Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour? $\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 22$

Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear. [i]Proposed by CDMX[/i]

$ \Delta_1,\ldots,\Delta_n$ are $ n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.

Let $a$ and $b$ real numbers. Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function. We say that f is "smp" if $[a,b]=[c_0,c_1]\cup[c_1,c_2]...\cup[c_{n-1},c_n]$ satisfying $c_0<c_1...<c_n$ and for each $i\in\{0,1,2...n-1\}$: $c_i<x<c_{i+1} \Rightarrow f(c_i)<f(x)<f(c_{i+1})$ or $c_i>x>c_{i+1} \Rightarrow f(c_i)>f(x)>f(c_{i+1})$ Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is continuous such that for each $v\in\mathbb{R}$ there are only finitely many $x$ satisfying $f(x)=v$, then $f$ is "smp".

The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is: $\textbf{(A)}\ 6\pi \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 36\pi$

At the altitude $AH_1$ of an acute non-isosceles triangle $ABC$ chose a point $X$ , from which draw the perpendiculars $XN$ and $XM$ on the sides $AB$ and $AC$ respectively. It turned out that $H_1A$ is the angle bisector $MH_1N$. Prove that $X$ is the point of intersection of the altitudes of the triangle $ABC$.

Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL \equal{} \angle XDC \equal{} 30^\circ$ and $ CX \equal{} O_1O_2$. [i] Author: Arnoldo Aguilar (El Salvador)[/i]

Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.

Evaluate $ \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx$.

Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.

$100$ students participate in an exam with $5$ questions. Every question is answered by exactly $50$ students. What is the least possible value of number of students who answered at most $2$ questions? $\textbf{(A)}\ 21 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None}$

Let P be a point in an acute triangle $ABC$, and $d_A, d_B, d_C$ be the distance from P to vertices of the triangle respectively. If the distance from P to the three edges are $d_1, d_2, d_3$ respectively, prove that \[d_A+d_B+d_C\geq 2(d_1+d_2+d_3)\]

Show that a triangle whose angles $A$, $B$, $C$ satisfy the equality \[ \frac{\sin^2 A + \sin^2 B + \sin^2 C}{\cos^2 A + \cos^2 B + \cos^2 C} = 2 \] is a rectangular triangle.

The row $x_1, x_2,…$ is defined by the following recursion $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n}$ Prove that $\sum_{n=1}^{2018}{\frac{1}{x_n}}<3$.

Rita the painter rolls a fair $6\text{-sided die}$that has $3$ red sides, $2$ yellow sides, and $1$ blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple?

The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$ .

Suppose that the roots of the quadratic $x^2 + ax + b$ are $\alpha$ and $\beta$. Then $\alpha^3$ and $\beta^3$ are the roots of some quadratic $x^2 + cx + d$. Find $c$ in terms of $a$ and $b$.

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$. [i]Proposed by Jacob Stavrianos[/i]

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? $\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $

Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.