Chandler wants to buy a $\$500$ dollar mountain bike. For his birthday, his grandparents send him $\$50$, his aunt sends him $\$35$ and his cousin gives him $\$15$. He earns $\$16$ per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 28$

$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$ I'm posting this problem for people to discuss

Let $r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}$. There exists a unique fifth-degree polynomial $P$ such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is $1$, and $P(r) = 0$. Evaluate $P(10)$.

Let $\displaystyle{f:{\cal R} \to {\cal R}}$ be a twice differentiable function. Suppose $\displaystyle{f\left( 0 \right) = 0}$. Prove there exists $\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}$ such that \[\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.\] [i]Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.[/i]

Let $ABCD$ be a quadrilateral such that $m(\widehat{A}) = m(\widehat{D}) = 90^\circ$. Let $M$ be the midpoint of $[DC]$. If $AC\perp BM$, $|DC|=12$, and $|AB|=9$, then what is $|AD|$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the above} $

Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.

Fill in each cell of the grid with a positive digit so that the following conditions hold: 1. each row and column contains ve distinct digits; 2. for any cage containing multiple cells of a row, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole row, and 3. for any cage containing multiple cells of a column, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole column. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [asy] unitsize(48); int[][] a = { {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}; for (int i = 0; i < 5; ++i) { for (int j = 0; j < 5; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(24pt)); } } real ep=0.1; real s=3; pen lw=linewidth(.12mm); real x=0.9; real y=1.2; draw((0+s*ep,0-ep)--(2-ep,0-ep)--(2-ep,-1+ep)--(0+ep,-1+ep)--(0+ep,0-s*ep),dashed+lw); draw((2+s*ep,0-ep)--(4-ep,0-ep)--(4-ep,-1+ep)--(2+ep,-1+ep)--(2+ep,0-s*ep),dashed+lw); draw((1+s*ep,-1-ep)--(3-ep,-1-ep)--(3-ep,-2+ep)--(1+ep,-2+ep)--(1+ep,-1-s*ep),dashed+lw); draw((2+s*ep,-3-ep)--(4-ep,-3-ep)--(4-ep,-4+ep)--(2+ep,-4+ep)--(2+ep,-3-s*ep),dashed+lw); draw((1+s*ep,-4-ep)--(3-ep,-4-ep)--(3-ep,-5+ep)--(1+ep,-5+ep)--(1+ep,-4-s*ep),dashed+lw); draw((3+s*ep,-4-ep)--(5-ep,-4-ep)--(5-ep,-5+ep)--(3+ep,-5+ep)--(3+ep,-4-s*ep),dashed+lw); label(scale(x)*"5", (0+ep,0-y*ep)); label(scale(x)*"7", (2+ep,0-y*ep)); label(scale(x)*"10", (1+ep,-1-y*ep)); label(scale(x)*"5", (2+ep,-3-y*ep)); label(scale(x)*"2", (1+ep, -4-y*ep)); label(scale(x)*"13", (3+ep, -4-y*ep)); draw((4+s*ep,0-ep)--(5-ep,0-ep)--(5-ep,-2+ep)--(4+ep,-2+ep)--(4+ep,0-s*ep),dashed+lw); draw((0+s*ep,-1-ep)--(1-ep,-1-ep)--(1-ep,-3+ep)--(0+ep,-3+ep)--(0+ep,-1-s*ep),dashed+lw); draw((3+s*ep,-1-ep)--(4-ep,-1-ep)--(4-ep,-3+ep)--(3+ep,-3+ep)--(3+ep,-1-s*ep),dashed+lw); draw((0+s*ep,-3-ep)--(1-ep,-3-ep)--(1-ep,-5+ep)--(0+ep,-5+ep)--(0+ep,-3-s*ep),dashed+lw); draw((1+s*ep,-2-ep)--(2-ep,-2-ep)--(2-ep,-4+ep)--(1+ep,-4+ep)--(1+ep,-2-s*ep),dashed+lw); draw((4+s*ep,-2-ep)--(5-ep,-2-ep)--(5-ep,-4+ep)--(4+ep,-4+ep)--(4+ep,-2-s*ep),dashed+lw); label(scale(x)*"10", (4+ep,0-y*ep)); label(scale(x)*"3", (0+ep,-1-y*ep)); label(scale(x)*"8", (3+ep,-1-y*ep)); label(scale(x)*"16", (1+ep,-2-y*ep)); label(scale(x)*"6", (4+ep,-2-y*ep)); label(scale(x)*"11", (0+ep,-3-y*ep)); [/asy]

The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.

Given $x=\phi(u,v)$ and $y=\psi(u,v)$, where $ \phi$ and $\psi$ are solutions of the partial differential equation $$(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.$$ By assuming that $x$ and $y$ are the independent variables, show that $(1)$ may be transformed to $$(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.$$ Integrate $(2)$ and show how this effects in general the solution of $(1)$. What other solutions does $(1)$ possess?

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation". Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than $10$". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?

Farmer Jim has an $8$ gallon bucket full with water. He has three empty buckets: $3$ gallons, $5$ gallons and $8$ gallons. How can he get two volumes of water, $4$ gallons each, using only the four buckets?

Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.

We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$

For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$

Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.

If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, what is the probability that this number is divisible by $11$? $\textbf{(A) }2/5\qquad\textbf{(B) }1/5\qquad\textbf{(C) }1/6\qquad\textbf{(D) }1/11\qquad \textbf{(E) }1/15$

Let's call several numbers written in a row a 'combination of numbers'. In the country of Robotland, some combinations of numbers have been declared prohibited. It is known that there are a finite number of forbidden combinations and there is an infinite decimal fraction that does not contain forbidden combinations. Prove that there is an infinite periodic decimal fraction that does not contain prohibited combinations.

Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.

Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.

Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$

Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$ Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.

Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.

Consider digits $\underline{A}, \underline{B}, \underline{C}, \underline{D}$, with $\underline{A} \neq 0,$ such that $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$ Compute the sum of all distinct possible values of $\underline{A} + \underline{B} + \underline{C} + \underline{D}$. [i]Proposed by Kyle Lee[/i]

The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.

Is it possible in a plane mark $10$ red, $10$ blue and $10$ green points (all distinct) such that three conditions hold: $i)$ For every red point $A$ there exists a blue point closer to point $A$ than any other green point $ii)$ For every blue point $B$ there exists a green point closer to point $B$ than any other red point $iii)$ For every green point $C$ there exists a red point closer to point $C$ than any other blue point