Find all real numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

Is there a positive integer divisible by the product of its digits such that this product is greater than $10^{2000}$?

$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.

A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is [I]small[/I] if $A, B, C$ are single-digit positive integers. It is [I]full[/I] if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.

Find the area of the domain expressed by the following system inequalities. \[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]

Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$ imply the equation $xy-12x+15y=0$.

Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$

Let $n$ be a positive integer. Given $n$ points in the plane, prove that it is possible to draw an angle with measure $\frac{2\pi}{n}$ with vertex as each one of the given points, such that any point in the plane is covered by at least one of the angles.

Let $ABC$ be a triangle with $AB < BC$, and let $E$ and $F$ be points in $AC$ and $AB$ such that $BF = BC = CE$, both on the same halfplane as $A$ with respect to $BC$. Let $G$ be the intersection of $BE$ and $CF$. Let $H$ be a point in the parallel through $G$ to $AC$ such that $HG = AF$ (with $H$ and $C$ in opposite halfplanes with respect to $BG$). Show that $\angle EHG = \frac{\angle BAC}{2}$.

The triangle $ABC$ is drawn on the board such that $AB + AC = 2BC$. The bisectors $AL_1, BL_2, CL_3$ were drawn in this triangle, after which everything except the points $L_1, L_2, L_3$ was erased. Use a compass and a ruler to reconstruct triangle $ABC$.

Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.

Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \] Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.

Let $P$ be the portion of the graph of $$y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8}$$ located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$. Find $\lfloor 1000d \rfloor$. [i]Proposed by [b] Th3Numb3rThr33 [/b][/i]

Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.

It is considered a regular polygon with $n$ sides, where $n(n>3)$ is an odd number that does not divide by 3. From the vertices of the polygon are arbitrarily chosen $m(0\leq m\leq n)$ vertices that are colored in red and the others in black. A triangle with the vertices at the vertices of the polygon it is considered $monocolor$ ,if all of its vertices are of the same color. Prove that the number of all $monocolor$ isosceles triangles with the vertices at the given polygon ends does not depend on the way of coloring of the vertices of the polygon. Determine the number of all these $monocolor$ isosceles triangles.

There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$. What is the difference between the two smallest such integers? $\textbf{(A) }2310\qquad\textbf{(B) }2311\qquad\textbf{(C) }27,720\qquad\textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}$

Let $k$ be a positive integer and $N_k$ be the number of sequences of length $2001$, all members of which are elements of the set $\{0,1,2,\ldots,2k+1\}$, and the number of zeroes among these is odd. Find the greatest power of $2$ which divides $N_k$.

The function $f:\mathbb{N} \rightarrow \mathbb{N}$ verifies: $1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};$ $2) f(20^{22})=f(22^{20});$ $3) f(2021)=2022$. Find all possible values of $f(2022)$.

Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove $2$, $3$, $7$, or $8$ marbles. If they can’t make a move, because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the game? [i]Proposed by Evin Liang[/i] [hide=Solution][i]Solution.[/i] $\boxed{573}$ Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such that the total number of marbles removed is $10$. Thus, when the number of marbles is a multiple of $10$, the first player loses the game. We analyse this game based on the number of marbles modulo $10$: If the number of marbles is $0$ modulo $10$, the first player loses the game If the number of marbles is $2$, $3$, $7$, or $8$ modulo $10$, the first player wins the game by moving to $0$ modulo 10 If the number of marbles is $5$ modulo $10$, the first player loses the game because every move leads to $2$, $3$, $7$, or $8$ modulo $10$ In summary, the first player loses if it is $0$ mod 5, and wins if it is $2$ or $3$ mod $5$. Now we solve the remaining cases by induction. The first player loses when it is $1$ modulo $5$ and wins when it is $4$ modulo $5$. The base case is when there is $1$ marble, where the first player loses because there is no move. When it is $4$ modulo $5$, then the first player can always remove $3$ marbles and win by the inductive hypothesis. When it is $1$ modulo $5$, every move results in $3$ or $4$ modulo $5$, which allows the other player to win by the inductive hypothesis. Thus, Evin loses the game if n is $0$ or $1$ modulo $5$. There are $\boxed{573}$ such values of $n$ from $1$ to $1434$.[/hide]

$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$. Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$. [i]Proposed by Kasra Ahmadi[/i]

A solid has a cylindrical middle with a conical cap at each end. The height of each cap equals the length of the middle. For a given surface area, what shape maximizes the volume?

A square $ ABCD$ of side length $ 2$ is given on a plane. The segment $ AB$ is moved continuously towards $ CD$ until $ A$ and $ C$ coincide with $ C$ and $ D$, respectively. Let $ S$ be the area of the region formed by the segment $ AB$ while moving. Prove that $ AB$ can be moved in such a way that $ S <\frac{5\pi}{6}$.

In this problem we consider so-called [i]cartesian triangles[/i], that is, triangles $ABC$ with integer sides $BC=a,CA=b,AB=c$ and $\angle A=\frac{2\pi}3$. Unless noted otherwise, $\triangle ABC$ is assumed to be cartesian. (a) If $U,V,W$ are the projections of the orthocenter $H$ to $BC,CA,AB$, respectively, specify which of the segments $AU$, $BV$, $CW$, $HA$, $HB$, $HC$, $HU$, $HV$, $HW$, $AW$, $AV$, $BU$, $BW$, $CV$, $CU$ have rational length. (b) If $I$ is the incenter, $J$ the excenter across $A$, and $P,Q$ the intersection points of the two bisectors at $A$ with the line $BC$, specify those of the segments $PB$, $PC$, $QB$, $QC$, $AI$, $AJ$, $AP$, $AQ$ having rational length. (c) Assume that $b$ and $c$ are prime. Prove that exactly one of the numbers $a+b-c$ and $a-b+c$ is a multiple of $3$. (d) Assume that $\frac{a+b-c}{3c}=\frac pq$, where $p$ and $q$ are coprime, and denote by $d$ the $\gcd$ of $p(3p+2q)$ and $q(2p+q)$. Compute $a,b,c$ in terms of $p,q,d$. (e) Prove that if $q$ is not a multiple of $3$, then $d=1$. (f) Deduce a necessary and sufficient condition for a triangle to be cartesian with coprime integer sides, and by geometrical observations derive an analogous characterization of triangles $ABC$ with coprime sides $BC=a$, $CA=b$, $AB=c$ and $\angle A=\frac\pi3$.