For a number $n$ in base $10$, let $f(n)$ be the sum of all numbers possible by removing some digits of $n$ (including none and all). For example, if $n = 1234$, $f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979$; this is formed by taking the sums of all numbers obtained when removing no digit from $n$ (1234), removing one digit from $n$ (123, 124, 134, 234), removing two digits from $n$ (12, 13, 14, 23, 24, 34), removing three digits from $n$ (1, 2, 3, 4), and removing all digits from $n$ (0). If $p$ is a 2011-digit integer, prove that $f(p)-p$ is divisible by $9$. Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number $101$, $1$ appears three times (by removing the first digit, giving $01$ which is equal to $1$, removing the first two digits, or removing the last two digits), so it is counted three times.

Find all function $f: \mathbb{R}^+ \rightarrow \mathbb{R}$,such that the inequality $$f(x) + f\left(\frac{y}{x}\right) \leqslant \frac{x^3}{y^2} + \frac{y}{x^3}$$ holds for all positive reals $x,y$ and for every positive real $x$, there exist positive reals $y$, such that the equality holds.

Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $ [i]Dumitru Acu[/i]

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? $\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$

In a given group of people $\mathcal{F}$, each member has at least two acquaintances from $\mathcal{F}$. Moreover, for each cycle $A_{1} \leftrightarrow A_{2} \leftrightarrow ... \leftrightarrow A_{n} \leftrightarrow A_{1}$ in $\mathcal{F}$ (here '$X \leftrightarrow Y$' means that $X$ and $Y$ are acquaintances), each $A_i$ knows exactly two other members $A_j$ of the cycle. Prove that there exist $X, Y \in \mathcal{F}$ such that each of them has exactly two acquaintances in $\mathcal{F}$, and $X, Y$ have at least one common acquaintance in the group. [i]Authored by Mirko Petrusevski[/i]

Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$

In right angled triangle $ABC$ point $D$ is midpoint of hypotenuse, and $E$ and $F$ are points on shorter sides $AC$ and $BC$, respectively, such that $DE \perp DF$. Prove that $EF^2=AE^2+BF^2$

A student is asked to measure $30.0 {\text{g}}$ of methanol $(d=0.7914 \text{g/mL at 25}^{\circ}\text{C})$ but has only a graduated cylinder with which to measure it. What volume of methanol should the student use to obtain the required ${30.0 \text{g}}$? ${ \textbf{(A)}\ 23.7 \text{mL} \qquad\textbf{(B)}\ 30.0 \text{mL} \qquad\textbf{(C)}\ 32.4 \text{mL} \qquad\textbf{(D)}\ 37.9 \text{mL} }$

Prove that $2^{147} - 1$ is divisible by $343$.

Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

Given is a square $ABCD$ with side length $1$. Points $K$, $L$, $M$, and $N$, distinct from the vertices of the square, lie on segments $AB$, $BC$, $CD$, and $DA$, respectively. Prove that the perimeter of at least one of the triangles $ANK$, $BKL$, $CLM$, $DMN$ is less than $2$.

Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?

$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.

Determine a) the smallest number b) the biggest number $n \ge 3$ of non-negative integers $x_1, x_2, ... , x_n$, having the sum $2011$ and satisfying: $x_1 \le | x_2 - x_3 | , x_2 \le | x_3 - x_4 | , ... , x_{n-2} \le | x_{n-1} -x_n | , x_{n-1} \le | x_n - x_1 |$ and $x_n \le | x_1 - x_2 | $.

Find the number of distinct permutations of $ITEST$. [i](.1 point)[/i]

If $p$ and $q$ are positive numbers with $p+q = 1$, knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$, show that $\frac{x+y}{2} \ge \sqrt{xy}$, $\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$, $\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$

For $1\leq i\leq 215$ let $a_i=\frac{1}{2^i}$ and $a_{216}=\frac{1}{2^{215}}$. Let $x_1,x_2,\ldots,x_{216}$ be positive real numbers such that \[ \sum\limits_{i=1}^{216} x_i=1 \text{\quad and \quad} \sum\limits_{1\leq i<j \leq 216} x_ix_j = \frac{107}{215}+ \sum\limits_{i=1}^{216} \frac{a_ix_i^2}{2(1-a_i)}.\] The maximum possible value of $x_2=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Consider a triangle $ABC$ with $AB = 13, BC = 14, CA = 15$. A line perpendicular to $BC$ divides the interior of $\vartriangle BC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\vartriangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$.

How many natural number triples $(x,y,z)$ are there such that $xyz = 10^6$? $ \textbf{(A)}\ 568 \qquad\textbf{(B)}\ 784 \qquad\textbf{(C)}\ 812 \qquad\textbf{(D)}\ 816 \qquad\textbf{(E)}\ 824 $

Let $ABC$ be an acute triangle, and let $H_A$, $H_B$, and $H_C$ be the feet of the altitudes relative to vertices $A$, $B$, and $C$, respectively. Define $I_A$, $I_B$, and $I_C$ as the incenters of triangles $AH_B H_C$, $BH_C H_A$, and $CH_A H_B$, respectively. Let $T_A$, $T_B$, and $T_C$ be the intersection of the incircle of triangle $ABC$ with $BC$, $CA$, and $AB$, respectively. Prove that the triangles $I_A I_B I_C$ and $T_A T_B T_C$ are congruent.

Let $ S$ be the set of permutations of the sequence $ 1, 2, 3, 4, 5$ for which the first term is not $ 1$. A permutation is chosen randomly from $ S$. The probability that the second term is $ 2$, in lowest terms, is $ a/b$. What is $ a \plus{} b$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 19$

Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }18$

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

There is a long-standing conjecture that there is no number with $2n + 1$ instances in Pascal’s triangle for $n \ge 2$. Assuming this is true, for how many $n \le 100, 000$ are there exactly $3$ instances of $n$ in Pascal’s triangle?