HMMT 2001

Find $x-y$, given that $x^{4}=y^{4}+24, x^{2}+y^{2}=6$, and $x+y=3$.

Find $(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right) \cdots$, where $|x|<1$.

How many times does 24 divide into 100 ! (factorial)?

Given that $7,999,999,999$ has at most two prime factors, find its largest prime factor.

Find the 6 -digit number beginning and ending in the digit 2 that is the product of three consecutive even integers.

What is the last digit of $1^{1}+2^{2}+3^{3}+\cdots+100^{100}$ ?

A polynomial $P$ has four roots, $\frac{1}{4}, \frac{1}{2}, 2,4$. The product of the roots is 1 , and $P(1)=1$. Find $P(0)$.

How many integers between 1 and 2000 inclusive share no common factors with 2001 ?

Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$.

Find the real solutions of $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=10$.

A circle of radius 3 crosses the center of a square of side length 2 . Find the positive difference between the areas of the nonoverlapping portions of the figures.

Square $A B C D$ is drawn. Isosceles Triangle $C D E$ is drawn with $E$ a right angle. Square $D E F G$ is drawn. Isosceles triangle $F G H$ is drawn with $H$ a right angle. This process is repeated infinitely so that no two figures overlap each other. If square $A B C D$ has area 1 , compute the area of the entire figure.

A circle has two parallel chords of length $x$ that are $x$ units apart. If the part of the circle included between the chords has area $2+\pi$, find $x$.

Equilateral triangle $A B C$ with side length 1 is drawn. A square is drawn such that its vertex at $A$ is opposite to its vertex at the midpoint of $B C$. Find the area enclosed within the intersection of the insides of the triangle and square.

Point D is drawn on side $B C$ of equilateral triangle $A B C$, and $A D$ is extended past $D$ to $E$ such that angles $E A C$ and $E B C$ are equal. If $B E=5$ and $C E=12$, determine the length of $A E$.

Parallelogram $A E C F$ is inscribed in square $A B C D$. It is reflected across diagonal $A C$ to form another parallelogram $A E^{\prime} C F^{\prime}$. The region common to both parallelograms has area $m$ and perimeter $n$. Compute the value of $\frac{m}{n^{2}}$ if $A F: A D=1: 4$.

$A$ is the center of a semicircle, with radius $A D$ lying on the base. $B$ lies on the base between $A$ and $D$, and $E$ is on the circular portion of the semicircle such that $E B A$ is a right angle. Extend $E A$ through $A$ to $C$, and put $F$ on line $C D$ such that $E B F$ is a line. Now $E A=1, A C=\sqrt{2}, B F=\frac{2-\sqrt{2}}{4}, C F=\frac{2 \sqrt{5}+\sqrt{10}}{4}$, and $D F=\frac{2 \sqrt{5}-\sqrt{10}}{4}$. Find $D E$.

A sequence of ants walk from $(0,0)$ to $(1,0)$ in the plane. The $n$th ant walks along $n$ semicircles of radius $\frac{1}{n}$ with diameters lying along the line from $(0,0)$ to ( 1,0 ). Let $L_{n}$ be the length of the path walked by the $n$th ant. Compute $\lim _{n \rightarrow \infty} L_{n}$.

The polynomial $3 x^{5}-250 x^{3}+735 x$ is interesting because it has the maximum possible number of relative extrema and points of inflection at integer lattice points for a quintic polynomial. What is the sum of the $x$-coordinates of these points?

A balloon that blows up in the shape of a perfect cube is being blown up at a rate such that at time $t$ fortnights, it has surface area $6 t$ square furlongs. At how many cubic furlongs per fortnight is the air being pumped in when the surface area is 144 square furlongs?

What is the size of the largest rectangle that can be drawn inside of a 3-4-5 right triangle with one of the rectangle's sides along one of the legs of the triangle?

Same as question 4, but now we want one of the rectangle's sides to be along the hypotenuse.

The graph of $x^{2}-(y-1)^{2}=1$ has one tangent line with positive slope that passes through $(x, y)=(0,0)$. If the point of tangency is $(a, b)$, find $\sin ^{-1}\left(\frac{a}{b}\right)$ in radians.

Find the coefficient of $x^{12}$ in the Maclaurin series (i.e. Taylor series around $x=0$ ) for $\frac{1}{1-3 x+2 x^{2}}$.

Evaluate $\sum_{n=0}^{\infty} \cot ^{-1}\left(n^{2}+n+1\right)$.

On the planet Lemniscate, the people use the elliptic table of elements, a far more advanced version of our periodic table. They're not very good at calculus, though, so they've asked for your help. They know that Kr is somewhat radioactive and deteriorates into Pl , a very unstable element that deteriorates to form the stable element As. They started with a block of Kr of size 10 and nothing else. (Their units don't translate into English, sorry.) and nothing else. At time $t$, they let $x(t)$ be the amount of $\mathrm{Kr}, y(t)$ the amount of Pl , and $z(t)$ the amount of As. They know that $x^{\prime}(t)=-x$, and that, in the absence of $\operatorname{Kr}, y^{\prime}(t)=-2 y$. Your job is to find at what time $t$ the quantity of Pl will be largest. You should assume that the entire amount of Kr that deteriorates has turned into Pl .

Evaluate the definite integral $\int_{-1}^{+1} \frac{2 u^{332}+u^{998}+4 u^{1664} \sin u^{691}}{1+u^{666}} \mathrm{~d} u$.

Find $\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)$ in terms of $n$.

Calculate the sum of the coefficients of $P(x)$ if $\left(20 x^{27}+2 x^{2}+1\right) P(x)=2001 x^{2001}$.

There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer without replacement, the probability they are of the same color is $\frac{1}{5}$. How many blue socks are there in the drawer?

Order these four numbers from least to greatest: $5^{56}, 10^{51}, 17^{35}, 31^{28}$.

Alex picks his favorite point $(x, y)$ in the first quadrant on the unit circle $x^{2}+y^{2}=1$, such that a ray from the origin through $(x, y)$ is $\theta$ radians counterclockwise from the positive $x$-axis. He then computes $\cos ^{-1}\left(\frac{4 x+3 y}{5}\right)$ and is surprised to get $\theta$. What is $\tan (\theta)$ ?

What is the last digit of $17^{103}+5$ ?

Find $x+y$, given that $x^{2}-y^{2}=10$ and $x-y=2$.

There are some red and blue marbles in a box. We are told that there are twelve more red marbles than blue marbles, and we experimentally determine that when we pick a marble randomly we get a blue marble one quarter of the time. How many marbles are there in the box?

Find $a+b+c+d+e$ if

$$\begin{gathered} 3 a+2 b+4 d=10, \\ 6 a+5 b+4 c+3 d+2 e=8, \\ a+b+2 c+5 e=3, \\ 2 c+3 d+3 e=4, \text { and } \\ a+2 b+3 c+d=7 . \end{gathered}$$

What is the sum of the coefficients of the expansion $(x+2 y-1)^{6}$ ?

A right triangle has a hypotenuse of length 2 , and one of its legs has length 1 . The altitude to its hypotenuse is drawn. What is the area of the rectangle whose diagonal is this altitude?

Boris was given a Connect Four game set for his birthday, but his color-blindness makes it hard to play the game. Still, he enjoys the shapes he can make by dropping checkers into the set. If the number of shapes possible modulo (horizontal) flips about the vertical axis of symmetry is expressed as $9(1+2+\cdots+n)$, find $n$. (Note: the board is a vertical grid with seven columns and eight rows. A checker is placed into the grid by dropping it from the top of a column, and it falls until it hits either the bottom of the grid or another checker already in that column. Also, $9(1+2+\cdots+n)$ is the number of shapes possible, with two shapes that are horizontal flips of each other counted as one. In other words, the shape that consists solely of 3 checkers in the rightmost row and the shape that consists solely of 3 checkers in the leftmost row are to be considered the same shape.)

A circle of radius 3 crosses the center of a square of side length 2 . Find the difference between the areas of the nonoverlapping portions of the figures.

Call three sides of an opaque cube adjacent if someone can see them all at once. Draw a plane through the centers of each triple of adjacent sides of a cube with edge length 1. Find the volume of the closed figure bounded by the resulting planes.

Some students are taking a math contest, in which each student takes one of four tests. One third of the students take one test, one fourth take another test, one fifth take the next test, and 26 students take the last test. How many students are taking the contest in total?

What is the area of a square inscribed in a semicircle of radius 1 , with one of its sides flush with the diameter of the semicircle?

You take a wrong turn on the way to MIT and end up in Transylvania, where $99 \%$ of the inhabitants are vampires and the rest are regular humans. For obvious reasons, you want to be able to figure out who's who. On average, nine-tenths of the vampires are correctly identified as vampires and nine-tenths of humans are correct identified as humans. What is the probability that someone identified as a human is actually a human?

A real numbers $x$ is randomly chosen in the interval $\left[-15 \frac{1}{2}, 15 \frac{1}{2}\right]$. Find the probability that the closest integer to $x$ is odd.

A point on a circle inscribed in a square is 1 and 2 units from the two closest sides of the square. Find the area of the square.

Two circles are concentric. The area of the ring between them is $A$. In terms of $A$, find the length of the longest chord contained entirely within the ring.

Find the volume of the tetrahedron with vertices $(5,8,10),(10,10,17),(4,45,46),(2,5,4)$.

How many digits are in the base two representation of 10 ! (factorial)?

On a certain unidirectional highway, trucks move steadily at 60 miles per hour spaced $1 / 4$ of a mile apart. Cars move steadily at 75 miles per hour spaced 3 seconds apart. A lone sports car weaving through traffic at a steady forward speed passes two cars between each truck it passes. How quickly is it moving in miles per hour?

What is the 18 th digit after the decimal point of $\frac{10000}{9899}$ ?

P is a polynomial. When P is divided by $x-1$, the remainder is -4 . When P is divided by $x-2$, the remainder is -1 . When $P$ is divided by $x-3$, the remainder is 4 . Determine the remainder when P is divided by $x^{3}-6 x^{2}+11 x-6$.

Find all $x$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ such that $1-\sin ^{4} x-\cos ^{2} x=\frac{1}{16}$.

What is the radius of the smallest sphere in which 4 spheres of radius 1 will fit?

The Fibonacci numbers are defined by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$ for $n \geq 1$. The Lucas numbers are defined by $L_{1}=1, L_{2}=2$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n \geq 1$. Calculate $\frac{\prod_{n=1}^{15} \frac{F_{2 n}}{F_{n}}}{\prod_{n=1}^{13} L_{n}}$.

Express $\frac{\sin 10+\sin 20+\sin 30+\sin 40+\sin 50+\sin 60+\sin 70+\sin 80}{\cos 5 \cos 10 \cos 20}$ without using trigonometric functions.

Compute $\sum_{i=1}^{\infty} \frac{a i}{a^{i}}$ for $a>1$.

Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of $\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)$ (for instance, $(x+1)^{2}$ has two prime factors)?

Define $a$ ? $=(a-1) /(a+1)$ for $a \neq-1$. Determine all real values $N$ for which $(N ?) ?=\tan 15$.

All subscripts in this problem are to be considered modulo 6 , that means for example that $\omega_{7}$ is the same as $\omega_{1}$. Let $\omega_{1}, \ldots \omega_{6}$ be circles of radius $r$, whose centers lie on a regular hexagon of side length 1 . Let $P_{i}$ be the intersection of $\omega_{i}$ and $\omega_{i+1}$ that lies further from the center of the hexagon, for $i=1, \ldots 6$. Let $Q_{i}, i=1 \ldots 6$, lie on $\omega_{i}$ such that $Q_{i}, P_{i}, Q_{i+1}$ are colinear. Find the number of possible values of $r$.

January 3 , 1911 was an odd date as its abbreviated representation, $1 / 3 / 1911$, can be written using only odd digits (note all four digits are written for the year). To the nearest month, how many months will have elapsed between the most recent odd date and the next odd date (today is $3 / 3 / 2001$, an even date).

Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar?

Find the number of triangulations of a general convex 7 -gon into 5 triangles by 4 diagonals that do not intersect in their interiors.

Find $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^{2}}\right)$.

Let $A B C$ be a triangle with incenter $I$ and circumcenter $O$. Let the circumradius be $R$. What is the least upper bound of all possible values of $I O$ ?

Six students taking a test sit in a row of seats with aisles only on the two sides of the row. If they finish the test at random times, what is the probability that some student will have to pass by another student to get to an aisle?

Suppose $a, b, c, d$, and $e$ are objects that we can multiply together, but the multiplication doesn't necessarily satisfy the associative law, i.e. ( $x y$ ) z does not necessarily equal $x(y z)$. How many different ways are there to interpret the product $a b c d e$ ?

Compute $1 \cdot 2+2 \cdot 3+\cdots+(n-1) n$.

Suppose $x$ satisfies $x^{3}+x^{2}+x+1=0$. What are all possible values of $x^{4}+2 x^{3}+$ $2 x^{2}+2 x+1$ ?

Two concentric circles have radii $r$ and $R>r$. Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find $\frac{R}{r}$.

12 points are placed around the circumference of a circle. How many ways are there to draw 6 non-intersecting chords joining these points in pairs?

How many distinct sets of 8 positive odd integers sum to 20 ?

Find the number of real zeros of $x^{3}-x^{2}-x+2$.

Find the exact value of $1+\frac{1}{1+\frac{2}{1+\frac{1}{1++\frac{2}{1+\ldots}}}}$.

A beaver walks from $(0,0)$ to $(4,4)$ in the plane, walking one unit in the positive $x$ direction or one unit in the positive $y$ direction at each step. Moreover, he never goes to a point $(x, y)$ with $y>x$. How many different paths can he walk?

After walking so much that his feet get really tired, the beaver staggers so that, at each step, his coordinates change by either $(+1,+1)$ or $(+1,-1)$. Now he walks from $(0,0)$ to $(8,0)$ without ever going below the $x$-axis. How many such paths are there?

Frank and Joe are playing ping pong. For each game, there is a $30 \%$ chance that Frank wins and a $70 \%$ chance Joe wins. During a match, they play games until someone wins a total of 21 games. What is the expected value of number of games played per match?

Find the largest prime factor of $-x^{10}-x^{8}-x^{6}-x^{4}-x^{2}-1$, where $x=2 i$, $i=\sqrt{-1}$.

Karen has seven envelopes and seven letters of congratulations to various HMMT coaches. If she places the letters in the envelopes at random with each possible configuration having an equal probability, what is the probability that exactly six of the letters are in the correct envelopes?

Evaluate $\sum_{i=1}^{\infty} \frac{(i+1)(i+2)(i+3)}{(-2)^{i}}$.

A man is standing on a platform and sees his train move such that after $t$ seconds it is $2 t^{2}+d_{0}$ feet from his original position, where $d_{0}$ is some number. Call the smallest (constant) speed at which the man have to run so that he catches the train $v$. In terms of $n$, find the $n$th smallest value of $d_{0}$ that makes $v$ a perfect square.

Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens' digits are different from each others' tens' digits and all of their numbers' ones digits are different from each others' ones' digits?

Square $A B C D$ has side length 1. A dilation is performed about point $A$, creating square $A B^{\prime} C^{\prime} D^{\prime}$. If $B C^{\prime}=29$, determine the area of triangle $B D C^{\prime}$.

What is the remainder when 100 ! is divided by 101 ?

A circle with center at $O$ has radius 1. Points $P$ and $Q$ outside the circle are placed such that $P Q$ passes through $O$. Tangent lines to the circle through $P$ hit the circle at $P_{1}$ and $P_{2}$, and tangent lines to the circle through $Q$ hit the circle at $Q_{1}$ and $Q_{2}$. If $\angle P_{1} P P_{2}=45^{\circ}$ and angle $Q_{1} Q Q_{2}=30^{\circ}$, find the minimum possible length of arc $P_{2} Q_{2}$.

Mona has 12 match sticks of length 1 , and she has to use them to make regular polygons, with each match being a side or a fraction of a side of a polygon, and no two matches overlapping or crossing each other. What is the smallest total area of the polygons Mona can make?

How many different combinations of 4 marbles can be made from 5 indistinguishable red marbles, 4 indistinguishable blue marbles, and 2 indistinguishable black marbles?

Count the number of sequences $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of integers such that $a_{i} \leq 1$ for all $i$ and all partial sums $\left(a_{1}, a_{1}+a_{2}\right.$, etc.) are non-negative.

How many roots does $\arctan x=x^{2}-1.6$ have, where the arctan function is defined in the range $-\frac{p i}{2}<\arctan x<\frac{p i}{2}$ ?

If two fair dice are tossed, what is the probability that their sum is divisible by 5 ?

Count the number of permutations $a_{1} a_{2} \ldots a_{7}$ of 1234567 with longest decreasing subsequence of length at most two (i.e. there does not exist $i<j<k$ such that $a_{i}>a_{j}>a_{k}$ ).

A line of soldiers 1 mile long is jogging. The drill sergeant, in a car, moving at twice their speed, repeatedly drives from the back of the line to the front of the line and back again. When each soldier has marched 15 miles, how much mileage has been added to the car, to the nearest mile?

Find all the values of $m$ for which the zeros of $2 x^{2}-m x-8$ differ by $m-1$.

Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \geq 5$.

Count the number of sequences $1 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{5}$ of integers with $a_{i} \leq i$ for all $i$.

Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee that he will be able to win?

The cafeteria in a certain laboratory is open from noon until 2 in the afternoon every Monday for lunch. Two professors eat 15 minute lunches sometime between noon and 2 . What is the probability that they are in the cafeteria simultaneously on any given Monday?

What is the remainder when $2^{2001}$ is divided by $2^{7}-1$ ?

A product of five primes is of the form $A B C, A B C$, where $A, B$, and $C$ represent digits. If one of the primes is 491, find the product $A B C, A B C$.

If $\left(a+\frac{1}{a}\right)^{2}=3$, find $\left(a+\frac{1}{a}\right)^{3}$ in terms of $a$.

Solve $x=\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}$ for $x$.

When a single number is added to each member of the seqence $20,50,100$, the sequence becomes expressable as $x, a x, a^{2} x$. Find $a$.

Through a point in the interior of a triangle $A B C$, three lines are drawn, one parallel to each side. These lines divide the sides of the triangle into three regions each. Let $a, b$, and $c$ be the lengths of the sides opposite $\angle A, \angle B$, and $\angle C$, respectively, and let $a^{\prime}, b^{\prime}$, and $c^{\prime}$ be the lengths of the middle regions of the sides opposite $\angle A, \angle B$, and $\angle C$, respectively. Find the numerical value of $a^{\prime} / a+b^{\prime} / b+c^{\prime} / c$.

A stacking of circles in the plane consists of a base, or some number of unit circles centered on the $x$-axis in a row without overlap or gaps, and circles above the $x$-axis that must be tangent to two circles below them (so that if the ends of the base were secured and gravity were applied from below, then nothing would move). How many stackings of circles in the plane have 4 circles in the base?

Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let $n$ be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and $m$ be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair $(n, m)$.

For the sequence of numbers $n_{1}, n_{2}, n_{3}, \ldots$, the relation $n_{i}=2 n_{i-1}+a$ holds for all $i>1$. If $n_{2}=5$ and $n_{8}=257$, what is $n_{5}$ ?

What is the smallest positive integer $x$ for which $x^{2}+x+41$ is not a prime?

If $\frac{1}{9}$ of 60 is 5 , what is $\frac{1}{20}$ of 80 ?

The Fibonacci sequence $F_{1}, F_{2}, F_{3}, \ldots$ is defined by $F_{1}=F_{2}=1$ and $F_{n+2}=$ $F_{n+1}+F_{n}$. Find the least positive integer $t$ such that for all $n>0, F_{n}=F_{n+t}$.

Some people like to write with larger pencils than others. Ed, for instance, likes to write with the longest pencils he can find. However, the halls of MIT are of limited height $L$ and width $L$. What is the longest pencil Ed can bring through the halls so that he can negotiate a square turn?

Find all ordered pairs $(m, n)$ of integers such that $231 m^{2}=130 n^{2}$.

Find the sum of the infinite series $\frac{1}{3^{2}-1^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)+\frac{1}{5^{2}-3^{2}}\left(\frac{1}{3^{2}}-\frac{1}{5^{2}}\right)+\frac{1}{7^{2}-5^{2}}\left(\frac{1}{5^{2}}-\frac{1}{7^{2}}\right)+$

The set of points $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ in $\mathbf{R}^{4}$ such that $x_{1} \geq x_{2} \geq x_{3} \geq x_{4}$ is a cone (or hypercone, if you insist). Into how many regions is this cone sliced by the hyperplanes $x_{i}-x_{j}=1$ for $1 \leq i<j \leq n$ ?

How many multiples of 7 between $10^{6}$ and $10^{9}$ are perfect squares?

A triangle has sides of length 888,925 , and $x>0$. Find the value of $x$ that minimizes the area of the circle circumscribed about the triangle.

Let $x=2001^{1002}-2001^{-1002}$ and $y=2001^{1002}+2001^{-1002}$. Find $x^{2}-y^{2}$.

Let $(x, y)$ be a point in the cartesian plane, $x, y>0$. Find a formula in terms of $x$ and $y$ for the minimal area of a right triangle with hypotenuse passing through $(x, y)$ and legs contained in the $x$ and $y$ axes.

Trevor and Edward play a game in which they take turns adding or removing beans from a pile. On each turn, a player must either add or remove the largest perfect square number of beans that is in the heap. The player who empties the pile wins. For example, if Trevor goes first with a pile of 5 beans, he can either add 4 to make the total 9 , or remove 4 to make the total 1, and either way Edward wins by removing all the beans. There is no limit to how large the pile can grow; it just starts with some finite number of beans in it, say fewer than 1000 .

Before the game begins, Edward dispatches a spy to find out how many beans will be in the opening pile, call this $n$, then "graciously" offers to let Trevor go first. Knowing that the first player is more likely to win, but not knowing $n$, Trevor logically but unwisely accepts, and Edward goes on to win the game. Find a number $n$ less than 1000 that would prompt this scenario, assuming both players are perfect logicians. A correct answer is worth the nearest integer to $\log _{2}(n-4)$ points.

Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\cdots \pm 1$ ! is prime. Be prepared to justify your answer for $\left\{\begin{array}{c}n, \\ {\left[\frac{n+225}{10}\right],} \\ n \leq 25\end{array}\right.$ points, where $[N]$ is the greatest integer less than $N$.