Find all primes $p$ such that the expression \[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\] is divisible by $p^3$.

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

Suppose that there is a set of $2016$ positive numbers, such that both their sum, and the sum of their reciprocals, are equal to $2017$. Let $x$ be one of those numbers. Find the maximum possible value of $x +\frac{1}{x}$. .

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?

How many permutations $a_1, a_2, a_3, a_4$ of $1, 2, 3, 4$ satisfy the condition that for $k = 1, 2, 3,$ the list $a_1,. . . , a_k$ contains a number greater than $k$?

Two $10$-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of $10$-digit integers with no two integers being neighbours.

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

The sequence of positive integers $a_1,a_2,a_3,\dots$ is [i]brazilian[/i] if $a_1=1$ and $a_n$ is the least integer greater than $a_{n-1}$ and $a_n$ is [b]coprime[/b] with at least half elements of the set $\{a_1,a_2,\dots, a_{n-1}\}$. Is there any odd number which does [b]not[/b] belong to the brazilian sequence?

A regular $n$-gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices.

In triangle $[ABC]$, the bissector of the angle $\angle{BAC}$ intersects the side $[BC]$ at $D$. Suppose that $\overline{AD}=\overline{CD}$. Find the lengths $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ that minimize the perimeter of $[ABC]$, given that all the sides of the triangles $[ABC]$ and $[ADC]$ have integer lengths.

Will has a sock drawer with $2$ socks of each color: red, green, blue, white, black (socks of the same color are indistinguishable). He absentmindedly grabs $2$ socks out of the drawer. What is the probability that he gets a pair of matching socks? [i]2019 CCA Math Bonanza Team Round #1[/i]

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$. Let $P$ and $Q$ be two points on segment $AB$ ($A$, $P$, $Q$, $B$ appear in that order and are distinct) such that $AP=QB$. Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$, respectively. Lines $AB$ and $EF$ intersect at $G$. Prove that line $DG$ is tangent to $\Gamma$.

$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least 50 cents? $\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}$

Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated. [asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy] The segments on the tiles define paths on the board. Determine the least possible number and the largest possible number of such paths. [i]For example, there are $9$ paths on the $4\times4$ board shown below.[/i] [asy]size(4cm);draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);draw((0,1)--(4,1));draw((0,2)--(4,2));draw((0,3)--(4,3));draw((1,0)--(1,4));draw((2,0)--(2,4));draw((3,0)--(3,4));draw((0,3.5)--(0.5,4),red);draw((0,2.5)--(1.5,4),red);draw((3.5,0)--(4,0.5),red);draw((2.5,0)--(4,1.5),red);draw((0.5,0)--(0,0.5),red);draw((2.5,4)--(3,3.5)--(3.5,4),red);draw((4,3.5)--(3.5,3)--(4,2.5),red);draw((0,1.5)--(1,2.5)--(1.5,2)--(0.5,1)--(1.5,0),red);draw((1.5,3)--(2,3.5)--(3.5,2)--(2,0.5)--(1.5,1)--(2.5,2)--cycle,red);[/asy]

Find all functions $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that for all positive real numbers $x,y$, \[x^2[f(x)+f(y)]=(x+y)f(yf(x)).\]

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$. [i]Proposed by Ankan Bhattacharya[/i]

A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ and $ b$ are positive integers. Find $ a\plus{}b$.

We consider an integer $n \ge 3,$ the set $S=\{1,2,3,\ldots,n\}$ and the set $\mathcal{F}$ of the functions from $S$ to $S.$ We say that $\mathcal{G} \subset \mathcal{F}$ is a generating set for $\mathcal{H} \subset \mathcal{F}$ if any function in $\mathcal{H}$ can be represented as a composition of functions from $\mathcal{G}.$ a) Let the functions $a:S \to S,$ $a(n-1)=n,$ $a(n)=n-1$ and $a(k)=k$ for $k \in S \setminus \{n-1,n\}$ and $b:S \to S,$ $b(n)=1$ and $b(k)=k+1$ for $k \in S \setminus \{n\}.$ Prove that $\{a,b\}$ is a generating set for the set $\mathcal{B}$ of bijective functions of $\mathcal{F}.$ b) Prove that the smallest number of elements that a generating set of $\mathcal{F}$ has is $3.$

An error of $ .02''$ is made in the measurement of a line $ 10''$ long, while an error of only $ .2''$ is made in a measurement of a line $ 100''$ long. In comparison with the relative error of the first measurement, the relative error of the second measurement is: $ \textbf{(A)}\ \text{greater by }.18 \qquad\textbf{(B)}\ \text{the same} \qquad\textbf{(C)}\ \text{less}$ $ \textbf{(D)}\ 10 \text{ times as great} \qquad\textbf{(E)}\ \text{correctly described by both (A) and (D)}$

Let $ I$ be an ideal of the ring $ R$ and $ f$ a nonidentity permutation of the set $ \{ 1,2,\ldots, k \}$ for some $ k$. Suppose that for every $ 0 \not\equal{} a \in R, \;aI \not\equal{} 0$ and $ Ia \not\equal{}0$ hold; furthermore, for any elements $ x_1,x_2,\ldots ,x_k \in I$, \[ x_1x_2\ldots x_k\equal{}x_{1f}x_{2f}\ldots x_{kf}\] holds. Prove that $ R$ is commutative. [i]R. Wiegandt[/i]

In a sequence ,first term is $2$ and after $2.$ term all terms is equal to sum of the previous number's digits' $5.$ power. (Like this $2.$term is $2^5=32$ , $3.$term is $3^5+2^5=243+32=275\dotsm$) Prove that, this infinite sequence has at least $2$ two numbers are equal.