Solve the equation \[\int_0^1(x+y)^2u(x)dx=\lambda u(y)+1\]

For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.

Prove that for any $n$ there exists a positive integer $k$ such that all the numbers $k\cdot2^s+1~(s=1,\ldots,n)$ are composite.

Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

Let $ABC$ be a triangle such that $AB = 13$, $BC = 14$, $AC = 15$. Let $M$ be the midpoint of $BC$ and define $P \ne B$ to be a point on the circumcircle of $ABC$ such that $BP \perp PM$. Furthermore, let $H$ be the orthocenter of $ABM$ and define $Q$ to be the intersection of $BP$ and $AC$. If $R$ is a point on $HQ$ such that $RB \perp BC$, find the length of $RB$.

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

Let $n$ be a positive integer. Prove that $a_1!a_2! ... a_n! < k!$, where $k$ is an integer which is greater than the sum of the positive integers $a_1, a_2,.., a_n$.

Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$? $\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$

In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment

Let a polynomial $ P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1$ $ (r > 0)$ such that the equation $ P(x) \equal{} 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n$. Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.

a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $, for which $x/y^2$ and $y/x^2$, then $xy/(x+y) ^k$ b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $, for which $x/y^2$, $y/z^2$ and $z/x^2$, then $xyz/(x+y+z)^l$

Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is a positive integer, find the sum of all possible values of $n.$

A regular dodecagon ($ 12$ sides) is inscribed in a circle with radius $ r$ inches. The area of the dodecagon, in square inches, is: $ \textbf{(A)}\ 3r^2 \qquad \textbf{(B)}\ 2r^2 \qquad \textbf{(C)}\ \frac{3r^2 \sqrt{3}}{4} \qquad \textbf{(D)}\ r^2 \sqrt{3} \qquad \textbf{(E)}\ 3r^2 \sqrt{3}$

Richard comes across an infinite row of magic hats, $H_1, H_2, \dots$ each of which may contain a dollar bill with probabilities $p_1, p_2, \dots$. If Richard draws a dollar bill from $H_i$, then $p_{i+1} = p_i$, and if not, $p_{i+1}=\frac{1}{2}p_i$. If $p_1 = \frac{1}{2}$ and $E$ is the expected amount of money Richard makes from all the hats, compute $\lfloor 100E \rfloor$. [i]Proposed by Alex Li[/i]

Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers $\{1, 2, 3\ldots , 10\}$ across his paper such that: [list] [*]The leftmost number is 1. [*]The rightmost number is 10. [*]Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list] How many such permutations are there?

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.

Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]

If $x=\frac{1}{3}$, what is the value, rounded to $100$ decimal digits, of $\sum_{n=0}^{7}\frac{2^n}{1+x^{2^n}}$?

Find all primes $p$ such that there exist an integer $n$ and positive integers $k, m$ which satisfies the following. $$ \frac{(mk^2+2)p-(m^2+2k^2)}{mp+2} = n^2$$

Let $n$ be a positive integer and let $S \subseteq \{0, 1\}^n$ be a set of binary strings of length $n$. Given an odd number $x_1, \dots, x_{2k + 1} \in S$ of binary strings (not necessarily distinct), their [i]majority[/i] is defined as the binary string $y \in \{0, 1\}^n$ for which the $i^{\text{th}}$ bit of $y$ is the most common bit among the $i^{\text{th}}$ bits of $x_1, \dots,x_{2k + 1}$. (For example, if $n = 4$ the majority of 0000, 0000, 1101, 1100, 0101 is 0100.) Suppose that for some positive integer $k$, $S$ has the property $P_k$ that the majority of any $2k + 1$ binary strings in $S$ (possibly with repetition) is also in $S$. Prove that $S$ has the same property $P_k$ for all positive integers $k$. [i]Proposed by Joshua Brakensiek[/i]

The Young’s modulus, $E$, of a material measures how stiff it is; the larger the value of $E$, the more stiff the material. Consider a solid, rectangular steel beam which is anchored horizontally to the wall at one end and allowed to deflect under its own weight. The beam has length $L$, vertical thickness $h$, width $w$, mass density $\rho$, and Young’s modulus $E$; the acceleration due to gravity is $g$. What is the distance through which the other end moves? ([i]Hint: you are expected to solve this problem by eliminating implausible answers. All of the choices are dimensionally correct.[/i]) (a) $h \exp\left( \frac{\rho gL}{E} \right)$ (b) $2\frac{\rho gh^2}{E}$ (c) $\sqrt{2Lh}$ (d) $\frac{3}{2}\frac{\rho gL^4}{Eh^2}$ (e) $\sqrt{3}\frac{EL}{\rho gh}$

Richard rolls a fair six-sided die repeatedly until he rolls his twentieth prime number or his second even number. Compute the probability that his last roll is prime.

Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$. Given that $a^2+b^2=17$, compute $c$.

After the schoolday is over, Juku must attend an extra math class. The teacher writes a quadratic equation $ x^2\plus{} p_1x\plus{}q_1 \equal{} 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 \plus{} p_2x \plus{} q_2 \equal{} 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.