Let $n \in \mathbb N$. Prove that for each factor $m \ge n$ of $n(n + 1)/2$, one can partition the set $\{1,2, 3,\cdots , n\}$ into disjoint subsets such that the sum of elements in each subset is equal to $m$.

The number of ordered pairs of integers $ (m,n)$ for which $ mn \ge 0$ and \[m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3\] is equal to $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 33\qquad \textbf{(D)}\ 35\qquad \textbf{(E)}\ 99$

The determinant \[\begin{vmatrix}3&-2&5\\ 7&1&-4\\ 5&2&3\end{vmatrix}\] has the same value as the determinant \[\begin{vmatrix}x&1+x&2+x\\ 3&0&1\\ 1&1&0\end{vmatrix}\] Find $x$.

The figure below is a $ 4 \times 4$ grid of points. [asy]unitsize(15); for ( int x = 1; x <= 4; ++x ) for ( int y = 1; y <= 4; ++y ) dot((x, y));[/asy]Each pair of horizontally adjacent or vertically adjacent points are distance 1 apart. In the plane of this grid, how many circles of radius 1 pass through exactly two of these grid points?

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins? [b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits? $\begin{tabular}{ccccc} & & & 9 & 5 \\ x & & & * & * \\ \hline & & & * & * \\ + & 1 & * & * & \\ \hline & * & * & * & * \\ \end{tabular}$ Find all solutions. [b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes. [b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all positive integers $n$ such that the decimal representation of the number $6^n + 1$ has all its digits the same.

Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner. A higher ranked team always wins against a lower-ranked team. If the probability that the team ranked $3$ (the third-best team) is one of the last four teams remaining can be written in simplest form as $\dfrac{m}{n}$, compute $m+n$.

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

Given a finite number of points in the plane as well as many different rays starting at the origin. It is always possible to pair the points with the rays so that they parallell displaced rays starting in respective points do not intersect?

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$. [i]Proposed by Anant Mudgal[/i]

Prove that if $u, v$ are any nonnegative real numbers, and $a,b$ positive real numbers such that $a + b = 1$, then $$u^a v^b \le au + bv.$$

A jar contains one white marble, two blue marbles, three red marbles, and four green marbles. If you select two of these marbles without replacement, the probability that both marbles will be the same color is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$ Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that $P(x,y)=G(x,y)\cdot H(x,y).$

Let $n$ be a positive integer and $A$ be a family of subsets of the set $\{1,2,...,n\},$ none of which contains another subset from A . Find the largest possible cardinality of $A$ .

For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

If \[ \begin {eqnarray*} x + y + z + w = 20 \\ y + 2z - 3w = 28 \\ x - 2y + z = 36 \\ -7x - y + 5z + 3w = 84 \]then what is $(x,y,z,w)$?

There are n stations $1,2,...,n$ in a broken road (like in Cars) in that order such that the distance between station $i$ and $i+1$ is one unit. The distance betwen two positions of cars is the minimum units needed to be fixed so that every car can go from its place in the first position to its place in the second (two cars can be in the same station in a position). Prove that for every $\alpha<1$ thre exist $n$ and $100^n$ positions such that the distance of every two position is at least $n\alpha$.

Let $a_1,a_2,...,a_{1994}$ be real numbers in the interval $[-1,1]$, $$S=\frac{a_1+a_2+...+a_{1994}}{1994}.$$ Prove that for an arbitrary natural , $1\le n \le 1994$, holds the inequality $$| a_1+a_2+...+a_n - nS | \le 997.$$

Triangles $ABC$ and $XYZ$ are similar, with $A$ corresponding to $X$ and $B$ to $Y$. If $AB=3$, $BC=4$, and $XY=5$, then $YZ$ is: $ \textbf{(A)}\ 3\frac 3 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 6\frac 1 4 \qquad \textbf{(D)}\ 6\frac 2 3 \qquad \textbf{(E)}\ 8$

a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$. [i]Dan Schwartz[/i]

$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Can the number of [b]clockwise[/b] triangles be exactly $2017$? [i]Proposed by Morteza Saghafian[/i]

If $ x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}$, then $ x^3 \plus{} 18x$ = ? $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10$