Upload AMC_questions.csv

AMC_questions.csv

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+ID,Difficulty,Label,Question
+1,Intermediate,"Combinatorics
+","Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \leq n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is?"
+2,Intermediate,"Combinatorics
+","Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B, A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum."
+3,Intermediate,"Combinatorics
+","The numbers 1447,1005 and 1231 have something in common: each is a 4-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?"
+4,Intermediate,"Combinatorics
+","For $\{1,2,3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1,2,3,6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply 5 . Find the sum of all such alternating sums for $n=7$."
+5,Intermediate,"Combinatorics
+","For $\{1,2,3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1,2,3,6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply 5 . Find the sum of all such alternating sums for $n=7$."
+6,Intermediate,"Combinatorics
+","Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?"
+7,Intermediate,"Combinatorics
+",What is the largest 2-digit prime factor of the integer $n=\left(\begin{array}{l}200 \\ 100\end{array}\right)$ ?
+8,Intermediate,"Combinatorics
+","A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$."
+9,Intermediate,"Combinatorics
+","Let $A, B, C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p=\frac{n}{729}$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$."
+10,Intermediate,"Combinatorics
+","In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?"
+11,Introductory,"Combinatorics
+",The number of diagonals that can be drawn in a polygon of 100 sides is:
+12,Introductory,"Combinatorics
+",Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:
+13,Introductory,"Combinatorics
+",One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is
+14,Introductory,"Combinatorics
+",A fair die is rolled six times. The probability of rolling at least a five at least five times is
+15,Introductory,"Combinatorics
+",Assume every 7-digit whole number is a possible telephone number except those that begin with 0 or 1 . What fraction of telephone numbers begin with 9 and end with 0 ?
+16,Introductory,"Combinatorics
+","The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately"
+17,Introductory,"Combinatorics
+","How many rearrangements of $a b c d$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $a b$ or $b a$."
+18,Introductory,"Combinatorics
+","How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?"
+19,Introductory,"Combinatorics
+","How many ways can a student schedule 3 mathematics courses - algebra, geometry, and number theory -- in a 6 -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)"
+20,Introductory,"Combinatorics
+","The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?"
+21,Olympiad,"Combinatorics
+","Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance 1 away. What is the least number of moves that Carina can make in order for triangle $A B C$ to have area $2021 ?$"
+22,Olympiad,"Combinatorics
+","There is an integer $n>1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car that starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed).
+Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies."
+23,Olympiad,"Combinatorics
+","There are $4 n$ pebbles of weights $1,2,3, \ldots, 4 n$. Each pebble is colored in one of $n$ colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:
+- The total weights of both piles are the same.
+- Each pile contains two pebbles of each color."
+24,Olympiad,"Combinatorics
+","Let $n$ be a positive integer. Eric and a squid play a turn-based game on an infinite grid of unit squares. Eric's goal is to capture the squid by moving onto the same square as it.
+Initially, all the squares are colored white. The squid begins on an arbitrary square in the grid, and Eric chooses a different square to start on. On the squid's turn, it performs the following action exactly 2020 times: it chooses an adjacent unit square that is white, moves onto it, and sprays the previous unit square either black or gray. Once the squid has performed this action 2020 times, all squares colored gray are automatically colored white again, and the squid's turn ends. If the squid is ever unable to move, then Eric automatically wins. Moreover, the squid is claustrophobic, so at no point in time is it ever surrounded by a closed loop of black or gray squares. On Eric's turn, he performs the following action at most $n$ times: he chooses an adjacent unit square that is white and moves onto it. Note that the squid can trap Eric in a closed region, so that Eric can never win.
+
+Eric wins if he ever occupies the same square as the squid. Suppose the squid has the first turn, and both Eric and the squid play optimally. Both Eric and the squid always know each other's location and the colors of all the squares. Find all positive integers $n$ such that Eric can win in finitely many moves."
+25,Olympiad,"Combinatorics
+","Let $k$ be the number of students in a circle. Then let $m$ be the number of ways they can rearrange ourselves so that each of them is in the same spot or within one spot of where they started, and no two people are ever on the same spot. If $m$ leaves a remainder of 1 when divided by 5 , how many possible values are there of $k$, where $k$ is at least 3 and at most $2008 ?$"
+26,Olympiad,"Combinatorics
+","A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column)."
+27,Olympiad,"Combinatorics
+","An $(n, k)$-tournament is a contest with $n$ players held in $k$ rounds such that:
+(i) Each player plays in each round, and every two players meet at most once.
+(ii) If player $A$ meets player $B$ in round $i$, player $C$ meets player $D$ on round $i$, and player $A$ meets player $C$ in round $j$, then player $B$ meets player $D$ in round $j$.
+Determine all pairs $(n, k)$ for which there exists an $(n, k)$-tournament."
+28,Olympiad,"Combinatorics
+","For a given positive integer $n>2$, let $C_1, C_2, C_3$ be the boundaries of three convex $n$-gons in the plane such that $C_1 \cap C_2, C_2 \cap C_3$, $C_3 \cap C_1$ are finite. Find the maximum number of points in the set $C_1 \cap C_2 \cap C_3$."
+29,Olympiad,"Combinatorics
+","The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer $n \geq 4$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from a complete graph on $n$ vertices (where each pair of vertices are joined by an edge)."
+30,Olympiad,"Combinatorics
+","The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer $n \geq 4$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from a complete graph on $n$ vertices (where each pair of vertices are joined by an edge)."
+31,Introductory,Geometry,"Triangle $A B C$ is equilateral with side length 6 . Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A, O$, and $C$ ?"
+32,Introductory,Geometry,"The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?"
+33,Introductory,Geometry,"Let $A B C$ be a triangle. The bisector of $\angle A B C$ intersects $\overline{A C}$ at $E$, and the bisector of $\angle A C B$ intersects $\overline{A B}$ at $F$. If $B F=1, C E=2$, and $B C=3$, then the perimeter of $\triangle A B C$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
+34,Introductory,Geometry,"The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point?"
+35,Introductory,Geometry,"A trapezoid has side lengths $3,5,7$, and 11 . The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1}+r_2 \sqrt{n_2}+r_3$, where $r_1, r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$ ?"
+36,Introductory,Geometry,"Square $P Q R S$ lies in the first quadrant. Points $(3,0),(5,0),(7,0)$, and $(13,0)$ lie on lines $S P, R Q, P Q$, and $S R$, respectively. What is the sum of the coordinates of the center of the square $P Q R S$ ?"
+37,Introductory,Geometry,"Triangle $A B C$ has $A B=27, A C=26$, and $B C=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle A B C$. What is $B I$ ?"
+38,Introductory,Geometry,"In rectangle $A B C D, A B=6, A D=30$, and $G$ is the midpoint of $\overline{A D}$. Segment $A B$ is extended 2 units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{E D}$ and $\overline{B C}$. What is the area of quadrilateral $B F D G$ ?"
+39,Introductory,Geometry,"In $\triangle A B C, A B=86$, and $A C=97$. A circle with center $A$ and radius $A B$ intersects $\overline{B C}$ at points $B$ and $X$. Moreover $\overline{B X}$ and $\overline{C X}$ have integer lengths. What is $B C$ ?"
+40,Introductory,Geometry,Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?
+41,Intermediate,Geometry,"Square $A B C D$ is inscribed in a circle. Point $P$ is on this circle such that $A P \cdot C P=56$, and $B P \cdot D P=90$. What is the area of the square?"
+42,Intermediate,Geometry,"In triangle $A B C$ we have $A B=7, A C=8, B C=9$. Point $D$ is on the circumscribed circle of the triangle so that $A D$ bisects angle $B A C$ What is the value of $A D / C D$ ?"
+43,Intermediate,Geometry,"A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{A B}$, has length 31 . Find the sum of the lengths of the three diagonals that can be drawn from $A$."
+44,Intermediate,Geometry,"A hexagon with sides of lengths $2,2,7,7,11$, and 11 is inscribed in a circle. Find the diameter of the circle"
+45,Intermediate,Geometry,"In a regular heptagon $A B C D E F G$, prove that: $\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A E}$."
+46,Intermediate,Geometry,"$\triangle A B C$ has point $D$ on $A B$, point $E$ on $B C$, and point $F$ on $A C . A E, C D$, and $B F$ intersect at point $G$. The ratio $A D: D B$ is $3: 5$ and the ratio $C E: E B$ is $8: 3$. Find the ratio of $F G: G B$"
+47,Intermediate,Geometry,"Consider a triangle $A B C$ with its three medians drawn, with the intersection points being $D, E, F$, corresponding to $A B, B C$, and $A C$ respectively. Thus, if we label point $A$ with a weight of $1, B$ must also have a weight of 1 since $A$ and $B$ are equidistant from $D$. By the same process, we find $C$ must also have a weight of 1 . Now, since $A$ and $B$ both have a weight of $1, D$ must have a weight of 2 (as is true for $E$ and $F$ ). Thus, if we label the centroid $P$, we can deduce that $D P: P C$ is $1: 2$ - the inverse ratio of their weights."
+48,Intermediate,Geometry,"In rectangle $A B C D, A B=6$ and $B C=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $B E=E F=F C$. Segments $\overline{A E}$ and $\overline{A F}$ intersect $\overline{B D}$ at $P$ and $Q$, respectively. The ratio $B P: P Q: Q D$ can be written as $r: s: t$ where the greatest common factor of $r, s$, and $t$ is 1 . What is $r+s+t$ ?"
+49,Intermediate,Geometry,"Triangle $A B C$ has $A C=450$ and $B C=300$. Points $K$ and $L$ are located on $\overline{A C}$ and $\overline{A B}$ respectively so that $A K=C K$, and $\overline{C L}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{B K}$ and $\overline{C L}$, and let $M$ be the point on line $B K$ for which $K$ is the midpoint of $\overline{P M}$. If $A M=180$, find $L P$."
+50,Intermediate,Geometry,"In parallelogram $A B C D$, point $M$ is on $\overline{A B}$ so that $\frac{A M}{A B}=\frac{17}{1000}$ and point $N$ is on $\overline{A D}$ so that $\frac{A N}{A D}=\frac{17}{2009}$. Let $P$ be the point of intersection of $\overline{A C}$ and $\overline{M N}$. Find $\frac{A C}{A P}$."
+51,Olympiad,Geometry,"Let $\triangle A B C$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $\overline{B C}, \overline{C A}$, and $\overline{A B}$ respectively. One of the intersection points of the line $\overline{E F}$ and the circumcircle is $P$. The lines $\overline{B P}$ and $\overline{D F}$ meet at point $Q$. Prove that $|A P|=|A Q|$. (IMO Shortlist $2010 \mathrm{G} 1$ )"
+52,Olympiad,Geometry,"Four circles $\omega, \omega_A, \omega_B$, and $\omega_C$ with the same radius are drawn in the interior of triangle $A B C$ such that $\omega_A$ is tangent to sides $A B$ and $A C, \omega_B$ to $B C$ and $B A, \omega_C$ to $C A$ and $C B$, and $\omega$ is externally tangent to $\omega_A, \omega_B$, and $\omega_C$. If the sides of triangle $A B C$ are 13,14, and 15 , the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
+53,Olympiad,Geometry,"The following problems are constructions involving only a straightedge (no compass).
+1. Construct the fourth harmonic line to three given lines through a point.
+2. Construct the fourth harmonic point to three points on a line.
+3. If a given right angle and a given arbitrary angle have their vertex and one side in common, double the given arbitrary angle.
+4. Draw a parallel through a given point $P$ to two given parallel lines $l_1$ and $l_2$."
+54,Olympiad,Geometry,"Given two similar right triangles $A B C$ and $A^{\prime} B^{\prime} C, k=\frac{A C}{B C}$, $\angle A C B=90^{\circ}, D=A A^{\prime} \cap B B^{\prime}$. Find $\angle A D B$ and $\frac{A A^{\prime}}{B B^{\prime}}$."
+55,Olympiad,Geometry,Let $\triangle A B C$ be an isosceles right triangle $(A C=B C)$. Let $S$ be a point on a circle with diameter $B C$. The line $\ell$ is symmetrical to $S C$ with respect to $A B$ and intersects $B C$ at $D$. Prove that $A S \perp D S$.
+56,Olympiad,Geometry,"$\triangle A B F \sim \triangle B C D \sim \triangle C A E$. Points $D, E, F$ are outside $\triangle A B C$. Prove that the centroids of triangles $\triangle A B C$ and $\triangle D E F$ are coinsite."
+57,Olympiad,Geometry,Let triangle $\triangle A B C$ and point $A^{\prime}$ on sideline $B C$ be given. Construct $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A C$ and $C^{\prime}$ lies on sideline $A B$.
+58,Olympiad,Geometry,"Let triangle $\triangle A B C$ and point $C^{\prime}\left(C^{\prime} \neq C, C^{\prime} \neq B\right)$ on sideline $B C$ be given. $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A B$ and $A^{\prime}$ lies on sideline $A C$. The spiral symilarity $T$ maps $\triangle A B C$ into $\triangle A^{\prime} B^{\prime} C^{\prime}$. Prove
+a) $\angle A B^{\prime} A^{\prime}=\angle B C^{\prime} B^{\prime}=\angle C A^{\prime} C^{\prime}$.
+b) Center of $T$ is the First Brocard point of triangles $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$."
+59,Olympiad,Geometry,"Let $\triangle A B C$ and point $A^{\prime}$ on sideline $B C$ be given. $\triangle A^{\prime} B^{\prime} C^{\prime} \sim \triangle A B C$ where $B^{\prime}$ lies on sideline $A C$ and $C^{\prime}$ lies on sideline $A B$.
+Denote $D=B B^{\prime} \cap C C^{\prime}, E=A A^{\prime} \cap C C^{\prime}, F=B B^{\prime} \cap A A^{\prime}$.
+Prove that circumcircles of triangles $\triangle A B F, \triangle A^{\prime} B^{\prime} F, \triangle B C D$, $\triangle B^{\prime} C^{\prime} D, \triangle A C E, \triangle A^{\prime} C^{\prime} E$ have the common point."
+60,Olympiad,Geometry,"Let triangle $\triangle A B C$ be given. The triangle $\triangle A C E$ is constructed using a spiral similarity of $\triangle A B C$ with center $A$, angle of rotation $\angle B A C$ and coefficient $\frac{A C}{A B}$
+A point $D$ is centrally symmetrical to a point $B$ with respect to $C$.
+Prove that the spiral similarity with center $E$, angle of rotation $\angle A C B$ and coefficient $\frac{B C}{A C}$ taking $\triangle A C E$ to $\triangle C D E$."
+61,Introductory,Algebra,"In a jar there are blue jelly beans and green jelly beans. Then, $15 \%$ of the blue jelly beans are removed and $40 \%$ of the green jelly beans are removed. If afterwards the total number of jelly beans is $80 \%$ of the original number of jelly beans, then determine the percent of the remaining jelly beans that are blue."
+62,Introductory,Algebra,"Find all values of $B$ that have the property that if $(x, y)$ lies on the hyperbola $2 y^2-x^2=1$, then so does the point $(3 x+4 y, 2 x+B y)$."
+63,Introductory,Algebra,"Joe has a collection of 23 coins, consisting of 5 -cent coins, 10 -cent coins, and 25 -cent coins. He has 3 more 10 -cent coins than 5 -cent coins, and the total value of his collection is 320 cents. How many more 25 -cent coins does Joe have than 5 -cent coins?"
+64,Introductory,Algebra,"Suppose that real number $x$ satisfies
+\begin{align*}
+\sqrt{49-x^2}-\sqrt{25-x^2}=3
+\end{align*}
+What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$ ?"
+65,Introductory,Algebra,"Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2 x^2 y+1$. What is $|a-b|$ ?"
+66,Introductory,Algebra,"Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by 1 , the value of the fraction is increased by $10 \%$ ?"
+67,Introductory,Algebra,"Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20 \%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5 \%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?"
+68,Introductory,Algebra,"Suppose $a, b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a b c}{|a b c|}$ ?"
+69,Introductory,Algebra,"A lattice point in an $x y$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y=m x+2$ passes through no lattice point with $0<x \leq 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$ ?"
+70,Introductory,Algebra,"The first four terms of an arithmetic sequence are $p, 9,3 p-q$, and $3 p+q$. What is the $2010^{\text {th }}$ term of this sequence?"
+71,Intermediate,Algebra,"The product $N$ of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$."
+72,Intermediate,Algebra,"It is given that $\log _6 a+\log _6 b+\log _6 c=6$, where $a, b$, and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c$."
+73,Intermediate,Algebra,"For $t=1,2,3,4$, define $S_t=\sum_{i=1}^{350} a_i^t$ where $a_i \in\{1,2,3,4\}$. If $S_1=513$ and $S_4=4745$, find the minimum possible value for $S_2$."
+74,Intermediate,Algebra,"Determine the value of $a$ so that the following fraction reduces to a quotient of two linear expressions:
+\begin{align*}
+\frac{x^3+(a-10) x^2-x+(a-6)}{x^3+(a-6) x^2-x+(a-10)}
+\end{align*}"
+75,Intermediate,Algebra,"For certain pairs $(m, n)$ of positive integers with $m \geq n$ there are exactly 50 distinct positive integers $k$ such that $|\log m-\log k|<\log n$. Find the sum of all possible values of the product $m n$."
+76,Intermediate,Algebra,"Suppose that $a, b$, and $c$ are positive real numbers such that $a^{\log _3 7}=27, b^{\log _7 11}=49$, and $c^{\log _{11} 25}=\sqrt{11}$. Find
+\begin{align*}
+a^{\left(\log _3 7\right)^2}+b^{\left(\log _7 11\right)^2}+c^{\left(\log _{11} 25\right)^2}
+\end{align*}"
+77,Intermediate,Algebra,"Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $x^2-2 x+2 \leq P(x) \leq 2 x^2-4 x+3$ for all real numbers $x$, and suppose $P(11)=181$. Find $P(16)$."
+78,Intermediate,Algebra,Find the smallest positive integer $n$ with the property that the polynomial $x^4-n x+63$ can be written as a product of two nonconstant polynomials with integer coefficients.
+79,Intermediate,Algebra,"Let $z_1, z_2, z_3, \ldots, z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\sum_{i=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$."
+80,Intermediate,Algebra,"Let $f(x)=\left(x^2+3 x+2\right)^{\cos (\pi x)}$. Find the sum of all positive integers $n$ for which
+\begin{align*}
+\left|\sum_{k=1}^n \log _{10} f(k)\right|=1
+\end{align*}"
+81,Olympiad,Algebra,"Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b$,
+\begin{align*}
+f\left(a^2+b^2\right)=f(a) f(b) \text { and } f\left(a^2\right)=f(a)^2 .
+\end{align*}"
+82,Olympiad,Algebra,"The real numbers $a, b, c, d$ are such that $a \geq b \geq c \geq d>0$ and $a+b+c+d=1$. Prove that
+\begin{align*}
+(a+2 b+3 c+4 d) a^a b^b c^c d^d<1
+\end{align*}"
+83,Olympiad,Algebra,"Let $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ (meaning $f$ takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers $x$ and $y$,
+\begin{align*}
+f(x) f(y) f(x+y)=f(x)+f(y)-f(x+y) .
+\end{align*}
+Prove that there is a constant $a>1$ such that
+\begin{align*}
+f(x)=\frac{a^x-1}{a^x+1}
+\end{align*}
+for all positive real numbers $x$."
+84,Olympiad,Algebra,"Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
+\begin{align*}
+(f(x)+x y) \cdot f(x-3 y)+(f(y)+x y) \cdot f(3 x-y)=(f(x+y))^2
+\end{align*}"
+85,Olympiad,Algebra,"Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that
+\begin{align*}
+f(x)+f(t)=f(y)+f(z)
+\end{align*}
+for all rational numbers $x<y<z<t$ that form an arithmetic progression. ( $\mathbb{Q}$ is the set of all rational numbers.)"
+86,Olympiad,Algebra,"Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)$
+for all real numbers $x$ and $y$."
+87,Olympiad,Algebra,"Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
+\begin{align*}
+x f(2 f(y)-x)+y^2 f(2 x-f(y))=\frac{f(x)^2}{x}+f(y f(y))
+\end{align*}
+for all $x, y \in \mathbb{Z}$ with $x \neq 0$."
+88,Olympiad,Algebra,"Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$(where $\mathbb{Z}^{+}$is the set of positive integers) such that $f(n !)=f(n)$ ! for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$ for all distinct positive integers $m, n$"
+89,Olympiad,Algebra,"Let $a_1, a_2, \ldots, a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\ldots+a_n$. A grasshopper is to jump along the real axis, starting at the point 0 and making $n$ jumps to the right with lengths $a_1, a_2, \ldots, a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$."
+90,Olympiad,Algebra,"Find the lowest possible values from the function
+\begin{align*}
+f(x)=x^{2008}-2 x^{2007}+3 x^{2006}-4 x^{2005}+5 x^{2004}-\cdots-2006 x^3+2007 x^2-2008 x+2009
+\end{align*}
+for any real numbers $x$."
+91,Introductory,Number Theory,"The least common multiple of a positive integer $n$ and 18 is 180 , and the greatest common divisor of $n$ and 45 is 15 . What is the sum of the digits of $n$ ?"
+92,Introductory,Number Theory,"Let $n$ be the least positive integer greater than 1000 for which
+\begin{align*}
+\operatorname{gcd}(63, n+120)=21 \text { and } \operatorname{gcd}(n+63,120)=60 .
+\end{align*}
+What is the sum of the digits of $n$ ?"
+93,Introductory,Number Theory,The sum of three consecutive integers is 54 . What is the smallest of the three integers?
+94,Introductory,Number Theory,"In the equation below, $A$ and $B$ are consecutive positive integers, and $A, B$, and $A+B$ represent number bases:
+\begin{align*}
+132_A+43_B=69_{A+B} \text {. }
+\end{align*}
+What is $A+B$ ?"
+95,Introductory,Number Theory,Let $N$ be the greatest integer multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by 1000 .
+96,Introductory,Number Theory,How many positive two-digit integers are factors of $2^{24}-1$ ?
+97,Introductory,Number Theory,"Find $a+b+c$, where $a, b$, and $c$ are the hundreds, tens, and units digits of the six-digit integer $123 a b c$, which is a multiple of 990 ."
+98,Introductory,Number Theory,"Let $a / b$ be the probability that a randomly selected divisor of 2007 is a multiple of 3 . If $a$ and $b$ are relatively prime positive integers, find $a+b$."
+99,Introductory,Number Theory,Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)=12$ ?
+100,Introductory,Number Theory,"Call a number prime-looking if it is composite but not divisible by 2,3 , or 5 . The three smallest prime-looking numbers are 49,77 , and 91 . There are 168 prime numbers less than 1000 . How many prime-looking numbers are there less than $1000 ?$"
+101,Intermediate,Number Theory,"Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55 r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$."
+102,Intermediate,Number Theory,For each positive integer $n$ let $a_n$ be the least positive integer multiple of 23 such that $a_n \equiv 1\left(\bmod 2^n\right)$. Find the number of positive integers $n$ less than or equal to 1000 that satisfy $a_n=a_{n+1}$.
+103,Intermediate,Number Theory,Let $A$ be an acute angle such that $\tan A=2 \cos A$. Find the number of positive integers $n$ less than or equal to 1000 such that $\sec ^n A+\tan ^n A$ is a positive integer whose units digit is 9 .
+104,Intermediate,Number Theory,"The sum of all positive integers $m$ such that $\frac{13 !}{m}$ is a perfect square can be written as $2^a 3^b 5^c 7^d 11^e 13^f$, where $a, b, c, d, e$, and $f$ are positive integers. Find $a+b+c+d+e+f$."
+105,Intermediate,Number Theory,"For positive integers $a, b$, and $c$ with $a<b<c$, consider collections of postage stamps in denominations $a, b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to 1000 cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c)=97$ for some choice of $a$ and $b$."
+106,Intermediate,Number Theory,"For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2,3,4,5,6,7,8,9$, and 10 . For example, $R(15)=1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n)=R(n+1) ?$"
+107,Intermediate,Number Theory,"For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma\left(a^n\right)-1$ is divisible by 2021 for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$."
+108,Intermediate,Number Theory,"The probability a randomly chosen positive integer $N<1000$ has more digits when written in base 7 than when written in base 8 can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
+109,Intermediate,Number Theory,"Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5, n$, and $n+1$ cents, 91 cents is the greatest postage that cannot be formed."
+110,Intermediate,Number Theory,"Determine all the sets of six consecutive positive integers such that the product of some two of them, added to the product of some other two of them is equal to the product of the remaining two numbers."
+111,Olympiad,Number Theory,"A deck of $n>1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
+For which $n$ does it follow that the numbers on the cards are all equal?"
+112,Olympiad,Number Theory,Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n+17 k$ for some positive integer $k$.
+113,Olympiad,Number Theory,"Two permutations $a_1, a_2, \ldots, a_{2010}$ and $b_1, b_2, \ldots, b_{2010}$ of the numbers $1,2, \ldots, 2010$ are said to intersect if $a_k=b_k$ for some value of $k$ in the range $1 \leq k \leq 2010$. Show that there exist 1006 permutations of the numbers $1,2, \ldots, 2010$ such that any other such permutation is guaranteed to intersect at least one of these 1006 permutations."
+114,Olympiad,Number Theory,"Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0, k_1 \ldots, k_n$, all strictly greater than 1 , such that $k_0 k_1 \cdots k_n-1$ is the product of two consecutive integers."
+115,Olympiad,Number Theory,$K>L>M>N$ are positive integers such that $K M+L N=(K+L-M+N)(-K+L+M+N)$. Prove that $K L+M N$ is not prime.
+116,Olympiad,Number Theory,"Let $k_1<k_2<k_3<\cdots$; be positive integers, no two consecutive, and let $s_m=k_1+k_2+\cdots+k_m$, for $m=1,2,3, \ldots$ Prove that, for each positive integer $n$, the interval $\left[s_n, s_{n+1}\right)$, contains at least one perfect square."
+117,Olympiad,Number Theory,"Let $a>b>c>d$ be positive integers and suppose that
+\begin{align*}
+a c+b d=(b+d+a-c)(b+d-a+c) .
+\end{align*}
+Prove that $a b+c d$ is not prime."
+118,Olympiad,Number Theory,"Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers satisfying $m, n \in\{1,2, \ldots, 1981\}$ and $\left(n^2-m n-m^2\right)^2=1$"
+119,Olympiad,Number Theory,"A permutation of the set of positive integers $[n]=\{1,2, \ldots, n\}$ is a sequence $\left(a_1, a_2, \ldots, a_n\right)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3,5,1,2,4)$ is a permutation of $[5]$. Let $P(n)$ be the number of permutations of $[n]$ for which $k a_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of 2010 ."
+120,Olympiad,Number Theory,"Consider an open interval of length $1 / n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p / q$, with $1 \leq q \leq n$, contained in the given interval is at most $(n+1) / 2$."