A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably) While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?

A $3\times 3\times 3$ cube is made of $1\times 1\times 1$ cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features: 1) Projection of this solid on each face of the original cube is a $3\times 3$ square, 2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces).

All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers \[f(n),f(2n),f(3n),\dots,f(10n^2)\] are perfect squares?

Let $ABCD$ be a square. On the segments $AB$, $BC$, $CD$ and $DA$, choose points $E, F, G$ and $H$, respectively, such that $AE = BF = CG = DH$. Let $P$ be the intersection point of $AF$ and $DE$, $Q$ be the intersection point of $BG$ and $AF$, $R$ the intersection point of $CH$ and $BG$, and $S$ the point of intersection of $DE$ and $CH$. Prove that $PQRS$ is a square.

Construct a square on one side of an equilateral triangle. One on non-adjacent side of the square, construct a regular pentagon, as shown. One a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have? [asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy] $\textbf{(A)} 21 \qquad \textbf{(B)} 23 \qquad \textbf{(C)} 25 \qquad \textbf{(D)} 27 \qquad \textbf{(E)} 29 $

Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.

Show that the length of a cycle that contains every edge of a connected graph is at most the sum between the vertices and nodes of the graph, minus $ 1. $

Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1$. [i]Proposed by Richard Stong[/i]

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring? [i]S. Berlov[/i]

There are $800$ marbles in a bag. Each marble is colored with one of $100$ colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops. Suppose Anna has not stopped after drawing $699$ marbles. Compute the probability that she stops immediately after drawing the $700$th marble.

There are $2n-1$ lamps in a row. In the beginning only the middle one is on (the $n$-th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight line from point A to some point C in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point C could possibly be located.

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

Find the sum of the prime factors of $67208001$, given that $23$ is one. [i]Proposed by Justin Stevens[/i]

A continuous function $f : R \to R$ satisfies the conditions $f(1000) = 999$ and $f(x)f(f(x)) = 1$ for all real $x$. Determine $f(500)$.

[i]The Algorithm.[/i] There are thirteen broken computers situated at the following set $S$ of thirteen points in the plane: \[\begin{array}{ccc}A=(1,10)&B=(976,9)&C=(666,87)\\D=(377,422)&E=(535,488)&F=(775,488) \\ G=(941,500) & H=(225,583)&I=(388,696)\\J=(3,713)&K=(504,872)&L=(560,934)\\&M=(22,997)&\end{array}\] At time $t=0$, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and $A$ and fixes computers instantly, what path does he take to minimize the [i]total downtime[/i] of the computers? List the points he visits in order. Your score will be $\left\lfloor \dfrac{N}{40}\right\rfloor$, where \[N=1000+\lfloor\text{the optimal downtime}\rfloor - \lfloor \text{your downtime}\rfloor ,\] or $0$, whichever is greater. By total downtime we mean the sum \[\sum_{P\in S}t_P,\] where $t_P$ is the time at which the repairman reaches $P$.

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

Positive integers (not necessarily unique) are written, one on each face, on two cubes such that when the two cubes are rolled, each integer $2\leq k\leq12$ appears as the sum of the upper faces with probability $\frac{6-|7-k|}{36}$. Compute the greatest possible sum of all the faces on one cube. [i]2015 CCA Math Bonanza Tiebreaker Round #3[/i]

In acute triangle $ABC$ ($AB<AC$) point $O$ is center of its circumcircle $\Omega$. Let the tangent to $\Omega$ drawn at point $A$ intersect the line $BC$ at point $D$. Let the line $DO$ intersects the segments $AB$ and $AC$ at points $E$ and $F$, respectively. Point $G$ is constructed such that $AEGF$ is a parallelogram. Let $K$ and $H$ be points of intersection of segment $BC$ with segments $EG$ and $FG$, respectively. Prove that the circle $(GKH)$ touches the circle $\Omega$. [i] Proposed by Dong Luu [/i]

Let $a,b,c$ be positive integers such that $a|b^5, b|c^5$ and $c|a^5$. Prove that $abc|(a+b+c)^{31}$.

Let $ABC$ be an acute triangle such that $BA = BC$. On the sides $BA$ and $BC$ points $D$ and $E$ are chosen respectively, such that $DE$ and $AC$ are parallel. Let $H$ be the orthocenter of the triangle $DBE$ and $M$ be the midpoint of $AE$. If $\angle HMC = 90^o$, determine the measure of angle $\angle ABC$.

Through the use of theorems on logarithms \[ \log{\frac{a}{b}} \plus{} \log{\frac{b}{c}} \plus{} \log{\frac{c}{d}} \minus{} \log{\frac{ay}{dx}}\] can be reduced to: $ \textbf{(A)}\ \log{\frac{y}{x}}\qquad \textbf{(B)}\ \log{\frac{x}{y}}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ 0\qquad \textbf{(E)}\ \log{\frac{a^2y}{d^2x}}$