Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

$a^3 + b^3 + 3abc \ge\ c^3$ prove that where a,b and c are sides of triangle.

The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy: \[ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd \].

Let $n$ be a fixed positive integer. We have a $n\times n$ chessboard. We call a pair of cells [b]good[/b] if they share a common vertex (May be common edge or common vertex). How many [b]good[/b] pairs are there on this chessboard?

The annual incomes of $1000$ families range from $8200$ dollars to $98000$ dollars. In error, the largest income was entered on the computer as $980000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is $ \text{(A)}\ \text{882 dollars}\qquad\text{(B)}\ \text{980 dollars}\qquad\text{(C)}\ \text{1078 dollars}\qquad\text{(D)}\ \text{482,000 dollars}\qquad\text{(E)}\ \text{882,000 dollars} $

The first $ 2007$ positive integers are each written in base $ 3$. How many of these base-$ 3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.) $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 101 \qquad \textbf{(C)}\ 102 \qquad \textbf{(D)}\ 103 \qquad \textbf{(E)}\ 104$

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

The players Alfred and Bertrand put together a polynomial $x^n + a_{n-1}x^{n- 1} +... + a_0$ with the given degree $n \ge 2$. To do this, they alternately choose the value in $n$ moves one coefficient each, whereby all coefficients must be integers and $a_0 \ne 0$ must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end. (a) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the right to the left, i.e. for $j = 0, 1,. . . , n - 1$, be determined? (b) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the left to the right, i.e. for $j = n -1, n - 2,. . . , 0$, be determined? (Theresia Eisenkölbl, Clemens Heuberger)

Find all functions $f: \mathbb{N} \rightarrow \mathbb {N}$ such that for all positive integers $m, n$, $f(m+n)\mid f(m)+f(n)$ and $f(m)f(n) \mid f(mn)$.

A $2$-kg rock is suspended by a massless string from one end of a uniform $1$-meter measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the $0.20$-meter mark? [asy] size(250); draw((0,0)--(7.5,0)--(7.5,0.2)--(0,0.2)--cycle); draw((1.5,0)--(1.5,0.2)); draw((3,0)--(3,0.2)); draw((4.5,0)--(4.5,0.2)); draw((6,0)--(6,0.2)); filldraw((1.5,0)--(1.2,-1.25)--(1.8,-1.25)--cycle, gray(.8)); draw((0,0)--(0,-0.4)); filldraw((0,-0.4)--(-0.05,-0.4)--(-0.1,-0.375)--(-0.2,-0.375)--(-0.3,-0.4)--(-0.3,-0.45)--(-0.4,-0.6)--(-0.35,-0.7)--(-0.15,-0.75)--(-0.1,-0.825)--(0.1,-0.84)--(0.15,-0.8)--(0.15,-0.75)--(0.25,-0.7)--(0.25,-0.55)--(0.2,-0.4)--(0.1,-0.35)--cycle, gray(.4)); [/asy] $ \textbf {(A) } 0.20 \, \text{kg} \qquad \textbf {(B) } 1.00 \, \text{kg} \qquad \textbf {(C) } 1.33 \, \text{kg} \qquad \textbf {(D) } 2.00 \, \text{kg} \qquad \textbf {(E) } 3.00 \, \text{kg} $

let $0<a,b<\pi/2$. Show that $\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a) $

Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a bijection $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-friendly[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$. (Note: A bijection is a one-to-one, onto function.) Does there exist a divisor-friendly bijection? Prove or disprove.

The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold: (a) $E$ is the mid-point of $AB$; (b) points $A,G$ and $F$ are on the same line; (c) there is a point $C$ at which $BG$ and $EF$ intersect; (d) $|CE|=1$ and $|AC|=|AE|=|FG|$. Show that if $|AG|=x$, then $|AB|=x^3$.

Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

Does there exist a triangle with length $a,b,c$ such that: [b]a)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$ [b]b)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$

Let $n\ge 3$ be a positive integer. Prove that a polynomial of the form \[x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,\] where at least one of the real coefficients $a_0,a_1,\ldots ,a_{n-3}$ is nonzero, cannot have all real roots.

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth. All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets. A rectangular city park is bounded on the north and south by Sixth Street and Eighth Street, and bounded on the east and west by Fourth Avenue and Twelfth Avenue. Although there are no breaks in the roads that bound the park, no road goes through the park. The city paints a crosswalk from every street corner across any adjacent road. Thus, where two roads cross such as at Second Street and Second Avenue, there are four crosswalks painted, while at corners such as First Street and First Avenue, there are only two crosswalks painted. How many crosswalks are there painted on the roads of this city?

In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.

For what positive integers $n\geq 4$ does there exist a set $S$ of $n$ points on the plane, not all collinear, such that for any three non-collinear points $A,B,C$ in $S$, either the incenter, $A$-excenter, $B$-excenter, or $C$-excenter of triangle $ABC$ is also contained in $S$?

Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$. [i]2017 CCA Math Bonanza Team Round #7[/i]

Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.

Real numbers $a_1,a_2,...,a_n$ ($n \ge 3$) satisfy the conditions $a_1 = a_n = 0$ and $$a_{k-1}+a_{k+1} \ge 2a_k$$ for $k = 2$,$3$$,...,$$n -1$. Prove that none of the numbers $a_1$,$...$,$a_n$ is positive.

A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor (respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle (respectively square) with vertices lattice points. We assign to each lattice point a real number, such that the sum of all the numbers in any square minor is less than $1$ in absolute value. Prove that the sum of all the numbers in any rectangle minor is less than $4$ in absolute value.

There are $2000$ apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than $100$ . (A. Razborov)