Can we divide an equilateral triangle $\vartriangle ABC$ into $2011$ small triangles using $122$ straight lines? (there should be $2011$ triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles)

Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? $\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $

Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$

Find all positive integers $n$, for which there exist positive integers $a>b$, satisfying $n=\frac{4ab}{a-b}$.

Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$. [i]Demetres Christofides, Cyprus[/i]

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

One moving point in the coordinate plane can move right or up one position. $N$ is a number of all paths : paths that moving point starts from $(0, 0)$, without passing $(1, 0), (2, 1), . . . , (n, n-1)$ and moves $2n$ times to $(n, n)$. $a_k$ is a number of special paths : paths include in $N$, but $k$th moves to the right, $k+1$th moves to the up. find $$\frac{1}{N} (a_1+a_2+ . . . + a_{2n-1})$$

What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?

In given triangle $\triangle ABC$, difference between sizes of each pair of sides is at least $d>0$. Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $$[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$$ where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$.

Line \(\ell\) touches circle $S_1$ in the point $X$ and circle $S_2$ in the point $Y$. We draw a line $m$ which is parallel to $\ell$ and intersects $S_1$ in a point $P$ and $S_2$ in a point $Q$. Prove that the ratio $XP/YQ$ does not depend on the choice of $m$.

A regular $ n\minus{}$gon $ A_1A_2A_3 \cdots A_k \cdots A_n$ inscribed in a circle of radius $ R$ is given. If $ S$ is a point on the circle, calculate \[ T \equal{} \sum^n_{k\equal{}1} SA^2_k.\]

William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder when $M$ is divided by $1000$.

For each $k$ , find the least $n$ in terms of $k$ st the following holds: There exists $n$ real numbers $a_1 , a_2 ,\cdot \cdot \cdot , a_n$ st for each $i$ : $$0 < a_{i+1} - a_{i} < a_i - a_{i-1}$$ And , there exists $k$ pairs $(i,j)$ st $a_i - a_j = 1$.

The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary? $ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$

There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? $ \text{(A)}\ 30\qquad\text{(B)}\ 40\qquad\text{(C)}\ 41\qquad\text{(D)}\ 60\qquad\text{(E)}\ 119 $

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

Masaru randomly paints $50\%$ of the area of a square. What is the probability that at least $60\%$ of the left side of the square is painted? [asy] size(2cm); defaultpen(fontsize(7)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle,linewidth(1.5)); fill(circle((0.3,0.3),0.3),paleblue); fill(circle((1,1),0.5),palered); fill(circle((0.8,2),0.8),purple); fill(circle((3,3),0.6),orange); fill(circle((3.4,1.5),0.6),mediumgreen); fill(circle((1.4,1.6),0.4),yellow); fill(circle((2.5,2.8),0.4),cyan); fill(circle((1,3.5),0.5),red); fill((3,0)--(4,0)--(4,0.4)--(3.5,0.8)--cycle,magenta); fill((2,4)--(3,4)--(3.1,3.8)--(2.7,3.5)--(2.4,3.1)--cycle,olive); draw((2,0)--(2,4),dashed); [/asy] $\textbf{(A) } 25\%\qquad\textbf{(B) } 30\%\qquad\textbf{(C) } 35\%\qquad\textbf{(D) } 40\%\qquad\textbf{(E) } 45\%$

Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D \equal{} 2 \angle B, AD \equal{} 5,$ and $ CD \equal{} 2.$ Then $ AB$ equals A. 7 B. 8 C. 13/2 D. 27/4 E. $ 5 \plus{} \frac{3 \sqrt{2}}{2}$

Suppose that $\displaystyle{{v_1},{v_2},...,{v_d}}$ are unit vectors in $\displaystyle{{{\Bbb R}^d}}$. Prove that there exists a unitary vector $\displaystyle{u}$ such that $\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for $\displaystyle{i = 1,2,...,d}$. [b]Note.[/b] Here $\displaystyle{ \cdot }$ denotes the usual scalar product on $\displaystyle{{{\Bbb R}^d}}$. [i]Proposed by Tomasz Tkocz, University of Warwick.[/i]

For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold: Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as \[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\] where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$. Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as \[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\] when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$. [i]Proposed by Cheng-Ying Chang and usjl[/i]

Fill in each blank unshaded cell with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence. 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] size(9cm); for (int x=0; x<=11; ++x) draw((x, 0) -- (x, 5), linewidth(.5pt)); for (int y=0; y<=5; ++y) draw((0, y) -- (11, y), linewidth(.5pt)); filldraw((0,4)--(0,3)--(2,3)--(2,4)--cycle, gray, gray); filldraw((1,1)--(1,2)--(3,2)--(3,1)--cycle, gray, gray); filldraw((4,1)--(4,4)--(5,4)--(5,1)--cycle, gray, gray); filldraw((7,0)--(7,3)--(6,3)--(6,0)--cycle, gray, gray); filldraw((7,4)--(7,5)--(6,5)--(6,4)--cycle, gray, gray); filldraw((8,1)--(8,2)--(10,2)--(10,1)--cycle, gray, gray); filldraw((9,4)--(9,3)--(11,3)--(11,4)--cycle, gray, gray); draw((0,0)--(11,0)--(11,5)--(0,5)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } foo(1, 2, "10"); foo(4, 0, "31"); foo(5, 0, "26"); foo(10, 0, "59"); foo(0, 4, "3"); foo(7, 4, "59"); [/asy]

Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of triangle $DFE$ to the area of quadrilateral $ABEF$? $ \textbf{(A)}\ 1:2 \qquad\textbf{(B)}\ 1:3 \qquad\textbf{(C)}\ 1:5 \qquad\textbf{(D)}\ 1:6 \qquad\textbf{(E)}\ 1:7 $

If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? $\textbf{(A) }\frac{x(x+2)(x+5)}{(x+1)(x+3)}\qquad\textbf{(B) }\frac{x(x+1)(x+5)}{(x+2)(x+3)}\qquad$ $\textbf{(C) }\frac{(x+1)(x+3)(x+5)}{x(x+2)}\qquad\textbf{(D) }\frac{(x+1)(x+3)}{x(x+2)(x+5)}\qquad \textbf{(E) }\text{none of these}$

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$. Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.