Find all real values of $u$ such that the curves $y=x^2+u$ and $y=\sqrt{x-u}$ intersect in exactly one point.

Let $ a\in\mathbb{R}. $ Prove the following proposition: $$ \left( x,y\in\mathbb{R}\implies x^4+y^4+axy+2\ge 0 \right)\iff |a|\le 4. $$

We define $ f: \mathbb{N} \rightarrow \mathbb{N}$, $ f(n) \equal{} \sum_{k \equal{} 1}^{n}(k,n)$. a) Show that if $ \gcd(m,n)\equal{}1$ then we have $ f(mn)\equal{}f(m)\cdot f(n)$; b) Show that $ \sum_{d|n}f(d) \equal{} nd(n)$.

Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$. [i]2020 CCA Math Bonanza Lightning Round #4.3[/i]

Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

Let $r, s,$ and $t$ be the distinct roots of $x^3- 2022x^2 + 2022x + 2022.$ Compute $$\frac{1}{1-r^2} + \frac{1}{1-s^2} + \frac{1}{1-t^2}.$$

A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences. If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.

Let $\odot O$ be the inscribed circle of $\vartriangle ABC$, with $D$, $E$, $N$ the touchpoints with sides $AB$, $AC$, $BC$ respectively. Extension of $NO$ intersects segment $DE$ at point $K$. Extension of $AK$ intersects segment $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/a503c500178551ddf9bdb1df0805ed22bc417d.png[/img]

Given a triangle \(ABC\), an arbitrary point \(D\) is chosen on the side \(AC\). In triangles \(ABD\) and \(CBD\), the angle bisectors \(BK\) and \(BL\) are drawn, respectively. The point \(O\) is the circumcenter of \(\triangle KBL\). Prove that the second intersection point of the circumcircles of triangles \(ABL\) and \(CBK\) lies on the line \(OD\). [i]Proposed by Anton Trygub[/i]

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

Let $a, b, c $ be distinct positive integers. a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$. b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that $$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$

Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

5. $A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types: Type $1$: Piece like Type $2$ but with one square at the right of the bottom square. Type $2$: Piece of $2$ consecutive squares, one over another. Type $3$: Piece of $1$ square. At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy. Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$, $2$ and $3$ can be put on exactly $3$, $2$ and $1$ squares of the board respectively.

A square with a side 1 is colored in 3 colors. What’s the greatest real number $a$ such that there can always be found 2 points of the same color at a distance $a$?

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

[b]Problem Section #2 a) If $$ax+by=7$$ $$ax^2+by^2=49$$ $$ax^3+by^3=133$$ $$ax^4+by^4=406$$ , find the value of $2014(x+y-xy)-100(a+b).$

Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by $ \textbf{(A)}\ p\%\qquad\textbf{(B)}\ \frac{p}{1+p}\%\qquad\textbf{(C)}\ \frac{100}{p}\%\qquad\textbf{(D)}\ \frac{p}{100+p}\%\qquad\textbf{(E)}\ \frac{100p}{100+p}\%$

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Find the sum of all perfect squares that divide 2016.

In a lattice, connected the elements $ a \wedge b$ and $ a \vee b$ by an edge whenever $ a$ and $ b$ are incomparable. Prove that in the obtained graph every connected component is a sublattice. [i]M. Ajtai[/i]

Solve the following system of equations in positive integers \[\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.\]

Let $r,s,t$ be the roots of the equation $x(x-2)(3x-7)=2$. Show that $r,s,t$ are real and positive and determine $\arctan r+\arctan s +\arctan t$.

Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andy Xu[/i]