ 12\sqrt\frac35 \qquad \textbf{(B)}\ 16\sqrt\frac35 \qquad \textbf{(C)}\ 20\sqrt\frac35 \qquad \textbf{(D)}\ 24\sqrt\frac35 \qquad \textbf{(E)}\ \text{None}$

There exists complex numbers $z=x+yi$ such that the point $(x, y)$ lies on the ellipse with equation $\frac{x^2}9+\frac{y^2}{16}=1$. If $\frac{z-1-i}{z-i}$ is real, compute $z$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)[/i]

[u]Round 1[/u] [b]p1.[/b] Jom and Terry both flip a fair coin. What is the probability both coins show the same side? [b]p2.[/b] Under the same standard air pressure, when measured in Fahrenheit, water boils at $212^o$ F and freezes at $32^o$ F. At thesame standard air pressure, when measured in Delisle, water boils at $0$ D and freezes at $150$ D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have $y = ax + b$. What is the value of $a + b$? (Ignore units.) [b]p3.[/b] What are the last two digits of $5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}$? [u]Round 2[/u] [b]p4.[/b] Compute the average of the magnitudes of the solutions to the equation $2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0$. [b]p5.[/b] How many integers between $1$ and $1000000$ inclusive are both squares and cubes? [b]p6.[/b] Simon has a deck of $48$ cards. There are $12$ cards of each of the following $4$ suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card? [u]Round 3[/u] [b]p7.[/b] Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability $p$ of shooting, and $1 - p$ of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, $p$ is constant. Once the ball has been shot, the game is over. What is the maximum value of $p$ that makes Christine’s total probability of shooting the ball $\frac{3}{20}$ ? [b]p8.[/b] If $x$ and $y$ are real numbers, then what is the minimum possible value of the expression $3x^2 - 12xy + 14y^2$ given that $x - y = 3$? [b]p9.[/b] Let $ABC$ be an equilateral triangle, let $D$ be the reflection of the incenter of triangle $ABC$ over segment $AB$, and let $E$ be the reflection of the incenter of triangle $ABD$ over segment $AD$. Suppose the circumcircle $\Omega$ of triangle $ADE$ intersects segment $AB$ again at $X$. If the length of $AB$ is $1$, find the length of $AX$. [u]Round 4[/u] [b]p10.[/b] Elaine has $c$ cats. If she divides $c$ by $5$, she has a remainder of $3$. If she divides $c$ by $7$, she has a remainder of $5$. If she divides $c$ by $9$, she has a remainder of $7$. What is the minimum value $c$ can be? [b]p11.[/b] Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a $10$-sided die (its faces are numbered from $1$ to $10$) $x$ times. If, within those $x$ rolls, the number $10$ appears, then you win. Assuming that you like winning, what is the highest value of $x$ where you would prefer to play the coin-flipping game over the die-rolling game? [b]p12.[/b] Let be the set $X = \{0, 1, 2, ..., 100\}$. A subset of $X$, called $N$, is defined as the set that contains every element $x$ of $X$ such that for any positive integer $n$, there exists a positive integer $k$ such that n can be expressed in the form $n = x^{a_1}+x^{a_2}+...+x^{a_k}$ , for some integers $a_1, a_2, ..., a_k$ that satisfy $0 \le a_1 \le a_2 \le ... \le a_k$. What is the sum of the elements in $N$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that \[ \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} \] if and only if the diagonals $ AC$ and $ BD$ are perpendicular. [i]Proposed by Dusan Djukic, Serbia[/i]

S-Corporation designs its logo by linking together $4$ semicircles along the diameter of a unit circle. Find the perimeter of the shaded portion of the logo. [img]https://cdn.artofproblemsolving.com/attachments/8/6/f0eabd46f5f3a5806d49012b2f871a453b9e7f.png[/img]

Solve the equation for primes $p$ and $q$: $$p^3-q^3=pq^3-1.$$

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.

Inside triangle \(ABC\), point \(P\) is marked. Point \(Q\) is on segment \(AB\), and point \(R\) is on segment \(AC\) such that the circumcircles of triangles \(BPQ\) and \(CPR\) are tangent to line \(AP\). Lines are drawn through points \(B\) and \(C\) passing through the center of the circumcircle of triangle \(BPC\), and through points \(Q\) and \(R\) passing through the center of the circumcircle of triangle \(PQR\). Prove that there exists a circle tangent to all four drawn lines.

There are $n$ players, $n\ge 2$, which are playing a card game with $np$ cards in $p$ rounds. The cards are coloured in $n$ colours and each colour is labelled with the numbers $1,2,\ldots ,p$. The game submits to the following rules: [list]each player receives $p$ cards. the player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card. the winner of the round is the player who has put the greatest card of the same colour as the first one. the winner of the round starts the next round with a card that he selects and the play continues with the same rules. the played cards are out of the game.[/list] Show that if all cards labelled with number $1$ are winners, then $p\ge 2n$. [i]Barbu Berceanu[/i]

Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$. Find the maximal possible value of $mn$.

Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$

Let $A,B \in \mathcal{M}_n(\mathbb{R}).$ We consider the function $f:\mathcal{M}_n(\mathbb{C}) \to \mathcal{M}_n(\mathbb{C}),$ defined by $f(Z)=AZ+B\overline{Z},$ $Z \in \mathcal{M}_n(\mathbb{C}),$ where $\overline{Z}$ is the matrix whose entries are the conjugates of the corresponding entries of $Z.$ Prove that the following statements are equivalent: $(1)$ the function $f$ is injective; $(2)$ the function $f$ is surjective; $(3)$ the matrices $A+B$ and $A-B$ are invertible.

Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$, \begin{align*} f(mn)&=f(m)f(n)\\ f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}

In right triangle $ \triangle ACE$, we have $ AC \equal{} 12$, $ CE \equal{} 16$, and $ EA \equal{} 20$. Points $ B$, $ D$, and $ F$ are located on $ \overline{AC}$, $ \overline{CE}$, and $ \overline{EA}$, respectively, so that $ AB \equal{} 3$, $ CD \equal{} 4$, and $ EF \equal{} 5$. What is the ratio of the area of $ \triangle DBF$ to that of $ \triangle ACE$? [asy] size(200);defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair C = (0,0); pair E = (16,0); pair A = (0,12); pair F = waypoint(E--A,0.25); pair B = waypoint(A--C,0.25); pair D = waypoint(C--E,0.25); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$",A,NW);label("$B$",B,W);label("$C$",C,SW);label("$D$",D,S);label("$E$",E,SE);label("$F$",F,NE); label("$3$",midpoint(A--B),W); label("$9$",midpoint(B--C),W); label("$4$",midpoint(C--D),S); label("$12$",midpoint(D--E),S); label("$5$",midpoint(E--F),NE); label("$15$",midpoint(F--A),NE); draw(A--C--E--cycle); draw(B--F--D--cycle);[/asy]$ \textbf{(A)}\ \frac {1}{4}\qquad \textbf{(B)}\ \frac {9}{25}\qquad \textbf{(C)}\ \frac {3}{8}\qquad \textbf{(D)}\ \frac {11}{25}\qquad \textbf{(E)}\ \frac {7}{16}$

Construct the semicircle $ h$ with the diameter $ AB$ and the midpoint $ M$. Now construct the semicircle $ k$ with the diameter $ MB$ on the same side as $ h$. Let $ X$ and $ Y$ be points on $ k$, such that the arc $ BX$ is $ \frac{3}{2}$ times the arc $ BY$. The line $ MY$ intersects the line $ BX$ in $ D$ and the semicircle $ h$ in $ C$. Show that $ Y$ ist he midpoint of $ CD$.

The quadrilateral $ABCD$ is an isosceles trapezoid, where $AB\parallel CD$. The trapezoid is inscribed in a circle with radius $R$ and center on side $AB$. Point $E$ lies on the circumscribed circle and is such that $\angle DAE = 90^o$. Given that $\frac{AE}{AB}=\frac34$, calculate the length of the sides of the isosceles trapezoid.