Problem: A mathematical moron is given the values b; c; A for a triangle ABC and is required to find a. He does this by using the cosine rule $ a^2 = b^2 + c^2 - 2bccosA$ and misapplying the low of the logarithm to this to get $ log a^2 = log b^2 + log c^2 - log(2bc cos A) $ He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$?

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]

Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n $ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins. [i]Proposed by amano_hina[/i]

Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$. Let $C$ be the circumcenter of the triangle $AOB$. Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$. $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$, cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$). Prove that $OP\perp QR$. [i]Proposed by Mehdi E'tesami Fard, Ali Khezeli[/i]

Let $ a$, $ b$, $ c$ and $ n$ be positive integers such that $ a^n$ is divisible by $ b$, such that $ b^n$ is divisible by $ c$, and such that $ c^n$ is divisible by $ a$. Prove that $ \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1}$ is divisible by $ abc$. An even broader [i]generalization[/i], though not part of the QEDMO problem and not quite number theory either: If $ u$ and $ n$ are positive integers, and $ a_1$, $ a_2$, ..., $ a_u$ are integers such that $ a_i^n$ is divisible by $ a_{i \plus{} 1}$ for every $ i$ such that $ 1\leq i\leq u$ (we set $ a_{u \plus{} 1} \equal{} a_1$ here), then show that $ \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1}$ is divisible by $ a_1a_2...a_u$.

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

(a) Prove that $\sqrt{n+1}-\sqrt n<\frac1{2\sqrt n}<\sqrt n-\sqrt{n-1}$ for all $n\in\mathbb N$. (b) Prove that the integer part of the sum $1+\frac1{\sqrt2}+\frac1{\sqrt3}+\ldots+\frac1{\sqrt{m^2}}$, where $m\in\mathbb N$, is either $2m-2$ or $2m-1$.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.

Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that \[f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.\]

In acute triangle $ABC$ let $H$ be its orthocenter and $I$ be its incenter. Let $D$ be the projection of point $I$ onto the line $BC$ and $E$ be the reflection of point $A$ in point $I$. Further, let $F$ be the projection of point $H$ onto the line $ED$. Prove that points $B, H, F$ and $C$ lie on circle.

Three squares $ABB_1B_2,BCC_1C_2,CAA_1A_2$ are constructed in the exterior of a triangle $ABC$. In the exterior of these squares, another three squares $A_1B_2B_3B_4,B_1C_2C_3C_4,C_1A_2A_3A_4$ are constructed. Prove that the area of a triangle with sides $C_3A_4,A_3B_4,B_3C_4$ is $16$ times the area of $\triangle ABC$.

Professor Shian Bray is buying CCA Math Bananas$^{\text{TM}}$. He starts with $\$500$. The first CCA Math Bananas$^{\text{TM}}$ he buys costs $\$1$. Each time after he buys a CCA Math Banana$^{\text{TM}}$, the cost of a CCA Math Bananas$^{\text{TM}}$ doubles with probability $\frac{1}{2}$ (otherwise staying the same). Professor Bray buys CCA Math Bananas$^{\text{TM}}$ until he cannot afford any more, ending with $n$ CCA Math Bananas$^{\text{TM}}$. Estimate the expected value of $n$. An estimate of $E$ earns $2^{1-0.25|E-A|}$ points, where $A$ is the actual answer. [i]2020 CCA Math Bonanza Lightning Round #5.1[/i]

Find the sum of all two-digit prime numbers whose digits are also both prime numbers. [i]Proposed by Nathan Xiong[/i]

Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.

A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there? [i]2018 CCA Math Bonanza Team Round #8[/i]

$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.

Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.

Let $n$ be an integer greater than $2$. Prove that the $n$th power of the length of the hypotenuse of a right triangle is greater than the sum of the $n$th powers of the lengths of the legs.

In January 1980 the Moana Loa Observation recorded carbon dioxide levels of 338 ppm (parts per million). Over the years the average carbon dioxide reading has increased by about 1.515 ppm each year. What is the expected carbon dioxide level in ppm in January 2030? Round your answer to the nearest integer. $\textbf{(A) } 399\qquad\textbf{(B) } 414\qquad\textbf{(C) } 420\qquad\textbf{(D) } 444\qquad\textbf{(E) } 459$

Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.