The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$

Bhairav runs a 15-mile race at 28 miles per hour, while Daniel runs at 15 miles per hour and Tristan runs at 10 miles per hour. What is the greatest length of time, in [i]minutes[/i], between consecutive runners' finishes? [i]2016 CCA Math Bonanza Lightning #2.1[/i]

A sequence $P_1, \dots, P_n$ of points in the plane (not necessarily diferent) is [i]carioca[/i] if there exists a permutation $a_1, \dots, a_n$ of the numbers $1, \dots, n$ for which the segments $$P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}$$ are all of the same length. Determine the greatest number $k$ such that for any sequence of $k$ points in the plane, $2023-k$ points can be added so that the sequence of $2023$ points is [i]carioca[/i].

In the morning, three people, $A$, $B$ and $C$ run in a same line at a beach in Albufeira. Some day, the three people were in the same point of the beach and then they started to run at the same time, but in different velocities. For each person, the velocity was constant. When someone arrived in an extreme of the beach, he/she turned back and runned in the opposite direction. In the moment in that the three people were in the same point of the beach again, the running finished. Not counting with the beginning and the final of the running, $A$ met $B$ six times and $A$ met $C$ eight times. How many times did $B$ and $C$ meet?

In the triangle $ABC$, where $<$ $ACB$ $=$ $45$, $O$ and $H$ are the center the circumscribed circle, respectively, the orthocenter. The line that passes through $O$ and is perpendicular to $CO$ intersects $AC$ and $BC$ in $K$, respectively $L$. Show that the perimeter of $KLH$ is equal to the diameter of the circumscribed circle of triangle $ABC$.

For any real number $t$, let $\lfloor t \rfloor$ denote the largest integer $\le t$. Suppose that $N$ is the greatest integer such that $$\left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4$$Find the sum of digits of $N$.

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

Real numbers $a$, $b$, $c$ which are differ from $1$ satisfies the following conditions; (1) $abc =1$ (2) $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$.

Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)?$ $\textbf{(A)}~0\qquad\textbf{(B)}~6\qquad\textbf{(C)}~10\qquad\textbf{(D)}~18\qquad\textbf{(E)}~24$

Ali puts $5$ points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would always be able to construct the desired triangles? (We say that triangle $T_1$ can be placed inside triangle $T_2$ if $T_1$ is congruent to a triangle that is located completely inside $T_2$.)

In a trapezoid $ABCD$ with bases $AD$ and $BC$, the bisector of the angle $\angle DAB$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $P$ and $S$, respectively, and the bisector of the angle $\angle BCD$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $Q$ and $R$, respectively. Prove that if $PS\parallel RQ$, then $AB = CD$.

Some towns in a country are connected by two–way roads, so that for any two towns there is a unique path along the roads connecting them. It is known that there is exactly 100 towns which are directly connected to only one town. Prove that we can construct 50 new roads in order to obtain a net in which every two towns will be connected even if one road gets closed.

For each non-negative integer $n$, let $u_n = \left( 2+\sqrt{5} \right)^n + \left( 2-\sqrt{5} \right)^n$. a) Prove that $u_n$ is a positive integer for all $n \geq 0$. When $n$ changes, what is the largest possible remainder when $u_n$ is divided by $24$? b) Find all pairs of positive integers $(a, b)$ such that $a, b < 500$ and for all odd positive integers $n$, $u_n \equiv a^n - b^n \pmod {1111}$.

The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$ $\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$

Dan has a fair $6$-sided die with faces labeled $1,2,3,4,+,$ and $-.$ In order to complete the equation \[ \underline{\qquad} \ \underline{\qquad} \ \underline{\qquad}=\underline{\qquad}, \] Dan repeatedly rolls his die and fills in a blank with the character he obtained, starting with the leftmost blank and progressing rightward. The probability that, when all blanks are filled, Dan forms a true equation, is $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p+q.$

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.

The incircle of a triangle $ABC$ is centered at $I$ and touches $AC,AB$ and $BC$ at points $K,L,M$, respectively. The median $BB_1$ of $\triangle ABC$ intersects $MN$ at $D$. Prove that the points $I,D,K$ are collinear.

The number of postive integers less than $1000$ divisible by neither $5$ nor $7$ is: $\text{(A)}\ 688 \qquad \text{(B)}\ 686\qquad \text{(C)}\ 684 \qquad \text{(D)}\ 658\qquad \text{(E)}\ 630$

Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.

What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\] $ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $

The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game? [i]Brian Hamrick.[/i]

Consider a non-zero real number $a$ such that $\{a\} + \left\{\frac{1}{a}\right\}=1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.

$\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. Find the angle $\angle EDF$ $ \textbf{(A) }64^{\circ}\qquad\textbf{(B) }66^{\circ}\qquad\textbf{(C) }68^{\circ}\qquad\textbf{(D) }70^{\circ}\qquad\textbf{(E) }72^{\circ} $

$F$* is the multiplicative group of the field $F$. $F$* is of finitely generated. Is it true that $F$* is cyclic? Additional question: (wasn’t at the olympiad) $K$* is the multiplicative group of the field $K$. $L \subseteq $$K$* is a finitely generated subgroup. Is it true that $L$ is cyclic?