Olympiad problems

2017 ASDAN Math Tournament, 9

Let $f(x)=x^3+ax^2+bx$ for some $a,b$. For some $c$, $f(c)$ achieves a local maximum of $539$ (in other words, $f(c)$ is the maximum value of $f$ for some open interval around $c$). In addition, at some $d$, $f(d)$ achieves a local minimum of $-325$. Given that $c$ and $d$ are integers, compute $a+b$.

2017 ASDAN Math Tournament, 8

Tags: 2017 , Discrete Math Test

Let $S=\{1,2,3,4,5,6\}$. Compute the number of functions $f:S\rightarrow S$ such that $f(f(f(s)))=2$ if $s$ is odd and $f(f(f(s)))=1$ if $s$ is even.

2017 ASDAN Math Tournament, 21

Tags: 2017 , Guts Round

In trapezoid $ABCD$, we have $\overline{AD}\parallel\overline{BC}$, $BC=3$, and $CD=4$. In addition, $\cos\angle ADC=\tfrac{1}{3}$ and $\angle ABC=2\angle ADC$. Compute $AC$.

2017 ASDAN Math Tournament, 9

Tags: 2017 , Discrete Math Test

Compute the number of positive integers $n\leq1330$ for which $\tbinom{2n}{n}$ is not divisible by $11$.

2017 CMIMC Individual Finals, 3

Tags: 2017 , Tiebreaker , number theory , algebra , polynomial

Say an integer polynomial is $\textit{primitive}$ if the greatest common divisor of its coefficients is $1$. For example, $2x^2+3x+6$ is primitive because $\gcd(2,3,6)=1$. Let $f(x)=a_2x^2+a_1x+a_0$ and $g(x) = b_2x^2+b_1x+b_0$, with $a_i,b_i\in\{1,2,3,4,5\}$ for $i=0,1,2$. If $N$ is the number of pairs of polynomials $(f(x),g(x))$ such that $h(x) = f(x)g(x)$ is primitive, find the last three digits of $N$.

2017 CMIMC Combinatorics, 2

Tags: 2017 , combinatorics

Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$.

2017 Macedonia JBMO TST, Source

Tags: Macedonia , JBMO TST , 2017 , number theory , geometry , inequalities , rectangle

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

2017 ASDAN Math Tournament, 1

Tags: 2017 , Calculus Test

Compute $$\int_0^13x^2dx.$$

2017 ASDAN Math Tournament, 3

Tags: 2017 , Algebra Tiebreaker

For some integers $b$ and $c$, neither of the equations below have real solutions: \begin{align*} 2x^2+bx+c&=0\\ 2x^2+cx+b&=0. \end{align*} What is the largest possible value of $b+c$?

2017 CMIMC Combinatorics, 4

Tags: 2017 , combinatorics

At a certain pizzeria, there are five different toppings available and a pizza can be ordered with any (possibly empty) subset of them on it. In how many ways can one order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither?

2017 CMIMC Team, 2

Tags: 2017 , team

Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute \[(x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right).\]

2017 CMI B.Sc. Entrance Exam, 4

Tags: CMI , Chennai Mathematical Institute , B.Sc , math , CS , 2017 , algebra

The domain of a function $f$ is $\mathbb{N}$ (The set of natural numbers). The function is defined as follows : $$f(n)=n+\lfloor\sqrt{n}\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$. Prove that, for every natural number $m$, the following sequence contains at least one perfect square $$m,~f(m),~f^2(m),~f^3(m),\cdots$$ The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times.

2017 CMIMC Combinatorics, 6

Tags: 2017 , combinatorics

Boris plays a game in which he rolls two standard four-sided dice independently and at random, and at the end of the game receives a number of dollars equal to the product of the two rolled numbers. After the initial roll of both dice, however, he can pay two dollars to reroll one die of choice, and he is allowed to pay to reroll as many times as he wishes. If Boris plays to maximize his expected gain, how much money, in dollars, can he expect to win by playing once?

2017 ASDAN Math Tournament, 4

Tags: 2017 , Calculus Test

A ladder $10\text{ m}$ long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $1\text{ m/s}$, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is $6\text{ m}$ from the wall?

2017 ASDAN Math Tournament, 1

Tags: 2017 , Guts Round

Alice and Bob are racing. Alice runs at a rate of $2\text{ m/s}$. Bob starts $10\text{ m}$ ahead of Alice and runs at a rate of $1.5\text{ m/s}$. How many seconds after the race starts will Alice pass Bob?

2017 ASDAN Math Tournament, 7

Tags: 2017 , Discrete Math Test

Alice and Bob play a game where on each turn, Alice rolls a die and Bob flips a coin. Bob wins the game if he flips $3$ heads before Alice rolls a $6$. What is the probability that Bob wins? Note that Bob does not in if he flips his third head the same turn Alice rolls her first $6$.

2017 Kazakhstan National Olympiad, 2

Tags: Kazakhstan , 2017 , inequalities

For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$

2017 CMIMC Combinatorics, 10

Tags: 2017 , combinatorics

Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds. [center][img]http://i.imgur.com/WNp59XW.png[/img][/center]

2017 ASDAN Math Tournament, 6

Tags: 2017 , Guts Round

You roll three six-sided dice. If the three dice and indistinguishable, how many combinations of numbers can result?

2017 CMIMC Geometry, 6

Tags: 2017 , geometry , cyclic quadrilateral

Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$, $\angle ADB=50^\circ$, and $BC=CD$. Suppose $AB$ intersects $CD$ at point $P$, while $AD$ intersects $BC$ at point $Q$. Compute $\angle APQ-\angle AQP$.

2017 CMIMC Geometry, 3

Tags: 2017 , geometry

In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$. Compute $EC - EB$.

2017 ASDAN Math Tournament, 3

Tags: 2017 , Guts Round

Four mathematicians, four physicists, and four programmers gather in a classroom. The $12$ people organize themselves into four teams, with each team having one mathematician, one physicist, and one programmer. How many possible arrangements of teams can exist?

2017 ASDAN Math Tournament, 17

Tags: 2017 , Guts Round

For $\triangle ABC$, $AB=BC=5$, and $AC=6$. Circle $O$ is inscribed in $\triangle ABC$, and circle $P$ is tangent to circle $O$, $AB$, and $AC$. Compute the area of $\triangle ABC$ not covered by circles $O$ and $P$.

2017 ASDAN Math Tournament, 10

Tags: 2017 , Geometry Test

Triangle $ABC$ is inscribed in circle $\gamma_1$ with radius $r_1$. Let $\gamma_2$ (with radius $r_2$) be the circle internally tangent to $\gamma_1$ at $A$ and tangent to $BC$ at $D$. Let $I$ be the incenter of $ABC$, and $P$ and $Q$ be the intersection of $\gamma_2$ with $AB$ and $AC$ respectively. Given that $P$, $I$, and $Q$ are collinear, $AI=25$, and the circumradius of triangle $BIC$ is $24$, compute the ratio of the radii $\tfrac{r_2}{r_1}$.

2017 CMIMC Algebra, 8

Tags: 2017 , algebra

Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15\qquad\text{and}\qquad \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.