Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$.

Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$. [i]Calvin Deng.[/i]

Chad and Chad2 run competing rare candy stores at Princeton. Chad has a large supply of boxes of candy, each box containing three candies and costing him \$ $3$ to purchase from his supplier. He charges \$ $1.50$ per candy per student. However, any rare candy in an opened box must be discarded at the end of the day at no profit. Chad knows that at each of $8$am, $10$am, noon, $2$pm, $4$pm, and $6$pm, there will be one person who wants to buy one candy, and that they choose between Chad and Chad2 at random. (He knows that those are the only times when he might have a customer.) Chad may refuse sales to any student who asks for candy. If Chad acts optimally, his expected daily profit can be written in simplest form as $\frac{m}{n}$. Find $m + n$. (Chad’s profit is \$ $1.50$ times the number of candies he sells, minus $3 per box he opens.)

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. [i]Soewono, Bandung[/i]

Show that for all real numbers $x, y$ satisfying $x+y \geq 0$ \[ (x^2+y^2)^3 \geq 32(x^3+y^3)(xy-x-y) \]

Given $4$ points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?

Circles $A$, $B$, and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside circle $C$ but outside circle $A$ and circle $B$? [asy] size(170); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; filldraw(arc((1,0),1,90,180)--arc((-1,0),1,0,90)--arc((0,1), 1, 180, 0)--cycle,gray); draw(circle((0,1),1)); draw(circle((1,0),1)); draw(circle((-1,0),1)); dot((-1,0)); dot((1,0)); dot((0,1)); label("$A$",(-1,0),SW); label("$B$",(1,0),SE); label("$C$",(0,1),N);[/asy] $ \textbf{(A)}\ 3-\frac{\pi}{2} \qquad \textbf{(B)}\ \frac{\pi}{2} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \frac{3\pi}{4} \qquad \textbf{(E)}\ 1+\frac{\pi}{2}$

In an acute triangle $ABC$ let $AH_a$ and $BH_b$ be altitudes. Let $H_aH_b$ intersect the circumcircle of $ABC$ at $P$ and $Q$. Let $A'$ be the reflection of $A$ in $BC$, and let $B'$ be the reflection of $B$ in $CA$. Prove that $A', B'$, $P$, $Q$ are concyclic.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.

Let $D$ and $E$ be points on side $[AB]$ of a right triangle with $m(\widehat{C})=90^\circ$ such that $|AD|=|AC|$ and $|BE|=|BC|$. Let $F$ be the second intersection point of the circumcircles of triangles $AEC$ and $BDC$. If $|CF|=2$, what is $|ED|$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ 1+\sqrt 2 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 2\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(xf(x + y)) = yf(x) + 1$$ holds for all $x, y \in \mathbb{R}^{+}$. [i]Proposed by Nikola Velov, North Macedonia[/i]

At Jefferson Summer Camp, $ 60\%$ of the children play soccer, $ 30\%$ of the children swim, and $ 40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? $ \textbf{(A)}\ 30\% \qquad \textbf{(B)}\ 40\% \qquad \textbf{(C)}\ 49\% \qquad \textbf{(D)}\ 51\% \qquad \textbf{(E)}\ 70\%$

$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called [i]chordal[/i] if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be $H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$ where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.

Solve in positive real numbers: $n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014$

Let $\phi (x, u)$ be the smallest positive integer $n$ so that $2^u$ divides $x^n + 95$ if it exists, or $0$ if no such positive integer exists. Determine$ \sum_{i=0}^{255} \phi(i, 8)$.

For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number. Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.

Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that: \[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]

Given a positive integer $n > 1$ and an angle $\alpha < 90^o$, Jaime draws a spiral $OP_0P_1...P_n$ of the following form (the figure shows the first steps): $\bullet$ First draw a triangle $OP_0P_1$ with $OP_0 = 1$, $\angle P_1OP_0 = \alpha$ and $P_1P_0O = 90^o$ $\bullet$ then for every integer $1 \le i \le n$ draw the point $P_{i+1}$ so that $\angle P_{i+1}OP_i = \alpha$, $\angle P_{i+1}P_iO = 90^o$ and $P_{i-1}$ and $P_{i+1}$ are in different half-planes with respect to the line $OP_i$ [img]https://cdn.artofproblemsolving.com/attachments/f/2/aa3913989dac1cf04f2b42b5d630b2e096dcb6.png[/img] a) If $n = 6$ and $\alpha = 30^o$, find the length of $P_0P_n$. b) If $n = 2018$ and $\alpha= 45^o$, find the length of $P_0P_n$.

In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively. Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ . [i]Proposed by Hojoo Lee, Korea[/i]

Construct a regular triangle using a plywood square. ([i]You can draw a line through pairs of points lying on the distance less than the side of the square, construct a perpendicular from a point to the line the distance between them does not exceed the side of the square, and measure segments on the constructed lines equal to the side or to the diagonal of the square[/i])

Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$, such that $AB||CD$. Let $P$ be a point on the circle $\omega$. Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle.

The polynomial $x^k + a_1x^{k-1} + a_2x^{k-2} +... + a_k$ has $k$ distinct real roots. Show that $a_1^2 > \frac{2ka_2}{k-1}$.

Suzanne went to the bank and withdrew \$$800$. The teller gave her this amount using \$$20$ bills, \$$50$ bills, and \$$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received? $\textbf{(A) }45\qquad\textbf{(B) }21\qquad\textbf{(C) }36\qquad\textbf{(D) }28\qquad\textbf{(E) }32$

Some domino pieces are placed in a chain according to standard rules. In each move, we may remove a sub-chain with equal numbers at its ends, turn the whole sub-chain around, and put it back in the same place. Prove that for every two legal chains formed from the same pieces and having the same numbers at their ends, we can transform one to another in a finite sequence of moves.

Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$ Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.