$ x \plus{} 19y \equiv 0 \pmod {23}$ and $ x \plus{} y < 69$. How many pairs of $ (x,y)$ are there in positive integers? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 102 \qquad\textbf{(C)}\ 105 \qquad\textbf{(D)}\ 109 \qquad\textbf{(E)}\ \text{None}$

[b](a)[/b] Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$. Prove that this altitude $DH_4$ intersects all the other three altitudes. [b](b)[/b] If we further know that a second altitude, say the one from vertex A to the face $BCD$, also passes through the orthocenter $H_1$ of $\triangle BCD$, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.

Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.

From a set of integers $\{1,...,100\}$, $k$ integers were deleted. Is it always possible to choose $k$ distinct integers from the remaining set such that their sum is $100$ if [b](a) $k=9$?[/b] [b](b) $k=8$?[/b]

trapezoid has side lengths $10, 10, 10$, and $22$. Each side of the trapezoid is the diameter of a semicircle with the two semicircles on the two parallel sides of the trapezoid facing outside the trapezoid and the other two semicircles facing inside the trapezoid as shown. The region bounded by these four semicircles has area $m + n\pi$, where m and n are positive integers. Find $m + n$. [img]https://3.bp.blogspot.com/-s8BoUPKVUQk/XoEaIYvaz4I/AAAAAAAALl0/ML0klwHogGYWkNhY6maDdI93_GkfL_eyQCK4BGAYYCw/s200/2018%2Bps%2Bhs9.png[/img]

Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle. [i]Alexandr Matveev[/i]

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.

Find all polynomials $P(x)$ and $Q(x)$ with real coefficients such that $$P(Q(x))=P(x)^{2017}$$ for all real numbers $x$.

Given are 10 quadric equations $x^2+a_1x+b_1=0$, $x^2+a_2x+b_2=0$,..., $x^2+a_{10}x+b_{10}=0$. It is known that each of these equations has two distinct real roots and the set of all solutions is ${1,2,...10,-1,-2...,-10}$. Find the minimum value of $b_1+b_2+...+b_{10}$

Complex number $z=\cos\theta+\text{i}\sin\theta(0\leq\theta\leq\pi)$. Points that three complex numbers $z,(1+\text{i})z,2\overline{z}$ refer to on complex plane are $P,Q,R$. When $P,Q,R$ are not collinear, $PQSR$ is a parallelogram. The longest distance between $S$ and the original point is________.

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

A circle centered at $A$ intersects sides $AC$ and $AB$ of $\triangle ABC$ at $E$ and $F$, and the circumcircle of $\triangle ABC$ at $X$ and $Y$. Let $D$ be the point on $BC$ such that $AD$, $BE$, $CF$ concur. Let $P=XE\cap YF$ and $Q=XF\cap YE$. Prove that the foot of the perpendicular from $D$ to $EF$ lies on $PQ$.

Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$.

Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.

Let $ABC$ a acute triangle. (a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$: \[\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}\] (b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$); (c) Show that the power of $X$ wrt the circumcircle of $ABC$ is: \[-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4\] Where $a=BC$ , $b=AC$ and $c=AB$.

A train consist of $2010$ wagons containing gold coins, all of the same shape. Any two coins have equal weight provided that they are in the same wagon, and differ in weight if they are in different ones. The weight of a coin is one of the positive reals \begin{align*} m_1 <m_2 <\ldots <m_{2010} \end{align*} Each wagon is marked by a label with one of the numbers $m_1,m_2, \ldots , m_{2010}$ (the numbers on different labels are different). A controller has a pair of scales (allowing only to compare masses) at his disposal. During each measurement he can use an arbitrary number of coins from any of the wagons. The controller has the task to establish: if all labels show rightly the common weight of the coins in a wagon or if there exists at least one wrong label. What is the least number of measurement that the controller has to perform to accomplish his task?

Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.

Let $m,n\geq 2$ be positive integers and $S\subseteq [1,m]\times [1,n]$ be a set of lattice points. Prove that if \[|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor\]then there exists a circle which passes through at least four distinct points of $S.$

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

Let be two quadrilaterals $ ABCD,A'B'C'D' $ with $ AB,BC,CD,AC,BD $ being perpendicular to $ A'B',B'C',C'D',A'C',B'D', $ respectively. Show that $ AD $ is perpendicular to $ A'D'. $

Show that $r = 2$ is the largest real number $r$ which satisfies the following condition: If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities \[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\] for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.

Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

Consider the following graph algorithm (where $V$ is the set of vertices and $E$ the set of edges in $G$). $\textbf{procedure }\textsc{s}(G)$ $\qquad \textbf{if } |V| = 0\textbf{ then return true}$ $\qquad \textbf{for }(u,v)\textbf{ in }E\textbf{ do}$ $\qquad\qquad H\gets G-u-v$ $\qquad\qquad\textbf{if } \textsc{s}(H)\textbf{ then return true}$ $\qquad\textbf{return false}$ Here $G - u - v$ means the subgraph of $G$ which does not contain vertices $u,v$ and all edges using them. How many graphs $G$ with vertex set $\{1,2,3,4,5,6\}$ and [i]exactly[/i] $6$ edges satisfy $s(G)$ being true?

A regular tetrahedron with $5$-inch edges weighs $2.5$ pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has $6$-inch edges? Express your answer as a decimal rounded to the nearest hundredth.