Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled $1$ to $n$. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles with the smallest label, say $i$, to either any empty box or the box containing marble $i+1$. Cathy wins if at any point there is a box containing only marble $n$. Determine all pairs of integers $(n,k)$ such that Cathy can win this game.

The base-ten expressions of all the positive integers are written on an infinite ribbon without spacing: $1234567891011\ldots$. Then the ribbon is cut up into strips seven digits long. Prove that any seven digit integer will: (a) appear on at least one of the strips; [i](3 points)[/i] (b) appear on an infinite number of strips. [i](1 point)[/i]

Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.

Given an equilateral triangle with side of length $s$, consider the locus of all points $\mathit{P}$ in the plane of the triangle such that the sum of the squares of the distances from $\mathit{P}$ to the vertices of the triangle is a fixed number $a$. This locus $\textbf{(A) }\text{is a circle if }a>s^2\qquad$ $\textbf{(B) }\text{contains only three points if }a=2s^2\text{ and is a circle if }a>2s^2\qquad$ $\textbf{(C) }\text{is a circle with positive radius only if }s^2<a<2s^2\qquad$ $\textbf{(D) }\text{contains only a finite number of points for any value of }a\qquad $ $\textbf{(E) }\text{is none of these}$

The Fibonacci numbers $F_1, F_2, F_3, \ldots$ are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for each integer $n \ge 1$. Let $P$ be the unique polynomial of least degree for which $P(n) = F_n$ for all integers $1 \le n \le 10$. Compute the integer $m$ for which \[P(100) - \sum_{k=11}^{98} P(k) = \frac{m}{10} \dbinom{98}{9} + 144.\] [i]Proposed by Michael Tang[/i]

Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$. [i]Author: Ray Li[/i]

For all $a,b,c\in \bb{R}^+ $ such that $a+b+c=1$ and $ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc$. Find min $M$

Let $M$ be an interior point of tetrahedron $V ABC$. Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$, and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$. [b](a)[/b] Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$. [b](b)[/b] Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$, where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$, and $M$ is the barycenter of $V ABC$.

Express $2.3\overline{57}$ as a common fraction. [i]2017 CCA Math Bonanza Lightning Round #3.1[/i]

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that: a) ${5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).$ b) ${F_{p - 1}} \equiv 0(\bmod p).$ c) $k(p)|p-1.$

A positive integer $n$ is called [i]good[/i] if for all positive integers $a$ which can be written as $a=n^2 \sum_{i=1}^n {x_i}^2$ where $x_1, x_2, \ldots ,x_n$ are integers, it is possible to express $a$ as $a=\sum_{i=1}^n {y_i}^2$ where $y_1, y_2, \ldots, y_n$ are integers with none of them is divisible by $n.$ Find all good numbers.

Let $ABC$ be a triangle with area $9$, and let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let $P$ be the point in side $BC$ such that $PC = \frac{1}{3}BC$. Let $O$ be the intersection point between $PN$ and $CM$. Find the area of the quadrilateral $BPOM$.

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD \equal{} \angle HFD \equal{} 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that: $ FG \equal{} t \equal{} \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a \equal{} \frac {t \sqrt {3}}{2 \plus{} \sqrt {3}}$.

Let $(P, P')$ and $(Q, Q')$ be two pairs of points isogonally conjugated with respect to a triangle $ABC$, and $R$ be the common point of lines $PQ$ and $P'Q'$. Prove that the pedal circles of points $P$, $Q$, and $R$ are coaxial.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

a) Given a convex hexagon $ABCDEF$, which has a center of symmetry. Prove that the perimeter of triangle $ACE$ is greater than half the perimeter of hexagon $ABCDEF$. b) Given a convex $(2n)$-gon $P$ having a center of symmetry, its vertices are colored alternately red and blue. Let $Q$ be an $n$-gon with red vertices. Is it possible to say that the perimeter of $Q$ is certainly greater than half the perimeter $P$? Solve the problem for $n = 4$ and $n = 5$.

Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus. [i]Proposed by Tommy Walker Mackay, United Kingdom[/i]

The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

Find the remainder of $$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$ when divided by $101$.

A table has $m$ rows and $n$ columns. The following permutations of its $mn$ elements are permitted: an arbitrary permutation leaving each element in the same row (a“horizontal move”) and an arbitrary permutation leaving each element in the same column (a “vertical move”). Find the number $k$ such that any permutation of $mn$ elements can be obtained by $k$ permitted moves but there exists a permutation that cannot be achieved in less than $k$ moves. (A Andjans, Riga0

Find the sum of all positive $30$-digit palindromes. The leading digit is not allowed to be $0$.

A computer program that works only with integer numbers reads the numbers on the screen, identifies the selected numbers and performs one of the following actions: • If button $A$ is pressed, the user selects $5$ numbers and then each selected number is changed to its successor; • If button $B$ is pressed, the user selects $5$ numbers and then each selected number is changed to its triple. Bento has this program on his computer with the numbers $1, 3, 3^2, · · ·, 3^{19}$ on the screen, each one appearing just once. a) By simply pressing button $A$ several times, is Bento able to make the sum of the numbers on the screen be $2024^{2025}$? b) What is the minimum number of times that Bento must press button $B$ to make all the numbers on the screen turn equal, without pressing button $A$?

For some integers $u$,$ v$, and $w$, the equation $$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$ holds for all real numbers a, b, and c that satisfy $$au + bv + cw = 0$$ Find the minimum possible value of $u^2 + v^2 + w^2$.