[u]Set 1[/u] [b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs? [b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it take, in minutes, for it travel $10d$ with a constant velocity of $2.5v$? [b]G3.[/b] Andy originally has $3$ times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has $2$ times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had. [u]Set 2[/u] [b]G4.[/b] A coin is weighted so that it is $3$ times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively? [b]G5.[/b] There are $n$ students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of $5$. When $8$ students are absent, the students can evenly be divided into groups of $7$. Find the minimum possible value of $n$. [b]G6.[/b] Trapezoid $ABCD$ has $AB \parallel CD$ such that $AB = 5$, $BC = 4$ and $DA = 2$. If there exists a point $M$ on $CD$ such that $AM = AD$ and $BM = BC$, find $CD$. [u]Set 3[/u] [b]G7.[/b] Angeline has $10$ coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has $62$ cents in total, then how many dimes does she have? [b]G8.[/b] Equilateral triangle $ABC$ has side length $6$. There exists point $D$ on side $BC$ such that the area of $ABD$ is twice the area of $ACD$. There also exists point $E$ on segment $AD$ such that the area of $ABE$ is twice the area of $BDE$. If $k$ is the area of triangle $ACE$, then find $k^2$. [b]G9.[/b] A number $n$ can be represented in base $ 6$ as $\underline{aba}_6$ and base $15$ as $\underline{ba}_{15}$, where $a$ and $b$ are not necessarily distinct digits. Find $n$. PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]Problem Section #2 a) If $$ax+by=7$$ $$ax^2+by^2=49$$ $$ax^3+by^3=133$$ $$ax^4+by^4=406$$ , find the value of $2014(x+y-xy)-100(a+b).$

Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$

[b]p1.[/b] Compute $$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$ [b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$? [b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area? [b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$ [b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box. [b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total? Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime. [b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion. Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning. [b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located. [b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? [b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. [b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. [b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. [b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$: $$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$ [i]Proposed by Sina Saleh[/i]

There are positive real numbers $a$, $b$, $c$, $d$, and $p$ such that $a$ is $62.5\%$ of $b$, $b$ is $64\%$ of $c$, c is $125\%$ of $d$, and $d$ is $p\%$ of $a$. Find $p$.

Evaluate $$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$

The sequence $ (x_n)$ of real numbers is defined by $ x_1\equal{}\frac{29}{10}$ and $ x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3}$ for all $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k\minus{}1}>a>x_{2k}$.

Suppose we have $w < x < y < z$, and each of the $6$ pairwise sums are distinct. The $4$ greatest sums are $4, 3, 2, 1$. What is the sum of all possible values of $w$?

Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

Let be two complex numbers $ a,b. $ Show that the following affirmations are equivalent: $ \text{(i)} $ there are four numbers $ x_1,x_2,x_3,x_4\in\mathbb{C} $ such that $ \big| x_1 \big| =\big| x_3 \big|, \big| x_2 \big| =\big| x_4 \big|, $ and $$ x_{j_1}^2-ax_{j_1}+b=0=x_{j_2}^2-bx_{j_2}+a,\quad\forall j_1\in\{ 1,2\} ,\quad\forall j_2\in\{ 3,4\} . $$ $ \text{(ii)} a^3=b^3 $ or $ b=\overline{a} $ (the conjugate of a).

Let $n$ be a positive integer and $a_1\le a_2\le\ldots\le a_{n+1}$ and $b_1\le b_2\le\ldots\le b_n$ be real numbers such that for all $k\le n$, \[\binom nk\sum_{\substack{1\le i_1<i_2<\ldots<i_k\le n+1,\\i_1,i_2,\ldots,i_k\in\mathbb N}}a_{i_1}a_{i_2}\ldots a_{i_k} = \binom{n+1}k\sum_{\substack{1\le j_1<j_2<\ldots<j_k\le n,\\j_1,j_2,\ldots,j_k\in\mathbb N}}b_{j_1}b_{j_2}\ldots b_{j_k}.\] Show that \[a_1\le b_1\le a_2\le b_2\le \ldots \le a_n\le b_n\le a_{n+1}.\] [i](Proposed by Ivan Chan Guan Yu)[/i]

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

On a board are written some positive reals (not necessarily distinct). For every two numbers in the frame $a$ and $b$ distinct such that $$\frac{1}{2}<\frac{a}{b}<2,$$ an allowed operation is to delete $a$ and $b$ and write $2a-b$ and $2b-a$ in their place. Show that we can do the operation only a finite number of times.

Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$

Alexa wrote the first $16$ numbers of a sequence: \[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\] Then she continued following the same pattern, until she had $2015$ numbers in total. What was the last number she wrote?

In a plane, the point $(x,y)$ has temperature $x^2+y^2-6x+4y$. Determine the coldest point of the plane and its temperature.

Let $ P(x)$ be a polynomial such that the following inequalities are satisfied: \[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\] and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\] Prove that for every positive natural number $ n,$ $ P(n)$ is positive.

Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]

Solve in real numbers the system: $$\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}$$

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following. For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$. Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

Prove the existence of a unique sequence $\{u_n\} \ (n = 0, 1, 2 \ldots )$ of positive integers such that \[u_n^2 = \sum_{r=0}^n \binom{n+r}{r} u_{n-r} \qquad \text{for all } n \geq 0\]

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]

For which values of the parameter $a$ does the equation \[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\] has three different real roots.