Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common? $\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $

A given natural number $N$ is being decomposed in a sum of some consecutive integers. [b]a.)[/b] Find all such decompositions for $N=500.$ [b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only? [b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$ [b]Note by Darij:[/b] The $0$ is not considered as a natural number.

In how many ways can $2^{n}$ be expressed as the sum of four squares of natural numbers?

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices are always on the $x$- and $y$-axes respectively. Find the locus of the midpoint of the square.

a) Each of the side of the convex hexagon is longer than $1$. Does it necessary have a diagonal longer than $2$? b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?

If a right triangle is drawn in a semicircle of radius $1/2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?

Let $ABC$ be an equilateral triangle with side length $2$, and let $M$ be the midpoint of $\overline{BC}$. Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$. What is the length of $\overline{XY}$?

2.Let $P(x) = x^2 + ax + b$ be a quadratic polynomial where a, b are real numbers. Suppose $P(-1)^2$ , $P(0)^2$, $P(1)^2$ is an Arithmetic progression of positive integers. Prove that a, b are integers.

Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

Study the convergence of the sequence $$ \left( \sum_{k=2}^{n+1} \sqrt[k]{n+1} -\sum_{k=2}^{n} \sqrt[k]{n} \right)_{n\ge 2} , $$ and calculate its limit. [i]Dan Negulescu[/i]

Given $n\geq2$, let $\mathcal{A}$ be a family of subsets of the set $\{1,2,\dots,n\}$ such that, for any $A_1,A_2,A_3,A_4 \in \mathcal{A}$, it holds that $|A_1 \cup A_2 \cup A_3 \cup A_4| \leq n -2$. Prove that $|\mathcal{A}| \leq 2^{n-2}.$

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.

Fix integers $n\geq 2$ and $1\leq m\leq n-1$. Let $a_0, a_1, \dots, a_n$ be non-negative real numbers satisfying $a_0+a_1+\dots +a_n=1$. Prove that, if $\sum_{k=0}^n a_kx^k < x^m$ for some $0<x<1$, then $$\sum_{k=0}^{m-1}(m-k)a_k < \sum_{k=m+1}^n (k-m)a_k.$$

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

[b]p1.[/b] If Clark wants to divide $100$ pizzas among $25$ people so that each person receives the same number of pizzas, how many pizzas should each person receive? [b]p2.[/b] In a group of $3$ people, every pair of people shakes hands once. How many handshakes occur? [b]p3.[/b] Dylan and Joey have $14$ costumes in total. Dylan gives Joey $4$ costumes, and Joey now has the number of costumes that Dylan had before giving Joey any costumes. How many costumes does Dylan have now? [b]p4.[/b] At Banjo Borger, a burger costs $7$ dollars, a soda costs $2$ dollars, and a cookie costs $3$ dollars. Alex, Connor, and Tony each spent $11$ dollars on their order, but none of them got the same order. If Connor bought the most cookies, how many cookies did Connor buy? [b]p5.[/b] Joey, James, and Austin stand on a large, flat field. If the distance from Joey to James is $30$ and the distance from Austin to James is $18$, what is the minimal possible distance from Joey to Austin? [b]p6.[/b] If the first and third terms of a five-term arithmetic sequence are $3$ and $8$, respectively, what is the sum of all $5$ terms in the sequence? [b]p7.[/b] What is the area of the $S$-shaped figure below, which has constant vertical height $5$ and width $10$? [img]https://cdn.artofproblemsolving.com/attachments/3/c/5bbe638472c8ea8289b63d128cd6b449440244.png[/img] [b]p8.[/b] If the side length of square $A$ is $4$, what is the perimeter of square $B$, formed by connecting the midpoints of the sides of $A$? [b]p9.[/b] The Chan Shun Auditorium at UC Berkeley has room number $2050$. The number of seats in the auditorium is a factor of the room number, and there are between $150$ and $431$ seats, inclusive. What is the sum of all of the possible numbers of seats in Chan Shun Auditorium? [b]p10.[/b] Krishna has a positive integer $x$. He notices that $x^2$ has the same last digit as $x$. If Krishna knows that $x$ is a prime number less than $50$, how many possible values of $x$ are there? [b]p11.[/b] Jing Jing the Kangaroo starts on the number $1$. If she is at a positive integer $n$, she can either jump to $2n$ or to the sum of the digits of $n$. What is the smallest positive integer she cannot reach no matter how she jumps? [b]p12.[/b] Sylvia is $3$ units directly east of Druv and runs twice as fast as Druv. When a whistle blows, Druv runs directly north, and Sylvia runs along a straight line. If they meet at a point a distance $d$ units away from Druv's original location, what is the value of $d$? [b]p13.[/b] If $x$ is a real number such that $\sqrt{x} + \sqrt{10} = \sqrt{x + 20}$, compute $x$. [b]p14.[/b] Compute the number of rearrangments of the letters in $LATEX$ such that the letter $T$ comes before the letter $E$ and the letter $E$ comes before the letter $X$. For example, $TLEAX$ is a valid rearrangment, but $LAETX$ is not. [b]p15.[/b] How many integers $n$ greater than $2$ are there such that the degree measure of each interior angle of a regular $n$-gon is an even integer? [b]p16.[/b] Students are being assigned to faculty mentors in the Berkeley Math Department. If there are $7$ students and $3$ mentors and each student has exactly one mentor, in how many ways can students be assigned to mentors given that each mentor has at least one student? [b]p17.[/b] Karthik has a paper square of side length $2$. He folds the square along a crease that connects the midpoints of two opposite sides (as shown in the left diagram, where the dotted line indicates the fold). He takes the resulting rectangle and folds it such that one of its vertices lands on the vertex that is diagonally opposite. Find the area of Karthik's final figure. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01aa386f6616cafeed5f95ababb27bf24657f6.png[/img] [b]p18.[/b] Sally is inside a pen consisting of points $(a, b)$ such that $0 \le a, b \le 4$. If she is currently on the point $(x, y)$, she can move to either $(x, y + 1)$, $(x, y - 1)$, or $(x + 1, y)$. Given that she cannot revisit any point she has visited before, find the number of ways she can reach $(4, 4)$ from $(0, 0)$. [b]p19.[/b] An ant sits on the circumference of the circular base of a party hat (a cone without a circular base for the ant to walk on) of radius $2$ and height $\sqrt{5}$. If the ant wants to reach a point diametrically opposite of its current location on the hat, what is the minimum possible distance the ant needs to travel? [img]https://cdn.artofproblemsolving.com/attachments/3/4/6a7810b9862fd47106c3c275c96337ef6d23c2.png[/img] [b]p20.[/b] If $$f(x) = \frac{2^{19}x + 2^{20}}{ x^2 + 2^{20}x + 2^{20}}.$$ find the value of $f(1) + f(2) + f(4) + f(8) + ... + f(220)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.

For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? ${ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8 $

Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$