We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?

Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.

What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$

In $1960$, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since $1960$, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached $21$ years. What is the age of each of the others two?

Let $ O$ be the circumcenter of an acute triangle $ ABC$ and let $ k$ be the circle with center $ P$ that is tangent to $ O$ at $ A$ and tangent to side $ BC$ at $ D$. Circle $ k$ meets $ AB$ and $ AC$ again at $ E$ and $ F$ respectively. The lines $ OP$ and $ EP$ meet $ k$ again at $ I$ and $ G$. Lines $ BO$ and $ IG$ intersect at $ H$. Prove that $ \frac{{DF}^2}{AF}\equal{}GH$.

The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?

Do either $(1)$ or $(2)$: $(1)$ $f$ is continuous on the closed interval $[a, b]$ and twice differentiable on the open interval $(a, b).$ Given $x_0 \in (a, b),$ prove that we can find $\xi \in (a, b)$ such that $\dfrac{ ( \dfrac{(f(x_0) - f(a))}{(x_0 - a)} - \dfrac{(f(b) - f(a))}{(b - a)} )}{(x_0 - b)} = \dfrac{f''(\xi)}{2}.$ $(2)$ $AB$ and $CD$ are identical uniform rods, each with mass $m$ and length $2a.$ They are placed a distance $b$ apart, so that $ABCD$ is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?

Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

$25$ boys and $25$ girls sit around a table. Show that there is a person who has a girl sitting on either side of them.

When three positive integers $a, b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen? $\textbf{(A)} ~0\qquad\textbf{(B)} ~1\qquad\textbf{(C)} ~2\qquad\textbf{(D)} ~3\qquad\textbf{(E)} ~4\qquad$

Triangle $\vartriangle ABC$ has side lengths $AB = 3$, $AC = 2$ and angle $\angle CBA = 30^o$. Let the possible lengths of $BC$ be $\ell_1$ and $\ell_2$, where $\ell_2 > \ell_1$. Compute $\frac{\ell_2}{\ell_1}$ .

Let $-1<a<0$, $\theta=\arcsin a$. Then the solution set to the inequality $\sin x<a$ is $\text{(A)}\{x|2n\pi+\theta<x<(2n+1)\pi-\theta,n\in\mathbb{Z}\}$ $\text{(B)}\{x|2n\pi-\theta<x<(2n+1)\pi+\theta,n\in\mathbb{Z}\}$ $\text{(C)}\{x|(2n-1)\pi+\theta<x<2n\pi-\theta,n\in\mathbb{Z}\}$ $\text{(D)}\{x|(2n-1)\pi-\theta<x<2n\pi+\theta,n\in\mathbb{Z}\}$

On the sides $AD , BC$ of the square $ABCD$ the points $M, N$ are selected $N$, respectively, such that $AM = BN$. Point $X$ is the foot of the perpendicular from point $D$ on the line $AN$. Prove that the angle $MXC$ is right. (Mirchev Borislav)

If $r\ge 0$, then for all $p$ and $q$ such that $pq\neq 0$ and $pr>qr$, we have $\textbf{(A) }-p>-q\qquad\textbf{(B) }-p>q\qquad\textbf{(C) }1>-q/p\qquad$ $\textbf{(D) }1<q/p\qquad \textbf{(E) }\text{None of These}$

Let $O$ be the centre of the incircle of $\triangle ABC$. Points $K,L$ are the intersection points of the circles circumscribed about triangles $BOC,AOC$ respectively with the bisectors of the angles at $A,B$ respectively $(K,L\not= O)$. Also $P$ is the midpoint of segment $KL$, $M$ is the reflection of $O$ with respect to $P$ and $N$ is the reflection of $O$ with respect to line $KL$. Prove that the points $K,L,M$ and $N$ lie on the same circle.

[b]a)[/b] Let be three vectorial spaces $ E,F,G, $ where $ F $ has finite dimension, and $ E $ is a subspace of $ F. $ Prove that if the function $ T:F\longrightarrow G $ is linear, then $$ \dim TF -\dim TE\le \dim F-\dim E. $$ [b]b)[/b] Let $ A,B,C $ be matrices of real numbers. Prove that $$ \text{rang} (AB) +\text{rang} (BC) \le \text{rang} (ABC) +\text{rang} (B) . $$

An ice-cream novelty item consists of a cup in the shape of a $4$-inch-tall frustum of a right circular cone, with a $2$-inch-diameter base at the bottom and a $4$-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height $4$ inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches? $\textbf{(A)}\ 8\pi\qquad\textbf{(B)}\ \frac{28\pi}{3}\qquad\textbf{(C)}\ 12\pi\qquad\textbf{(D)}\ 14\pi\qquad\textbf{(E)}\ \frac{44\pi}{3}$

Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$

Let $ABC$ be a triangle with $AB=5$, $AC=12$ and incenter $I$. Let $P$ be the intersection of $AI$ and $BC$. Define $\omega_B$ and $\omega_C$ to be the circumcircles of $ABP$ and $ACP$, respectively, with centers $O_B$ and $O_C$. If the reflection of $BC$ over $AI$ intersects $\omega_B$ and $\omega_C$ at $X$ and $Y$, respectively, then $\frac{O_BO_C}{XY}=\frac{PI}{IA}$. Compute $BC$. [i]2016 CCA Math Bonanza Individual #15[/i]

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

Find $p+r$ if $p$ and $q$ are primes and $r$ is an integer such that \[ \left( r^2 + pr + 1 \right) \cdot \left( r^2 + \left( p^2 - q \right) r - p \right) = pq. \]

Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of $a_1^2 + a_2^2 + \cdots + a_{100}^2$, and find all possible sequences $a_1, a_2, \ldots , a_{100}$ which achieve this maximum.

In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly $2025$ days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers $n>0$ for which we can be absolutely certain that she cannot receive the trophy again in $n$ days, given the above information.

Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon. [i]Greece[/i]

Find all functions $f:(0,\infty)\to(0,\infty)$ such that $$f(f(f(x)))+4f(f(x))+f(x)=6x.$$