[b]p1.[/b] If Bob takes $6$ hours to build $4$ houses, how many hours will he take to build $ 12$ houses? [b]p2.[/b] Compute the value of $\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}$. [b]p3.[/b] Given a line $2x + 5y = 170$, find the sum of its $x$- and $y$-intercepts. [b]p4.[/b] In some future year, BmMT will be held on Saturday, November $19$th. In that year, what day of the week will April Fool’s Day (April $1$st) be? [b]p5.[/b] We distribute $78$ penguins among $10$ people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the $10$ people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take? [b]p6.[/b] A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry? [b]p7. [/b]Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct? [b]p8.[/b] The pages of a book are consecutively numbered from $1$ through $480$. How many times does the digit $8$ appear in this numbering? [b]p9.[/b] A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length $2$, what is the area of the flower? [b]p10.[/b] There are two non-consecutive positive integers $a, b$ such that $a^2 - b^2 = 291$. Find $a$ and $b$. [b]p11.[/b] Let $ABC$ be an equilateral triangle. Let $P, Q, R$ be the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Suppose the area of triangle $PQR$ is $1$. Among the $6$ points $A, B, C, P, Q, R$, how many distinct triangles with area $1$ have vertices from that set of $6$ points? [b]p12.[/b] A positive integer is said to be binary-emulating if its base three representation consists of only $0$s and $1$s. Determine the sum of the first $15$ binary-emulating numbers. [b]p13.[/b] Professor $X$ can choose to assign homework problems from a set of problems labeled $ 1$ to $30$, inclusive. No two problems in his assignment can share a common divisor greater than $ 1$. What is the maximum number of problems that Professor $X$ can assign? [b]p14.[/b] Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length $5$ and $6$ respectively, and there exists a point $X$ on $CD$ such that $\angle XBC = \angle XAD = \angle AXB = 90^o$ . Find the area of trapezoid $ABCD$. [b]p15.[/b] Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.) [b]p16.[/b] Let $ABC$ be a triangle and M be the midpoint of $BC$. If $AB = AM = 5$ and $BC = 12$, what is the area of triangle $ABC$? [b]p17. [/b] A positive integer $n$ is called good if it can be written as $5x+ 8y = n$ for positive integers $x, y$. Given that $42$, $43$, $44$, $45$ and $46$ are good, what is the largest n that is not good? [b]p18.[/b] Below is a $ 7 \times 7$ square with each of its unit squares labeled $1$ to $49$ in order. We call a square contained in the figure [i]good [/i] if the sum of the numbers inside it is odd. For example, the entire square is [i]good [/i] because it has an odd sum of $1225$. Determine the number of [i]good [/i] squares in the figure. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 [hide][img]https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.png[/hide] [b]p19.[/b] A circle of integer radius $ r$ has a chord $PQ$ of length $8$. There is a point $X$ on chord $PQ$ such that $\overline{PX} = 2$ and $\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\overline{AX}$ and $\overline{XB}$ are integers. What is the minimal possible integer $ r$ such that there exist $6$ euphonic chords for $X$? [b]p20.[/b] On planet [i]Silly-Math[/i], two individuals may play a game where they write the number $324000$ on a whiteboard and take turns dividing the number by prime powers – numbers of the form $p^k$ for some prime $p$ and positive integer $k$. Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide $324000$ by in order to ensure a win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all polynomials $ P_n(x)\equal{}x^n\plus{}a_1 x^{n\minus{}1}\plus{}...\plus{}a_{n\minus{}1} x\plus{}a_n$ with integer coefficients whose $ n$ zeros are precisely the numbers $ a_1,...,a_n$ (counted with their respective multiplicities).

For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers. Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one. R. Salimov

Determine the number of functions $f : Z^+ \to Z^+$ so that for all positive integers $x$ we have $f(f(x)) = f(x + 1)$, and $\max (f(2), . . . , f(14)) \le f(1) - 2 = 12$.

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

Given $x,y,z\in \mathbb{R^+}$. Find all sets of $x,y,z$ that correspond to $$x+y+z=x^2+y^2+z^2+18xyz=1$$ [i](Zhuge Liang)[/i]

Find all functions $f$ from the reals to the reals such that \[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\] for all real $x,y$.

Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $​​y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $A, B, X$ be real $n\times n$ matrices for which $AXB + A + B = 0$ holds. Prove that $AXB = BXA$.

Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.

Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares.

Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.

Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\]

Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.

We build a modified version of Pascal’s triangle as follows: in the first row we write a $2$ and a $3$, and in the further rows, every number is the sum of the two numbers directly above it (and rows always begin with a $2$ and end with a $3$). In the $13$th row, what is the $5$th number from the left? [img]https://cdn.artofproblemsolving.com/attachments/7/2/58e1a9f43fa7c304bfd285fc1b73bed883e9a6.png[/img]

Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that \[ xyz(x+y+z)+2021\geqslant 2024xyz\]

Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that \[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

Lakshay chooses two numbers, $m$ and $n$, and draws two lines, $y = mx + 3$ and $y = nx + 23$. Given that the two lines intersect at $(20, 23)$, compute $m + n$.