At CMIMC headquarters, there is a row of $n$ lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things: [list] [*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken). [*] Check if a given lightbulb is on or off. [*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on. [/list] Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most $n$ seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value $n$ such that he can complete his task on time? [i]Proposed by Adam Bertelli[/i]

Determine whether there exists a function $ f: \mathbb{N}\longrightarrow \mathbb{N}$ such that $ f(n)\equal{}f(f(n\minus{}1))\plus{}f(f(n\plus{}1))$ for all natural numbers $ n\ge 2$.

Let $f(n)=\sum_{k=0}^{2010}n^k$. Show that for any integer $m$ satisfying $2\leqslant m\leqslant 2010$, there exists no natural number $n$ such that $f(n)$ is divisible by $m$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)[/i]

Prove that for natural number $n$ it's possible to find complex numbers $\omega_1,\omega_2,\cdots,\omega_n$ on the unit circle that $$\left\lvert\sum_{j=1}^{n}\omega_j\right\rvert=\left\lvert\sum_{j=1}^{n}\omega_j^2\right\rvert=n-1$$ iff $n=2$ occurs.

$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

How many ways are there to cover this region with dominoes? [asy] unitsize(20); int[][] a = { {999, 999, 000, 000, 000, 999, 999, 999}, {999, 999, 000, 888, 000, 999, 999, 999}, {999, 999, 000, 000, 000, 000, 000, 000}, {000, 000, 000, 888, 888, 000, 888, 000}, {000, 888, 000, 888, 888, 000, 000, 000}, {000, 000, 000, 000, 000, 000, 999, 999}, {999, 999, 999, 000, 888, 000, 999, 999}, {999, 999, 999, 000, 000, 000, 999, 999}}; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { if (a[j][i] != 999) draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] == 888) fill((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); } } [/asy]

Suppose two functions $ f(x)\equal{}x^4\minus{}x,\ g(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ satisfy $ f(1)\equal{}g(1),\ f(\minus{}1)\equal{}g(\minus{}1)$. Find the values of $ a,\ b,\ c,\ d$ such that $ \int_{\minus{}1}^1 (f(x)\minus{}g(x))^2dx$ is minimal.

Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars? $ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$

Let the bisector of the outside angle of $A$ of triangle $ABC$ and the circumcircle of triangle $ABC$ meet at point $P$. The circle passing through points $A$ and $P$ intersects segments $BP$ and $CP$ at points $E$ and $F$ respectively. Let $AD$ is the angle bisector of triangle $ABC$. Prove that $\angle PED = \angle PFD$. [img]https://cdn.artofproblemsolving.com/attachments/0/3/0638429a220f07227703a682479ed150302aae.png[/img]

There are $ 16$ pupils in a class. Every month, the teacher divides the pupils into two groups. Find the smallest number of months after which it will be possible that every two pupils were in two different groups during at least one month.

Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$. [I]Proposed by Italy[/i]

In the grid below, draw horizontal and vertical segments of unit length joining pairs of adjacent dots (some have been given to you) so that $1.$ every dot is connected by line segments to exactly $1$ or $3$ adjacent dots, $2.$ any dot can be reached from any other dot by following a path of segments, and $3.$ no area is completely enclosed by segments. Note: “Unit length” is the length between two adjacent dots when there is no missing dot between them. For example, we cannot draw a vertical line segment down from the dot in the top right corner because the length of this segment would be 2 units. There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(0.5cm); for (int i = 0; i <= 6; ++i) { for (int j = 0; j <= 6; ++j) { if ((i == 2 && j == 0) != true && (i == 6 && j == 0) != true && (i == 6 && j == 3) != true && (i == 0 && j == 5) != true && (i == 6 && j == 5) != true) { dot((i, j)); } } } draw((0, 3) -- (0,4)); draw((0, 3) -- (1,3)); draw((1,3) -- (1,4)); draw((2,2) -- (3,2)); draw((3,2) -- (3,3)); draw((3,3) -- (2,3)); draw((2,3) -- (2,4)); draw((2,4) -- (3,4)); draw((4,2) -- (4,1)); draw((4,1) -- (5,1)); draw((5,1) -- (5,2)); draw((5,2) -- (5,3)); draw((5,3) -- (4,3)); [/asy]

Determine positive integers $n\geq 3$ such that there exists a set $M$ of $n$ complex numbers and a positive integer $m$ such that $(1+z_1z_2z_3)^m=1$ for all pairwise distinct $z_1,z_2,z_3\in M$. [i]Vlad Matei[/i]

Prove that the number $9^n+8^n+7^n+6^n-4^n-3^n-2^n-1^n$ is divisible by $10$ for all non-negative $n$.

Let $S$ be the set of all numbers of the form $2^m 3^n$, where $m$ and $n$ are integers. Is $S$ dense in the set of positive real numbers?

A unit square block is attached to any place on the group of seven unit square blocks below such that it shares a side with at least one block. [asy] defaultpen(linewidth(0.5)); size(80); draw((-1.2,0)--(-0.2,0)--(-0.2,1)--(-1.2,1)--cycle); draw((0.8,0.5)--(2.9,0.5),EndArrow); draw((4,-1)--(5,-1)--(5,0)--(4,0)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((4,1)--(5,1)--(5,2)--(4,2)--cycle); draw((5,-1)--(6,-1)--(6,0)--(5,0)--cycle); draw((5,0)--(6,0)--(6,1)--(5,1)--cycle); draw((6,0)--(7,0)--(7,1)--(6,1)--cycle); draw((6,1)--(7,1)--(7,2)--(6,2)--cycle); [/asy]What is the minimum possible perimeter of this new group of blocks? $\textbf{(A) }11\qquad\textbf{(B) }12\qquad\textbf{(C) }13\qquad\textbf{(D) }14\qquad\textbf{(E) }15$

Let $ ABC$ be a triangle and $ AB\ne AC$ . $ D$ is a point on $ BC$ such that $ BA \equal{} BD$ and $ B$ is between $ C$ and $ D$ . Let $ I_{c}$ be center of the circle which touches $ AB$ and the extensions of $ AC$ and $ BC$ . $ CI_{c}$ intersect the circumcircle of $ ABC$ again at $ T$ . If $ \angle TDI_{c} \equal{} \frac {\angle B \plus{} \angle C}{4}$ then find $ \angle A$

In triangle $ABC$, $90^{o}> \angle A> \angle B> \angle C$. Let the circumcenter and orthocenter of the triangle be $O$ and $H$. $OH$ intersects $BC$ at $T$ and the circumcenter of $(AHO)$ is $X$. Prove that the reflection of $H$ over $XT$ lies on the circumcircle of triangle $ABC$.

Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.

What is the sum of the infinite series $\frac{20}{3} +\frac{17}{9} + \frac{20}{27} + \frac{17}{81} + \frac{20}{243} + \frac{17}{729} + ...$ ?

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties? [list] [*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$; [*] $a_2 - a_1$ is not divisible by $m$. [/list] [i]Holden Mui[/i]

It takes a steamer $5$ days to go from Gorky to Astrakhan downstream the Volga river and $7$ days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?

Let $ABC$ be an acute triangle and suppose points $A, A_b, B_a, B, B_c, C_b, C, C_a,$ and $A_c$ lie on its perimeter in this order. Let $A_1 \neq A$ be the second intersection point of the circumcircles of triangles $AA_bC_a$ and $AA_cB_a$. Analogously, $B_1 \neq B$ is the second intersection point of the circumcircles of triangles $BB_cA_b$ and $BB_aC_b$, and $C_1 \neq C$ is the second intersection point of the circumcircles of triangles $CC_aB_c$ and $CC_bA_c$. Suppose that the points $A_1, B_1,$ and $C_1$ are all distinct, lie inside the triangle $ABC$, and do not lie on a single line. Prove that lines $AA_1, BB_1, CC_1,$ and the circumcircle of triangle $A_1B_1C_1$ all pass through a common point. [i]Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)[/i]

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?$\phantom{h}$ [asy] unitsize(0.4 cm); pair transx, transy; int i, j; int x, y; transx = (13,0); transy = (0,-9); for (i = 0; i <= 2; ++i) { for (j = 0; j <= 1; ++j) { if (i <= 1 || j <= 0) { for (x = 1; x <= 10; ++x) { draw(shift(i*transx + j*transy)*((x,0)--(x,5)),gray(0.7) + dashed); } for (y = 1; y <= 5; ++y) { draw(shift(i*transx + j*transy)*((0,y)--(10,y)),gray(0.7) + dashed); } draw(shift(i*transx + j*transy)*((0,0)--(11,0)),Arrow(6)); draw(shift(i*transx + j*transy)*((0,0)--(0,6)),Arrow(6)); label("time", (5,-0.5) + i*transx + j*transy); label(rotate(90)*"distance", (-0.5,2.5) + i*transx + j*transy); } }} draw((0,0)--(1.5,2.5)--(7.5,2.5)--(9,5),linewidth(1.5*bp)); draw((0,0)--(10,5),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(2.5,2.5)--(7.5,2.5)--(10,5)),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(2.5,3)--(7,2)--(10,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(2.5,2.5)--(6.5,2.5)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(7.5,2)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(2.5,2)--(7.5,3)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(9,5)),linewidth(1.5*bp)); label("(A)", (-1,6)); label("(B)", (-1,6) + transx); label("(C)", (-1,6) + 2*transx); label("(D)", (-1,6) + transy); label("(E)", (-1,6) + transx + transy); [/asy]

Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^\circ$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.