Found problems: 83
2018 PUMaC Algebra A, 6
Let $a, b, c$ be non-zero real numbers that satisfy $\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}$. The expression $\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}$ has a maximum value $M$. Find the sum of the numerator and denominator of the reduced form of $M$.
2018 PUMaC Algebra A, 2
If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$,
$$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$
find the smallest $n$ such that $a_n < \frac{1}{2018}$.
2018 PUMaC Algebra B, 2
For what value of $n$ is $\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot 11}+\frac{1}{n(n+3)}=\frac{25}{154}$?
2018 PUMaC Algebra B, 1
Find the sum of the solutions to $\dfrac{1}{1+\dfrac{1}{|x-25|}}=\frac{49}{50}$.
2018 PUMaC Combinatorics B, 2
There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?
2018 PUMaC Algebra B, 8
Let $a, b, c$ be non-zero real numbers that satisfy $\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}$. The expression $\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}$ has a maximum value $M$. Find the sum of the numerator and denominator of the reduced form of $M$.
2018 PUMaC Live Round, Calculus 2
Three friends are trying to meet for lunch at a cafe. Each friend will arrive independently at random between $1\!:\!00$ pm and $2\!:\!00$ pm. Each friend will only wait for $5$ minutes by themselves before leaving. However, if another friend arrives within those $5$ minutes, the pair will wait $15$ minutes from the time the second friend arrives. If the probability that the three friends meet for lunch can be expressed in simplest form as $\tfrac{m}{n}$, what is $m+n$?
2018 PUMaC Live Round, 4.2
Some number of regular polygons meet at a point on the plane such that the polygons' interiors do not overlap, but the polygons fully surround the point (i.e. a sufficiently small circle centered at the point would be contained in the union of the polygons). What is the largest possible number of sides in any of the polygons?
2018 PUMaC Combinatorics A, 8
Let $S_5$ be the set of permutations of $\{1,2,3,4,5\}$, and let $C$ be the convex hull of the set
$$\{(\sigma(1),\sigma(2),\ldots,\sigma(5))\,|\,\sigma\in S_5\}.$$
Then $C$ is a polyhedron. What is the total number of $2$-dimensional faces of $C$?
2012 Princeton University Math Competition, A2
Let $a, b, c$ be real numbers such that $a+b+c=abc$. Prove that $\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge \frac{3}{4}$.
2018 PUMaC Algebra B, 6
Suppose real numbers $a, b, c, d$ satisfy $a + b + c + d = 17$ and $ab + bc + cd + da = 46$. If the minimum possible value of $a^2 + b^2 + c^2 + d^2$ can be expressed as a rational number $\frac{p}{q}$ in simplest form, find $p + q$.
2011 Princeton University Math Competition, Team Round
[hide=Rules]Time Limit: 25 minutes
Maximum Possible Score: 81
The following is a mathematical Sudoku puzzle which is also a crossword. Your job is to fill in as many blanks as you possibly can, including all shaded squares. You do not earn extra points for showing your work; the only points you get are for correctly filled-in squares. You get one point for each correctly filled-in square. You should read through the following rules carefully before starting.
$\bullet$ Your time limit for this round is $25$ minutes, in addition to the five minutes you get for reading the rules. So make use of your time wisely. The round is based more on speed than on perfect reasoning, so use your intuition well, and be fast.
$\bullet$ This is a Sudoku puzzle; all the squares should be filled in with the digits $1$ through $9$ so that every row and column contains each digit exactly once. In addition, each of the nine $3\times 3$ boxes that compose the grid also contains each digit exactly once. Furthermore, this is a super-Sudoku puzzle; in addition to satisfying all these conditions, the four $3\times 3$ boxes with red outlines also contain each of $1,..., 9$ exactly once. This last property is important to keep in mind – it may help you solve the puzzle faster.
$\bullet$ Just to restate the idea, you can use the digits $1$ through $9$, but not $0$. You may not use any other symbol, such as $\pi$ or $e$ or $\epsilon$. Each square gets exactly one digit.
$\bullet$ The grid is also a crossword puzzle; the usual rules apply. The shaded grey squares are the “black” squares of an ordinary crossword puzzle. The white squares as well as the shaded yellow ones count as the “white” crossword squares. All squares, white or shaded, count as ordinary Sudoku squares.
$\bullet$ If you obtain the unique solution to the crossword puzzle, then this solution extends to a unique solution to the Sudoku puzzle.
$\bullet$ You may use a graphing calculator to help you solve the clues.
The following hints and tips may prove useful while solving the puzzle.
$\bullet$ Use the super-Sudoku structure described in the first rule; use all the symmetries you have. Remember that we are not looking for proofs or methods, only for correctly filled-in squares.
$\bullet$ If you find yourself stuck on a specific clue, it is nothing to worry about. You can obtain the solution to that clue later on by solving other clues and figuring out certain digits of your desired solution. Just move on to the rest of the puzzle.
$\bullet$ As you progress through the puzzle, keep filling in all squares you have found on your solution sheet, including the shaded ones. Remember that for scoring, the shaded grey squares count the same as the white ones.
Good luck!
[/hide]
[asy]
// place label "s" in row i, column j
void labelsq(int i, int j, string s) { label("$"+s+"$",(j-0.5,7.5-i),fontsize(14)); }
// for example, use the command
// labelsq(1,7,"2");
// to put the digit 2 in the top right box
// **** rest of code ****
size(250); defaultpen(linewidth(1));
pair[] labels = {(1,1),(1,4),(1,6),(1,7),(1,9),(2,1),(2,6),(3,4),(4,1),(4,8),(5,1),(6,3),(6,5),(6,6),(7,1),(7,2),(7,7),(7,9),(8,1),(8,4),(9,1),(9,6)};
pair[] blacksq = {(1,5),(2,5),(3,2),(3,3),(3,8),(5,5),(5,6),(5,7),(5,9),(6,2),(6,7),(6,9),(8,3),(9,5),(9,8)};
path peachsq = shift(1,1)*scale(3)*unitsquare; pen peach = rgb(0.98,0.92,0.71); pen darkred = red + linewidth(2);
fill(peachsq,peach); fill(shift(4,0)*peachsq,peach); fill(shift(4,4)*peachsq,peach); fill(shift(0,4)*peachsq,peach);
for(int i = 0; i < blacksq.length; ++i) fill(shift(blacksq[i].y-1, 9-blacksq[i].x)*unitsquare, gray(0.6));
for(int i = 0; i < 10; ++i) { pen sudokuline = linewidth(1); if(i == 3 || i == 6) sudokuline = linewidth(2); draw((0,i)--(9,i),sudokuline); draw((i,0)--(i,9),sudokuline); }
draw(peachsq,darkred); draw(shift(4,0)*peachsq,darkred); draw(shift(4,4)*peachsq,darkred); draw(shift(0,4)*peachsq,darkred);
for(int i = 0; i < labels.length; ++i) label(string(i+1), (labels[i].y-1, 10-labels[i].x), SE, fontsize(10));
// **** draw letters ****
draw(shift(.5,.5)*((0,6)--(0,8)--(2,8)--(2,7)--(0,7)^^(3,8)--(3,6)--(5,6)--(5,8)^^(6,6)--(6,8)--(7,8)--(7,7)--(7,8)--(8,8)--(8,6)^^(0,3)--(0,5)--(2,5)--(2,3)--(2,4)--(0,4)^^(5,3)--(3,3)--(3,5)--(5,5)),linewidth(1)+rgb(0.94,0.74,0.58));
// **** end rest of code ****[/asy]
[b][u][i]Across[/i][/u][/b]
[b]1 Across.[/b] The following is a normal addition where each letter represents a (distinct) digit: $$GOT + TO + GO + TO = TOP$$This certainly does not have a unique solution. However, you discover suddenly that $G = 2$ and $P \notin \{4, 7\}$. Then what is the numeric value of the expression $GOT \times TO$?
[b]3 Across.[/b] A strobogrammatic number which reads the same upside down, e.g. $619$. On the other hand, a triangular number is a number of the form $n(n + 1)/2$ for some $n \in N$, e.g. $15$ (therefore, the $i^{th}$ triangular number $T_i$ is the sum of $1$ through $i$). Let $a$ be the third strobogrammatic prime number. Let $b$ be the smaller number of the smallest pair of triangular numbers whose sum and difference are also triangular numbers. What is the value of $ab$?
[b]6 Across.[/b] A positive integer $m$ is said to be palindromic in base $\ell$ if, when written in base $\ell$ , its digits are the same front-to-back and back-to-front. For $j, k \in N$, let $\mu (j, k)$ be the smallest base-$10$ integer that is palindromic in base $j$ as well as in base$ k$, and let $\nu (j, k) := (j + k) \cdot \mu (j, k)$. Find the value of $\nu (5, 9)$.
[b]7 Across.[/b] Suppose you have the unique solution to this Sudoku puzzle. In that solution, let $X$ denote the sum of all digits in the shaded grey squares. Similarly, let $Y$ denote the sum of all numbers in the shaded yellow squares on the upper left block (i.e. the $3 \times 3$ box outlined red towards the top left). Concatenate $X$ with $Y$ in that order, and write that down.
[b]8 Across.[/b] For any $n \in N$ such that $1 < n < 10$, define the sequence $X_{n,1}$,$X_{n,2}$,$ ...$ by $X_{n,1} = n$, and for $r \ge 2$, X_{n,r} is smallest number $k \in N$ larger than X_{n,r-1} such that $k$ and the sum of digits of $k$ are both powers of $n$. For instance, $X_{3,1 = 3}$, $X_{3,2} = 9$, $X_{3,3} = 27$, and so on. Concatenate $X_{2,2}$ with $X_{2,4}$, and write down the answer.
[b]9 Across.[/b] Find positive integers $x, y,z$ satisfying the following properties: $y$ is obtained by subtracting $93$ from $x$, and $z$ is obtained by subtracting $183$ from $y$, furthermore, $x, y$ and $z$ in their base-$10$ representations contain precisely all the digits from $1$ through $9$ once (i.e. concatenating $x, y$ and $z$ yields a valid $9$-digit Sudoku answer). Obviously, write down the concatenation of $x, y$ and $z$ in that order.
[b]11 Across.[/b] Find the largest pair of two-digit consecutive prime numbers $a$ and $b$ (with $a < b$) such that the sum of the digits of a plus the sum of the digits of b is also a prime number. Write the concatenation of $a$ and $b$.
[b]12 Across.[/b] Suppose you have a strip of $2n + 1$ squares, with n frogs on the $n$ squares on the left, and $n$ toads on the $n$ squares on the right. A move consists either of a toad or a frog sliding to an adjacent square if it is vacant, or of a toad or a frog jumping one square over another one and landing on the next square if it is vacant. For instance, the starting position
[img]https://cdn.artofproblemsolving.com/attachments/a/a/6c97f15304449284dc282ff86014f526322e4a.png[/img]
has the position
[img]https://cdn.artofproblemsolving.com/attachments/e/6/e2c9520731bd94dc0aa37f540c2b9d1bce6432.png[/img]
or the position
[img]https://cdn.artofproblemsolving.com/attachments/3/f/06868eca80d649c4f80425dc9dc5c596cb2a4e.png[/img]
as results of valid first moves. What is the minimum number of moves needed to swap the toads with the frogs if $n = 5$? How about $n = 6$? Concatenate your answers.
[b]15 Across.[/b] Let $w$ be the largest number such that $w$, $2w$ and $3w$ together contain every digit from $1$ through $9$ exactly once. Let $x$ be the smallest integer with the property that its first $5$ multiples contain the digit $9$. A Leyland number is an integer of the form $m^n + n^m$ for integers $m, n > 1$. Let $y$ be the fourth Leyland number. A Pillai prime is a prime number $p$ for which there is an integer $n > 0$ such that $n! \equiv - 1 (mod \,\, p)$, but $p \not\equiv 1 (mod \,\, n)$. Let $z$ be the fourth Pillai prime. Concatenate $w$, $x, y$ and $z$ in that order to obtain a permutation of $1,..., 9$. Write down this permutation.
[b]19 Across.[/b] A hoax number $k \in N$ is one for which the sum of its digits (in base $10$) equals the sum of the digits of its distinct prime factors (in base $10$). For instance, the distinct prime factors of $22$ are $2$ and $11$, and we have $2+2 = 2+(1+1)$. In fact, $22$ is the first hoax number. What is the second?
[b]20 Across.[/b] Let $a, b$ and $c$ be distinct $2$-digit numbers satisfying the following properties:
– $a$ is the largest integer expressible as $a = x^y = y^x$, for distinct integers $x$ and $y$.
– $b$ is the smallest integer which has three partitions into three parts, which all give the same product (which turns out to be $1200$) when multiplied.
– $c$ is the largest number that is the sum of the digits of its cube.
Concatenate $a, b$ and $c$, and write down the resulting 6-digit prime number.
[b]21 Across.[/b] Suppose $N = \underline{a}\, \underline{b} \, \underline{c} \, \underline{d}$ is a $4$-digit number with digits $a, b, c$ and $d$, such that $N = a \cdot b \cdot c \cdot d^7$. Find $N$.
[b]22 Across.[/b] What is the smallest number expressible as the sum of $2, 3, 4$, or $5$ distinct primes?
[b][u][i]Down [/i][/u][/b]
[b]1 Down.[/b] For some $a, b, c \in N$, let the polynomial $$p(x) = x^5 - 252x^4 + ax^3 - bx^2 + cx - 62604360$$ have five distinct roots that are positive integers. Four of these are 2-digit numbers, while the last one is single-digit. Concatenate all five roots in decreasing order, and write down the result.
[b]2 Down.[/b] Gene, Ashwath and Cosmin together have $2511$ math books. Gene now buys as many math books as he already has, and Cosmin sells off half his math books. This leaves them with $2919$ books in total. After this, Ashwath suddenly sells off all his books to buy a private jet, leaving Gene and Cosmin with a total of $2184$ books. How many books did Gene, Ashwath and Cosmin have to begin with? Concatenate the three answers (in the order Gene, Ashwath, Cosmin) and write down the result.
[b]3 Down.[/b] A regular octahedron is a convex polyhedron composed of eight congruent faces, each of which is an equilateral triangle; four of them meet at each vertex. For instance, the following diagram depicts a regular octahedron:
[img]https://cdn.artofproblemsolving.com/attachments/c/1/6a92f12d5e9f56b0699531ae8369a0ab8ab813.png[/img]
Let $T$ be a regular octahedron of edge length $28$. What is the total surface area of $T$ , rounded to the nearest integer?
[b]4 Down.[/b] Evaluate the value of the expression $$\sum^{T_{25}}_{k=T_{24}+1}k, $$ where $T_i$ denotes the $i^{th}$ triangular number (the sum of the integers from $1$ through $i$).
[b]5 Down.[/b] Suppose $r$ and $s$ are consecutive multiples of$ 9$ satisfying the following properties:
– $r$ is the smallest positive integer that can be written as the sum of $3$ positive squares in $3$ different ways.
– $s$ is the smallest $2$-digit number that is a Woodall number as well as a base-$10$ Harshad number. A Woodall number is any number of the form $n \cdot 2^n - 1$ for some $n \in N$. A base-$10$ Harshad number is divisible by the sum of its digits in base $10$.
Concatenate $r$ and $s$ and write down the result.
[b]10 Down.[/b] For any $k \in N$, let $\phi_p(k)$ denote the sum of the distinct prime factors of $k$. Suppose $N$ is the largest integer less than $50000$ satisfying $\phi_p(N) =\phi_p(N + 1)$, where the common value turns out to be a meager $55$. What is$ N$?
[b]13 Down.[/b] The $n^{th}$ $s$-gonal number $P(s, n)$ is defined as $$P(s, n) = (s - 3)T_{n-1} + T_n$$ where $T_i$ is the $i^{th}$ triangular number (recall that the $i^{th}$ triangular number is the sum of the numbers $1$ through $i$). Find the least $N$ such that $N$ is both a $34$-gonal number, and a $163$-gonal number.
[b]14 Down.[/b] A biprime is a positive integer that is the product of precisely two (not necessarily distinct) primes. A cluster of biprimes is an ordered triple $(m,m + 1,m + 2)$ of consecutive integers that are biprimes. There are precisely three clusters of biprimes below 100. Denote these by, say, $$\{(p, p + 1, p + 2), (q, q + 1,q + 2), (r, r + 1, r + 2)\}$$ and add the condition that $p + 2 < q < r - 2$ to fix the three clusters. Interestingly, $p + 1$ and $q$ are both multiples of $17$. Concatenate $q$ with $p + 1$ in that order, and write down the result.
[b]16 Down.[/b] Find the least positive integer $m$ (written in base $10$ as $m = \underline{a} \, \underline{b} \, \underline{c} $, with digits $a, b,c$), such that $m = (b + c)^a$.
[b]17 Down.[/b] Let $X$ be a set containing $32$ elements, and let $Y\subseteq X$ be a subset containing $29$ elements. How many $2$-element subsets of $X$ are there which have nonempty intersection with $Y$?
[b]18 Down.[/b] Find a positive integer $K < 196$, which is a strange twin of the number $196$, in the sense that $K^2$ shares the same digits as $196^2$, and $K^3$ shares the same digits as $196^3$.
PS. You should use hide for answers.
2018 PUMaC Live Round, 4.1
The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.
2018 PUMaC Team Round, 12
In right triangle $\triangle{ABC}$, a square $WXYZ$ is inscribed such that vertices $W$ and $X$ lie on hypotenuse $\overline{AB}$, vertex $Y$ lies on leg $\overline{BC}$, and vertex $Z$ lies on leg $\overline{CA}$. Let $\overline{AY}$ and $\overline{BZ}$ intersect at some point $P$. If the length of each side of square $WXYZ$ is $4$, the length of the hypotenuse $\overline{AB}$ is $60$, and the distance between point $P$ and point $G$, where $G$ denotes the centroid of $\triangle{ABC}$, is $\tfrac{a}{b}$, compute the value of $a+b$.
2018 PUMaC Team Round, 4
For how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13?$
2018 PUMaC Geometry B, 2
Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.
2018 PUMaC Combinatorics B, 3
In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.
2018 PUMaC Algebra A, 5
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
2018 PUMaC Algebra B, 7
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
2018 PUMaC Combinatorics A, 7
Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of
$$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$
then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?
2018 PUMaC Live Round, Estimation 1
A $2$-by-$2018$ grid is completely covered by non-overlapping L-tiles (see diagram below) and $1$-by-$1$ squares. If the L-tiles can be rotated and flipped, there are a total of $M$ such tilings.
[asy]
size(1cm);
draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);
draw((0,1)--(1,1)--(1,0));
[/asy]
What is $\ln M?$
Give your answer as an integer or decimal. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor7.5-\tfrac{|A-C|^{1.5}}{20}\rfloor,0\}.$
2018 PUMaC Algebra A, 3
Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$.
Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.
2018 PUMaC Live Round, Misc. 1
Consider all cubic polynomials $f(x)$ such that $f(2018)=2018$, the graph of $f$ intersects the $y$-axis at height $2018$, the coefficients of $f$ sum to $2018$, and $f(2019)>(2018)$.
We define the infinimum of a set $S$ as follows. Let $L$ be the set of lower bounds of $S$. That is, $\ell\in L$ if and only if for all $s\in S$, $\ell\leq s$. Then the infinimum of $S$ is $\max(L)$.
Of all such $f(x)$, what is the infinimum of the leading coefficient (the coefficient of the $x^3$ term)?
2018 PUMaC Algebra A, 4
Suppose real numbers $a, b, c, d$ satisfy $a + b + c + d = 17$ and $ab + bc + cd + da = 46$. If the minimum possible value of $a^2 + b^2 + c^2 + d^2$ can be expressed as a rational number $\frac{p}{q}$ in simplest form, find $p + q$.
2018 PUMaC Combinatorics B, 5
Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.