Find the smallest positive integer $n$ with the following property: in every sequence of $n$ positive integers such that the sum of the $n$ numbers is equal to $2013$, there are some consecutive terms whose sum is equal to $31$.

Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$. A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove, Points $B,Q,R,P$ are concyclic.

Let $f(x)=(x-1)^3$. Find $f'(0)$.

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

[b]p1.[/b] What is the units digit of the product of the first seven primes? [b]p2. [/b]In triangle $ABC$, $\angle BAC$ is a right angle and $\angle ACB$ measures $34$ degrees. Let $D$ be a point on segment $ BC$ for which $AC = CD$, and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$. What is the measure of $\angle BED$? [b]p3.[/b] Chad numbers five paper cards on one side with each of the numbers from $ 1$ through $5$. The cards are then turned over and placed in a box. Jordan takes the five cards out in random order and again numbers them from $ 1$ through $5$ on the other side. When Chad returns to look at the cards, he deduces with great difficulty that the probability that exactly two of the cards have the same number on both sides is $p$. What is $p$? [b]p4.[/b] Only one real value of $x$ satisfies the equation $kx^2 + (k + 5)x + 5 = 0$. What is the product of all possible values of $k$? [b]p5.[/b] On the Exeter Space Station, where there is no effective gravity, Chad has a geometric model consisting of $125$ wood cubes measuring $ 1$ centimeter on each edge arranged in a $5$ by $5$ by $5$ cube. An aspiring carpenter, he practices his trade by drawing the projection of the model from three views: front, top, and side. Then, he removes some of the original $125$ cubes and redraws the three projections of the model. He observes that his three drawings after removing some cubes are identical to the initial three. What is the maximum number of cubes that he could have removed? (Keep in mind that the cubes could be suspended without support.) [b]p6.[/b] Eric, Meena, and Cameron are studying the famous equation $E = mc^2$. To memorize this formula, they decide to play a game. Eric and Meena each randomly think of an integer between $1$ and $50$, inclusively, and substitute their numbers for $E$ and $m$ in the equation. Then, Cameron solves for the absolute value of $c$. What is the probability that Cameron’s result is a rational number? [b]p7.[/b] Let $CDE$ be a triangle with side lengths $EC = 3$, $CD = 4$, and $DE = 5$. Suppose that points $ A$ and $B$ are on the perimeter of the triangle such that line $AB$ divides the triangle into two polygons of equal area and perimeter. What are all the possible values of the length of segment $AB$? [b]p8.[/b] Chad and Jordan are raising bacteria as pets. They start out with one bacterium in a Petri dish. Every minute, each existing bacterium turns into $0, 1, 2$ or $3$ bacteria, with equal probability for each of the four outcomes. What is the probability that the colony of bacteria will eventually die out? [b]p9.[/b] Let $a = w + x$, $b = w + y$, $c = x + y$, $d = w + z$, $e = x + z$, and $f = y + z$. Given that $af = be = cd$ and $$(x - y)(x - z)(x - w) + (y - x)(y - z)(y - w) + (z - x)(z - y)(z - w) + (w - x)(w - y)(w - z) = 1,$$ what is $$2(a^2 + b^2 + c^2 + d^2 + e^2 + f^2) - ab - ac - ad - ae - bc - bd - bf - ce - cf - de - df - ef ?$$ [b]p10.[/b] If $a$ and $b$ are integers at least $2$ for which $a^b - 1$ strictly divides $b^a - 1$, what is the minimum possible value of $ab$? Note: If $x$ and $y$ are integers, we say that $x$ strictly divides $y$ if $x$ divides $y$ and $|x| \ne |y|$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$. created by kunny

Find all positive integers $a, b, c, d$, such that the following equality takes place for an infinity of positive integers $n$ $$(1^a + 2^a +...+ n^a)^b = (1^c + 2^c +...+ n^c)^d$$

Real numbers $x_1,x_2,\ldots ,x_{1996}$ have the following property: For any polynomial $W$ of degree $2$ at least three of the numbers $W(x_1),W(x_2),\ldots ,W(x_{1996})$ are equal. Prove that at least three of the numbers $x_1,x_2,\ldots ,x_{1996}$ are equal.

Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that [b]a)[/b] $f(p)=1$ for every prime $p$. [b]b)[/b] $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$ Find the least number $n \ge 2021$ such that $f(n)=n$

Given a sequence of eight integers $x_{1},x_{2},...,x_{8}$ in a single operation one replaces these numbers with $|x_{1}-x_{2}|,|x_{2}-x_{3}|,...,|x_{8}-x_{1}|$. Find all the eight-term sequences of integers which reduce to a sequence with all the terms equal after finitely many single operations.

Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying \[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)} \] for all $ x \neq y$.

(Revised 2-4-2000) Let $P$ be a point interior to square $ABCD$ so that $PA=a$, $PB=b$, $PC=c$, and $c^2=a^2+2b^2$. Given only the lengths $a$, $b$, and $c$, and using only a compass and a straightedge, construct a square congruent to square $ABCD$.

One thousand students participate in the $2011$ Canadian Closed Mathematics Challenge. Each student is assigned a unique three-digit identification number $abc,$ where each of $a, b$ and $c$ is a digit between $0$ and $9,$ inclusive. Later, when the contests are marked, a number of markers will be hired. Each of the markers will be given a unique two-digit identification number $xy,$ with each of $x$ and $y$ a digit between $0$ and $9,$ inclusive. Marker $xy$ will be able to mark any contest with an identification number of the form $xyA$ or $xAy$ or $Axy,$ for any digit $A.$ What is the minimum possible number of markers to be hired to ensure that all contests will be marked?

If $x$ is a real number so $3^x=27x$, compute $\log_3 \left(\tfrac{3^{3^x}}{x^{3^3}}\right)$.

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

Determine the maximum number of three-term arithmetic progressions which can be chosen from a sequence of $n$ real numbers \[a_1<a_2<\cdots<a_n.\]

In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB|=|AC|$ such that $[CD]$ is an angle bisector and $m(\widehat{ABC})=40^\circ$. Let $F$ be a point on the extension of $[AB]$ after $B$ such that $|BC|=|AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is $m(\widehat{FBG})$? $ \textbf{(A)}\ 150^\circ \qquad\textbf{(B)}\ 135^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 105^\circ \qquad\textbf{(E)}\ \text{None of above} $

Consider the sequence $u_0, u_1, u_2, ...$ defined by $u_0 = 0, u_1 = 1,$ and $u_n = 6u_{n - 1} + 7u_{n - 2}$ for $n \ge 2$. Show that there are no non-negative integers $a, b, c, n$ such that $$ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.$$

Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$

Two medians drawn from vertices A and B of triangle ABC are perpendicular. Prove that side AB is the shortest side of ABC.

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

Brandon wants to split his orchestra of $20$ violins, $15$ violas, $10$ cellos, and $5$ basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?