Found problems: 85335
ABMC Team Rounds, 2020
[u]Round 5[/u]
[b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$.
[b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. )
[b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$?
[u]Round 6[/u]
[b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$?
[b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally)
[b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$.
[u]Round 7[/u]
[b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit?
[b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$.
[b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$.
[u]Round 8[/u]
[b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of
people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of
times the letter c appears in the Speed Round. Estimate
$$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2015 ASDAN Math Tournament, 1
Four unit circles are placed on a square of side length $2$, with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles.
2023 Sharygin Geometry Olympiad, 8.5
The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.
LMT Accuracy Rounds, 2021 F9
There exist some number of ordered triples of real numbers $(x,y,z)$ that satisfy the following system of equations:
\begin{align*}
x+y+2z &= 6\\
x^2+y^2+2z^2 &= 18\\
x^3+y^3+2z^3&=54
\end{align*}
Given that the sum of all possible positive values of $x$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$,$b$,$c$, and $d$ are positive integers, $c$ is squarefree, and $\gcd(a,b,d)=1$, find the value of $a+b+c+d$.
2007 Indonesia TST, 2
Let $ ABCD$ be a convex quadrtilateral such that $ AB$ is not parallel with $ CD$. Let $ \Gamma_1$ be a circle that passes through $ A$ and $ B$ and is tangent to $ CD$ at $ P$. Also, let $ \Gamma_2$ be a circle that passes through $ C$ and $ D$ and is tangent to $ AB$ at $ Q$. Let the circles $ \Gamma_1$ and $ \Gamma_2$ intersect at $ E$ and $ F$. Prove that $ EF$ passes through the midpoint of $ PQ$ iff $ BC \parallel AD$.
2018 IFYM, Sozopol, 3
Let $p$ be some prime number.
a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$.
b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.
1995 Austrian-Polish Competition, 7
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal:
(i) $|x|$ is a square of an integer;
(ii) $y$ is a squarefree number.
1976 IMO Longlists, 11
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
2013 Princeton University Math Competition, 8
Eight all different sushis are placed evenly on the edge of a round table, whose surface can rotate around the center. Eight people also evenly sit around the table, each with one sushi in front. Each person has one favorite sushi among these eight, and they are all distinct. They find that no matter how they rotate the table, there are never more than three people who have their favorite sushis in front of them simultaneously. By this requirement, how many different possible arrangements of the eight sushis are there? Two arrangements that differ by a rotation are considered the same.
STEMS 2021-22 Math Cat A-B, A2 B4 C1
If there are integers $a,b,c$ such that $a^2+b^2+c^2-ab-bc-ca$ is divisible by a prime $p$ such that $\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$, then prove that there are integers $x,y,z$ such that $p=x^2+y^2+z^2-xy-yz-zx$.
2019 Jozsef Wildt International Math Competition, W. 21
Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.
1993 Vietnam National Olympiad, 3
Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.
2005 Putnam, B3
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
JOM 2015 Shortlist, A4
Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $, $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.
2014 Online Math Open Problems, 12
The points $A$, $B$, $C$, $D$, $E$ lie on a line $\ell$ in this order. Suppose $T$ is a point not on $\ell$ such that $\angle BTC = \angle DTE$, and $\overline{AT}$ is tangent to the circumcircle of triangle $BTE$. If $AB = 2$, $BC = 36$, and $CD = 15$, compute $DE$.
[i]Proposed by Yang Liu[/i]
2024 Iran Team Selection Test, 5
Suppose that we have two natural numbers $x , y \le 100!$ with undetermined values. Prove that there exist natural numbers $m , n$ such that values of $x , y$ get uniquely determined according to value of $\varphi(d(my))+d(\varphi(nx))$. ( for each natural number $n$ , $d(n)$ is number of its positive divisors and $\varphi(n)$ is the number of the numbers less that $n$ which are relatively prime to $n$. )
[i]Proposed by Mehran Talaei[/i]
2013 Saudi Arabia IMO TST, 4
Determine if there exists an infinite sequence of positive integers $a_1,a_2, a_3, ...$ such that
(i) each positive integer occurs exactly once in the sequence, and
(ii) each positive integer occurs exactly once in the sequence $ |a_1 - a_2|, |a_2 - a_3|, ..., |a+k - a_{k+1}|, ...$
1949-56 Chisinau City MO, 23
Inside the angle $ABC$ of $60^o$, point $O$ is selected, which is located at distances from the sides of the angle $a$ and $b$, respectively. Determine the distance from the top of the angle to this point.
KoMaL A Problems 2019/2020, A.756
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$
[i]Based on a problem of Romanian Masters of Mathematics[/i]
2021 AMC 12/AHSME Spring, 9
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
2003 Belarusian National Olympiad, 5
Let $m,n,k$ be positive integers, $m> n> k$. An $1 \times m$ strip of paper is divided into the $1 \times 1$ cells. A teacher asks Bill and Pit to place numbers $0$ and $1$ in the cells of the strip so that the sum of the numbers in any $n$ consecutive cells is equal to $k$. After the task was performed it turned out that the sum $S(B)$ of all numbers on the strip of Bill was different from the sum $S(P)$ of Pit.
Find the largest possible value of $|S(B) - S(P) |$.
(I. Voronovich)
2021 Bangladesh Mathematical Olympiad, Problem 9
A positive integer $n$ is called nice if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is nice because its largest proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of nice integers not greater than $3000$.
1968 German National Olympiad, 4
Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.
1973 Poland - Second Round, 1
Prove that if positive numbers $ x, y, z $ satisfy the inequality
$$
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$
then they are the lengths of the sides of a certain triangle.
2016 Saudi Arabia GMO TST, 1
Let $S = x + y +z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities:
$$xyz > 1$$
$$x + y + z >\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
Prove that $S$ can take on any real values when $x, y, z$ vary