Found problems: 85335
LMT Accuracy Rounds, 2023 S Tie
Estimate the value of $$\sum^{2023}_{n=1} \left(1+ \frac{1}{n} \right)^n$$ to $3$ decimal places.
2013 NIMO Summer Contest, 4
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]
2006 Princeton University Math Competition, 8
Find all integers $n$ (not necessarily positive) such that $7n^3-3n^2-3n-1$ is a perfect cube.
LMT Guts Rounds, 2013
[u]Round 9[/u]
[b]p25.[/b] Define a hilly number to be a number with distinct digits such that when its digits are read from left to right, they strictly increase, then strictly decrease. For example, $483$ and $1230$ are both hilly numbers, but $123$ and $1212$ are not. How many $5$-digit hilly numbers are there?
[b]p26.[/b] Triangle ABC has $AB = 4$ and $AC = 6$. Let the intersection of the angle bisector of $\angle BAC$ and $\overline{BC}$ be $D$ and the foot of the perpendicular from C to the angle bisector of $\angle BAC$ be $E$. What is the value of $AD/AE$?
[b]p27.[/b] Given that $(7+ 4\sqrt3)^x+ (7-4\sqrt3)^x = 10$, find all possible values of $(7+ 4\sqrt3)^x-(7-4\sqrt3)^x$.
[u]Round 10[/u]
Note: In this set, the answers for each problem rely on answers to the other problems.
[b]p28.[/b] Let X be the answer to question $29$. If $5A + 5B = 5X - 8$ and $A^2 + AB - 2B^2 = 0$, find the sum of all possible values of $A$.
[b]p29.[/b] Let $W$ be the answer to question $28$. In isosceles trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$, line segments $ \overline{AC}$ and $ \overline{BD}$ split each other in the ratio $2 : 1$. Given that the length of $BC$ is $W$, what is the greatest possible length of $\overline{AB}$ for which there is only one trapezoid $ABCD$ satisfying the given conditions?
[b]p30.[/b] Let $W$ be the answer to question $28$ and $X$ be the answer to question $29$. For what value of $Z$ is $ |Z - X| + |Z - W| - |W + X - Z|$ at a minimum?
[u]Round 11[/u]
[b]p31.[/b] Peijin wants to draw the horizon of Yellowstone Park, but he forgot what it looked like. He remembers that the horizon was a string of $10$ segments, each one either increasing with slope $1$, remaining flat, or decreasing with slope $1$. Given that the horizon never dipped more than $1$ unit below or rose more than $1$ unit above the starting point and that it returned to the starting elevation, how many possible pictures can Peijin draw?
[b]p32.[/b] DNA sequences are long strings of $A, T, C$, and $G$, called base pairs. (e.g. AATGCA is a DNA sequence of 6 base pairs). A DNA sequence is called stunningly nondescript if it contains each of A, T, C, G, in some order, in 4 consecutive base pairs somewhere in the sequence. Find the number of stunningly nondescript DNA sequences of 6 base pairs (the example above is to be included in this count).
[b]p33.[/b] Given variables s, t that satisfy $(3 + 2s + 3t)^2 + (7 - 2t)^2 + (5 - 2s - t)^2 = 83$, find the minimum possible value of $(-5 + 2s + 3t) ^2 + (3 - 2t)^2 + (2 - 2s - t)^2$.
[u]Round 12[/u]
[b]p34.[/b] Let $f(n)$ be the number of powers of 2 with n digits. For how many values of n from $1$ to $2013$ inclusive does $f(n) = 3$? If your answer is N and the actual answer is $C$, then the score you will receive on this problem is $max\{15 - \frac{|N-C|}{26039} , 0\}$, rounded to the nearest integer.
[b]p35.[/b] How many total characters are there in the source files for the LMT $2013$ problems? If your answer is $N$ and the actual answer is $C$, then the score you receive on this problem is $max\{15 - \frac{|N - C|}{1337}, 0\}$, rounded to the nearest integer.
[b]p36.[/b] Write down two distinct integers between $0$ and $300$, inclusive. Let $S$ be the collection of everyone’s guesses. Let x be the smallest nonnegative difference between one of your guesses and another guess in $S$ (possibly your other guess). Your team will be awarded $min(15, x)$ points.
PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here [/url] and 6-8 [url=https://artofproblemsolving.com/community/c3h3136014p28427163]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2020 Caucasus Mathematical Olympiad, 5
Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$, $b\leq 100\,000$, and
$$
\frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}.
$$
MOAA Accuracy Rounds, 2022
[b]p1.[/b] Find the last digit of $2022^{2022}$.
[b]p2.[/b] Let $a_1 < a_2 <... < a_8$ be eight real numbers in an increasing arithmetic progression. If $a_1 + a_3 + a_5 + a_7 = 39$ and $a_2 + a_4 + a_6 + a_8 = 40$, determine the value of $a_1$.
[b]p3.[/b] Patrick tries to evaluate the sum of the first $2022$ positive integers, but accidentally omits one of the numbers, $N$, while adding all of them manually, and incorrectly arrives at a multiple of $1000$. If adds correctly otherwise, find the sum of all possible values of $N$.
[b]p4.[/b] A machine picks a real number uniformly at random from $[0, 2022]$. Andrew randomly chooses a real number from $[2020, 2022]$. The probability that Andrew’s number is less than the machine’s number is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[b]p5.[/b] Let $ABCD$ be a square and $P$ be a point inside it such that the distances from $P$ to sides $AB$ and $AD$ respectively are $2$ and $4$, while $PC = 6$. If the side length of the square can be expressed in the form $a +\sqrt{b}$ for positive integers $a, b$, then determine $a + b$.
[b]p6.[/b] Positive integers $a_1, a_2, ..., a_{20}$ sum to $57$. Given that $M$ is the minimum possible value of the quantity $a_1!a_2!...a_{20}!$, find the number of positive integer divisors of $M$.
[b]p7.[/b] Jessica has $16$ balls in a box, where $15$ of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.
[b]p8.[/b] The Lucas sequence is defined by these conditions: $L_0 = 2$, $L_1 = 1$, and $L_{n+2} =L_{n+1} +L_n$ for all $n \ge 0$. Determine the remainder when $L^2_{2019} +L^2_{2020}$ is divided by $L_{2023}$.
[b]p9.[/b] Let $ABCD$ be a parallelogram. Point $P$ is selected in its interior such that the distance from $P$ to $BC$ is exactly $6$ times the distance from $P$ to $AD$, and $\angle APB = \angle CPD = 90^o$. Given that $AP = 2$ and $CP = 9$, the area of $ABCD$ can be expressed as $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.
[b]p10.[/b] Consider the polynomial $P(x) = x^{35} + ... + x + 1$. How many pairs $(i, j)$ of integers are there with $0 \le i < j \le 35$ such that if we flip the signs of the $x^i$ and $x^j$ terms in $P(x)$ to form a new polynomial $Q(x)$, then there exists a nonconstant polynomial $R(x)$ with integer coefficients dividing both $P(x)$ and $Q(x)$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1972 All Soviet Union Mathematical Olympiad, 173
One-round hockey tournament is finished (each plays with each one time, the winner gets $2$ points, looser -- $0$, and $1$ point for draw). For arbitrary subgroup of teams there exists a team (may be from that subgroup) that has got an odd number of points in the games with the teams of the subgroup. Prove that there was even number of the participants.
2022 Taiwan TST Round 1, A
Find all $f:\mathbb{Z}\to\mathbb{Z}$ such that
\[f\left(\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\right)+f(x)=f(f(y))+\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\]
holds for all $x,y\in\mathbb{Z}$.
[i]Proposed by usjl[/i]
2020 Miklós Schweitzer, 8
Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying
\[D(f\circ g)=(D(f)\circ g)\cdot D(g)\]
for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.
2014 China Team Selection Test, 5
Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.
1957 Polish MO Finals, 3
Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.
2021 Czech-Polish-Slovak Junior Match, 2
An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.
2008 Indonesia TST, 1
Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.
2023 EGMO, 2
We are given an acute triangle $ABC$. Let $D$ be the point on its circumcircle such that $AD$ is a diameter. Suppose that points $K$ and $L$ lie on segments $AB$ and $AC$, respectively, and that $DK$ and $DL$ are tangent to circle $AKL$.
Show that line $KL$ passes through the orthocenter of triangle $ABC$.
KoMaL A Problems 2018/2019, A. 739
Let $a_1,a_2,\dotsc$ be a sequence of real numbers from the interval $[0,1]$. Prove that there is a sequence $1\leqslant n_1<n_2<\dotsc$ of positive integers such that
$$A=\lim_{\substack{i,j\to \infty \\ i\neq j}} a_{n_i+n_j}$$exists, i.e., for every real number $\epsilon >0$, there is a constant $N_{\epsilon}$ that $|a_{n_i+n_j}-A|<\epsilon$ is satisfied for any pair of distinct indices $i,j>N_{\epsilon}$.
DMM Individual Rounds, 2015
[b]p1.[/b] Find the minimum value of $x^4 +2x^3 +3x^2 +2x+2$, where x can be any real number.
[b]p2.[/b] A type of digit-lock has $5$ digits, each digit chosen from $\{1,2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s?
[b]p3.[/b] Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have $80\%$ chance to be good balls and MIO balls have $50\%$ chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball?
[b]p4.[/b] What is the smallest positive integer that is one-ninth of its reverse?
[b]p5.[/b] When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many di erent orders can Michael get dressed?
[b]p6.[/b] The numbers $1, 2,..., 2015$ are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number?
[b]p7.[/b] Let $A = (a_1b_1a_2b_2... a_nb_n)_{34}$ and $B = (b_1b_2... b_n)_{34}$ be two numbers written in base $34$. If the sum of the base-$34$ digits of $A$ is congruent to $15$ (mod $77$) and the sum of the base $34$ digits of $B$ is congruent to $23$ (mod $77$). Then if $(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x$ (mod $77$) and $0 \le x \le 76$, what is $x$? (you can write $x$ in base $10$)
[b]p8.[/b] What is the sum of the medians of all nonempty subsets of $\{1, 2,..., 9\}$?
[b]p9.[/b] Tony is moving on a straight line for $6$ minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes?
[b]p10.[/b] Find the number of nonnegative integer triplets $a, b, c$ that satisfy $$2^a3^b + 9 = c^2.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1988 Bundeswettbewerb Mathematik, 3
Prove that all acute-angled triangles with the equal altitudes $h_c$ and the equal angles $\gamma$ have orthic triangles with same perimeters.
2017 European Mathematical Cup, 4
The real numbers $x,y,z$ satisfy $x^2+y^2+z^2=3.$ Prove that the inequality
$x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}.$
and find all triples $(x,y,z)$ for which equality holds.
OMMC POTM, 2023 8
Find all polygons $P$ that can be covered completely by three (possibly overlapping) smaller dilated versions of itself.
[i]Proposed by Evan Chang (squareman), USA[/i]
2019 Germany Team Selection Test, 3
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
2020 AMC 12/AHSME, 15
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
$\textbf{(A) } 11 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$
2020 JHMT, 7
Quadrilateral $ABCD$ is cyclic and has positive integer side lengths. Suppose $AC \cdot BD = 53$ and $CD < DA$. The value of $\frac{AB /BC}{AD /DC}$ can be expressed as a common fraction $\frac{p}{q}$ , where $p$ and $q$ are relatively prime. Compute $p + q$.
2006 Sharygin Geometry Olympiad, 6
a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$.
Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$.
b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$.
Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.
2019 MOAA, Sets 6-9
[u]Set 6[/u]
[b]p16.[/b] Let $n! = n \times (n - 1) \times ... \times 2 \times 1$. Find the maximum positive integer value of $x$ such that the quotient $\frac{160!}{160^x}$ is an integer.
[b]p17.[/b] Let $\vartriangle OAB$ be a triangle with $\angle OAB = 90^o$ . Draw points $C, D, E, F, G$ in its plane so that $$\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,$$ and none of these triangles overlap. If points $O, A, G$ lie on the same line, then let $x$ be the sum of all possible values of $\frac{OG}{OA }$. Then, $x$ can be expressed in the form $m/n$ for relatively prime positive integers $m, n$. Compute $m + n$.
[b]p18.[/b] Let $f(x)$ denote the least integer greater than or equal to $x^{\sqrt{x}}$. Compute $f(1)+f(2)+f(3)+f(4)$.
[u]Set 7[/u]
The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \ge 0$.
[b]p19.[/b] Find the least odd prime factor of $(F_3)^{20} + (F_4)^{20} + (F_5)^{20}$.
[b]p20.[/b] Let
$$S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...$$ Compute $420S$.
[b]p21.[/b] Consider the number $$Q = 0.000101020305080130210340550890144... ,$$ the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to $\frac{1}{Q}$.
[u]Set 8[/u]
[b]p22.[/b] In five dimensional hyperspace, consider a hypercube $C_0$ of side length $2$. Around it, circumscribe a hypersphere $S_0$, so all $32$ vertices of $C_0$ are on the surface of $S_0$. Around $S_0$, circumscribe a hypercube $C_1$, so that $S_0$ is tangent to all hyperfaces of $C_1$. Continue in this same fashion for $S_1$, $C_2$, $S_2$, and so on. Find the side length of $C_4$.
[b]p23.[/b] Suppose $\vartriangle ABC$ satisfies $AC = 10\sqrt2$, $BC = 15$, $\angle C = 45^o$. Let $D, E, F$ be the feet of the altitudes in $\vartriangle ABC$, and let $U, V , W$ be the points where the incircle of $\vartriangle DEF$ is tangent to the sides of $\vartriangle DEF$. Find the area of $\vartriangle UVW$.
[b]p24.[/b] A polynomial $P(x)$ is called spicy if all of its coefficients are nonnegative integers less than $9$. How many spicy polynomials satisfy $P(3) = 2019$?
[i]The next set will consist of three estimation problems.[/i]
[u]Set 9[/u]
Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed $1,000,000$.
[b]p25.[/b] Suppose a circle of radius $20192019$ has area $A$. Let s be the side length of a square with area $A$. Compute the greatest integer less than or equal to $s$.
If $n$ is the correct answer, an estimate of $e$ gives $\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}$ points.
[b]p26.[/b] Given a $50 \times 50$ grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue?
If $n$ is the correct answer, an estimate of $e$ gives $\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor$ points.
[b]p27.[/b] The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is $\frac{\pi}{3\sqrt2} \approx 74.05\%$ of space (confirmed as recently as $2017!$), so we say that the packing density of spheres in three dimensions is about $0.74$. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than $10^8 \times d$.
If $n$ is the correct answer, an estimate of e gives $\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}$ points.
PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2002 AMC 10, 20
Points $ A,B,C,D,E$ and $ F$ lie, in that order, on $ \overline{AF}$, dividing it into five segments, each of length 1. Point $ G$ is not on line $ AF$. Point $ H$ lies on $ \overline{GD}$, and point $ J$ lies on $ \overline{GF}$. The line segments $ \overline{HC}, \overline{JE},$ and $ \overline{AG}$ are parallel. Find $ HC/JE$.
$ \text{(A)}\ 5/4 \qquad
\text{(B)}\ 4/3 \qquad
\text{(C)}\ 3/2 \qquad
\text{(D)}\ 5/3 \qquad
\text{(E)}\ 2$