An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. The aquarium is filled with water to a depth of $ 37$ cm. A rock with volume $ 1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? $ \textbf{(A)}\ 0.25 \qquad \textbf{(B)}\ 0.5 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 1.25 \qquad \textbf{(E)}\ 2.5$

[i]Triangular numbers[/i] are numbers of the form $1 + 2 + . . . + n$ with positive integer $n$, that is $1, 3, 6, 10$, . . . . Find the largest non-triangular positive integer number that cannot be represented as the sum of distinct triangular numbers. [i]Proposed by A. Golovanov[/i]

Calculate the limit of the following sequences: [b]a)[/b] n^{n!}/(n!)^n [b]b)[/b] n^{ln n}/n! [i]Adrian Troie[/i]

In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.

The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)

Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

Fix a positive integer $n$ and a finite graph with at least one edge; the endpoints of each edge are distinct, and any two vertices are joined by at most one edge. Vertices and edges are assigned (not necessarily distinct) numbers in the range from $0$ to $n-1$, one number each. A vertex assignment and an edge assignment are [i]compatible[/i] if the following condition is satisfied at each vertex $v$: The number assigned to $v$ is congruent modulo $n$ to the sum of the numbers assigned to the edges incident to $v$. Fix a vertex assignment and let $N$ be the total number of compatible edge assignments; compatibility refers, of course, to the fixed vertex assignment. Prove that, if $N \neq 0$, then the prime divisors of $N$ are all at most $n$.

Given a natural number $k$, find the smallest natural number $C$ such that $$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$.

Lizzie writes a list of fractions as follows. First, she writes $\frac11$, the only fraction whose numerator and denominator add to $2$. Then she writes the two fractions whose numerator and denominator add to $3$, in increasing order of denominator. Then she writes the three fractions whose numerator and denominator sum to $4$ in increasing order of denominator. She continues in this way until she has written all the fractions whose numerator and denominator sum to at most $1000$. So Lizzie's list looks like: $$\frac11, \frac21, \frac12 , \frac31 , \frac22, \frac13, \frac41, \frac32, \frac23, \frac14, ..., \frac{1}{999}.$$ Let $p_k$ be the product of the first $k$ fractions in Lizzie's list. Find, with proof, the value of $p_1 + p_2 + ...+ p_{499500}$.

Given an acute-angled triangle $ABC$, in which $P$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $AC$, respectively. A point $H$ is taken inside the triangle and perpendiculars $HK$, $HS$, $HQ$ are lowered from it to the sides $AB$, $BC$, $AC$, respectively ($K \in AB$, $S \in BC$, $Q \in AC$). It turned out that $MK = MQ$, $NS = NK$, $PS=PQ$. Prove that $H$ is the point of intersection of the altitudes of triangle $ABC$.

Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.

The functions $f(x)-x$ and $f(x^2)-x^6$ are defined for all positive $x$ and increase. Prove that the function $f(x^3) -\frac{\sqrt3}{2} x^6$ also increases for all positive $x$.

Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.

Find the number of sets with $10$ distinct positive integers such that the average of its elements is less than 6.

For all $x> \sqrt 2$, $y> \sqrt 2$ numbers, prove that $$x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2$$

For each positive integer $m$, denote by $d(m)$ the number of positive divisors of $m$. We say that a positive integer $n$ is [i]Burapha[/i] integer if it satisfy the following condition [list] [*] $d(n)$ is an odd integer. [*] $d(k) \leqslant d(\ell)$ holds for every positive divisor $k, \ell$ of $n$, such that $k < \ell$ [/list] Find all Burapha integer.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions \[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\] and $f(-1) \neq 0$.

For a positive integer $n$, ($n\ge 2$), find the number of sets with $2n + 1$ points $P_0, P_1,..., P_{2n}$ in the coordinate plane satisfying the following as its elements: - $P_0 = (0, 0),P_{2n}= (n, n)$ - For all $i = 1,2,..., 2n - 1$, line $P_iP_{i+1}$ is parallel to $x$-axis or $y$-axis and its length is $1$. - Out of $2n$ lines$P_0P_1, P_1P_2,..., P_{2n-1}P_{2n}$, there are exactly $4$ lines that are enclosed in the domain $y \le x$.

[b]i.)[/b] The polynomial $x^{2 \cdot k} + 1 + (x+1)^{2 \cdot k}$ is not divisible by $x^2 + x + 1.$ Find the value of $k.$ [b]ii.)[/b] If $p,q$ and $r$ are distinct roots of $x^3 - x^2 + x - 2 = 0$ the find the value of $p^3 + q^3 + r^3.$ [b]iii.)[/b] If $r$ is the remainder when each of the numbers 1059, 1417 and 2312 is divided by $d,$ where $d$ is an integer greater than one, then find the value of $d-r.$ [b]iv.)[/b] What is the smallest positive odd integer $n$ such that the product of \[ 2^{\frac{1}{7}}, 2^{\frac{3}{7}}, \ldots, 2^{\frac{2 \cdot n + 1}{7}} \] is greater than 1000?

Let $\Gamma_1$ and $\Gamma_2$ be two circles with respective centres $O_1$ and $O_2$ intersecting in two distinct points $A$ and $B$ such that $\angle{O_1AO_2}$ is an obtuse angle. Let the circumcircle of $\Delta{O_1AO_2}$ intersect $\Gamma_1$ and $\Gamma_2$ respectively in points $C (\neq A)$ and $D (\neq A)$. Let the line $CB$ intersect $\Gamma_2$ in $E$ ; let the line $DB$ intersect $\Gamma_1$ in $F$. Prove that, the points $C, D, E, F$ are concyclic.

Two circles meeting at points $P$ and $R$ are given. Let $\ell_1$, $\ell_2$ be two lines passing through $P$. The line $\ell_1$ meets the circles for the second time at points $A_1$ and $B_1$. The tangents at these points to the circumcircle of triangle $A_1RB_1$ meet at point $C_1$. The line $C_1R$ meets $A_1B_1$ at point $D_1$. Points $A_2$, $B_2, C_2, D_2$ are defined similarly. Prove that the circles $D_1D_2P$ and $C_1C_2R$ touch.

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each. [i]Radu Gologan and Dan Schwartz[/i]

Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25,$ and $36$. What is the sum of the areas of the six rectangles? $\textbf{(A) }91\qquad\textbf{(B) }93\qquad\textbf{(C) }162\qquad\textbf{(D) }182\qquad \textbf{(E) }202$

The [i]distance[/i] between two vertices in a connected graph is defined to be the length of the shortest path between them. How many graphs with the vertex set $\{0,1,2,\dots,6\}$ satisfy the following property: there are $3$ vertices of distance $1$ away from vertex $0$, $2$ of distance $2$ away, and $1$ of distance $3$ away?