Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. $(\text a)$ Show that if $a_1,a_2,\ldots,a_k$ are distinct nonnegative integers, then $f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})$ depends only on the sum $a_1+a_2+\ldots+a_k$. $(\text b)$ Assuming part $(\text a)$, we can define $$g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),$$where $s=a_1+a_2+\ldots+a_k$. Find the least positive integer $p$ for which $$g(s)=g(s+p),\enspace\text{for all }s\ge1,$$or show that no such $p$ exists.

Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}\plus{}b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that a) if $ m\plus{}n > p$ then $ k \equal{} p$; b) if $ m\plus{}n\leq p$ then $ k\geq m\plus{}n\minus{}1$.

Positive real numbers $a_1,a_2,\ldots, a_n$ satisfy the equation $$2a_1+a_2+\ldots+a_{n-1}=a_n+\frac{n^2-3n+2}{2}$$ For every positive integer $n \geq 3$ find the smallest possible value of the sum $$\frac{(a_1+1)^2}{a_2}+\ldots+\frac{(a_{n-1}+1)^2}{a_n}$$ [i]M. Zorka[/i]

In the $8 \times 8$ grid below, label $8$ squares with $X$ and 8 squares with $Y$ such that: 1. No square can be labeled with both an $X$ and a $Y$. 2. Each row and each column must contain exactly one square labeled $X$ and one square labeled $Y$. 3. Any square marked with a $?$ or a $\heartsuit$ cannot be labeled with an $X$ or a $Y$. 4. We say that a square marked with a $?$ or a $\heartsuit$ sees a label ($X$ or $Y$) if one can move in a straight line horizontally or vertically from the marked square to the square with the label, without crossing any other squares with $X$’s or $Y$’s. It is OK to cross other squares marked with a $?$ or $\heartsuit$. Using this definition: (a) Each square marked with a $?$ must see exactly 2 $X$’s and 1 $Y$. (b) Each square marked with a $\heartsuit$ must see exactly 1 $X$ and 2 $Y$’s. \begin{tabular}{ | c | c | c | c | c | c | c | c | } \hline & & $\star$ & $\star$ & $\star$ & $\star$ & $\star$ & \\ \hline & & & & & & & $\star$ \\ \hline & & & & & & & $\star$ \\ \hline & $\heartsuit$ & & & & & & $\star$ \\ \hline & & & & & & & $\star$ \\ \hline & & & & & & & $\heartsuit$ \\ \hline & & & & $\star$ & & & \\ \hline & & & & & & & \\ \hline \end{tabular} 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 conditions of the problem. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Prove that every positive rational number $m/n$ can be represented as a sum of reciprocals of distinct positive integers.

Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent. Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \overline{AD}$ and $ \overline{BC}$ are common external tangents to the circles. What is the area of hexagon $ AOBCPD$? [asy] unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11)); pair A, B, C, D; pair[] O; O[1] = (6,0); O[2] = (12,0); A = (32/6,8*sqrt(2)/6); B = (32/6,-8*sqrt(2)/6); C = 2*B; D = 2*A; draw(Circle(O[1],2)); draw(Circle(O[2],4)); draw((0.7*A)--(1.2*D)); draw((0.7*B)--(1.2*C)); draw(O[1]--O[2]); draw(A--O[1]); draw(B--O[1]); draw(C--O[2]); draw(D--O[2]); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SW); label("$D$", D, NW); dot("$O$", O[1], SE); dot("$P$", O[2], SE); label("$2$", (A + O[1])/2, E); label("$4$", (D + O[2])/2, E);[/asy] $ \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}$

In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made $100$ left turns, how many right turns must it have made? [i](4 points)[/i]

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

Show that the equation $ y^{37}\equiv x^3\plus{}11 \pmod p$ is solvable for every prime $ p$, where $ p\leq100$.

An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

Does there exist a surjective function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which $f(x+y)-f(x)-f(y)$ takes only 0 and 1 for values for random $x$ and $y$?

Let $ABC$ be an acute triangle. Points $B'$ and $C'$ are located on the interior of sides $AB$ and $AC$, respectively. Let $M$ denote the second intersection of the circumcircles of triangles $ABC$ and $AB'C'$, while let $N$ denote the second intersection of the circumcircles of triangles $ABC'$ and $AB'C$. Reflect $M$ across lines $AB$ and $AC$, and let $l$ denote the line through the reflections. a) Prove that the line through $M$ perpendicular to $AM$, the line $AK$, and $l$ are either concurrent or all parallel. b) Show that if the three lines are concurrent at $S$, then triangles $SBC'$ and $SCB'$ have equal areas. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Given the equation $x^4-x^3-1=0$ [b](a)[/b] Find the number of its real roots. [b](b)[/b] We denote by $S$ the sum of the real roots and by $P$ their product. Prove that $P< - \frac{11}{10}$ and $S> \frac {6}{11}$.

In the quadrilateral $ABCD$, $AB=AD$, $CB=CD$, $\angle ABC =90^\circ$. $E$, $F$ are on $AB$, $AD$ and $P$, $Q$ are on $EF$($P$ is between $E, Q$), satisfy $\frac{AE}{EP}=\frac{AF}{FQ}$. $X, Y$ are on $CP, CQ$ that satisfy $BX \perp CP, DY \perp CQ$. Prove that $X, P, Q, Y$ are concyclic.

Suppose that the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are distinct. Show that \[(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\odot O.$ The lines tangent to $\odot O$ at $A,B$ intersect at $L.$ $M$ is the midpoint of the segment $AB.$ The line passing through $D$ and parallel to $CM$ intersects $ \odot (CDL) $ at $F.$ Line $CF$ intersects $DM$ at $K,$ and intersects $\odot O$ at $E$ (different from point $C$). Prove that $EK=DK.$

Let $a,b\in \mathbb{R}$ and $z\in \mathbb{C}\backslash \mathbb{R}$ so that $\left| a-b \right|=\left| a+b-2z \right|$. a) Prove that the equation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$, has a unique solution. b) Solve the following inequation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}\le {{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$. The Mathematical Gazette

Equilateral triangle $ABC$ has $AD = DB = FG = AE = EC = 4$ and $BF = GC = 2$. From $D$ and $G$ are drawn perpendiculars to $EF$ intersecting at $H$ and $I$, respectively. The three polygons $ECGI$, $FGI$, and $BFHD$ are rearranged to $EANL$, $MNK$, and $AMJD$ so that the rectangle $HLKJ$ is formed. Find its area. [img]https://cdn.artofproblemsolving.com/attachments/d/4/7e6667f0f0544b6fbc860f8d86c8ceaaf85cc1.png[/img]

Three circumferences are tangent to each other, as shown in the figure. The region of the outer circle that is not covered by the two inner circles has an area equal to $2 \pi$. Determine the length of the $PQ$ segment . [img]https://cdn.artofproblemsolving.com/attachments/a/e/65c08c47d4d20a05222a9b6cf65e84a25283b7.png[/img]

Which of the following are true? $\textbf{(A)}~GL(n,\mathbb R)\text{ is connected}$ $\textbf{(B)}~GL(n,\mathbb C)\text{ is connected}$ $\textbf{(C)}~O(n,\mathbb R)\text{ is connected}$ $\textbf{(D)}~O(n,\mathbb C)\text{ is connected}$

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$