Found problems: 229
Evaluate
\[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\]
[i]Proposed by Nathan Xiong[/i]
Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find
\[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]
Given a triangle, $AB=78, BC=50, AC=112,$ construct squares $ABXY, BCPQ, ACMN$ outside the triangle. Let $L_1, L_2, L_3$ be the midpoints of $\overline{MP}, \overline{QX}, \overline{NY},$ respectively. Find the area of $L_1L_2L_3.$
Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.
David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?
Compute the value of
$$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$
Your answer should not involve any trigonometric functions.
[i]Proposed by Howard Halim[/i]
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
[asy]
import olympiad;
size(120);
defaultpen(linewidth(0.8));
pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A;
pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y);
draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4"));
draw(rightanglemark(A,B,C,3));
draw(A--B--C--cycle);
draw((0,2)--(0,0)--(2,0),Arrows(size=8));
label("$6$",(PAx+PBx)/2,S);
label("$8$",(PAy+PCy)/2,W);
[/asy]
Consider the set of all permutations, $\mathcal{P}$, of $\{1,2,\ldots,2022\}$. For permutation $P\in \mathcal{P}$, let $P_1$ denote the first element in $P$. Let $\text{sgn}(P)$ denote the sign of the permutation. Compute the following number modulo 1000: $$\displaystyle\sum_{P\in\mathcal{P}}\dfrac{P_1\cdot\text{sgn}(P)^{P_1}}{2020!}.$$
(The [i]sign[/i] of a permutation $P$ is $(-1)^k$, where $k$ is the minimum number of two-element swaps needed to reach that permutation).
[i]Proposed by Nairit Sarkar[/i]
$1296$ CMU Students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the `score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?
[i]Proposed by Dilhan Salgado[/i]
Equilateral triangle $T_0$ with side length $3$ is on a plane. Given triangle $T_n$ on the plane, triangle $T_{n+1}$ is constructed on the plane by translating $T_n$ by $1$ unit, in one of six directions parallel to one of the sides of $T_n$. The direction is chosen uniformly at random.
Let $a$ be the least integer such that at most one point on the plane is in or on all of $T_0, T_1, T_2, \ldots, T_a$. It can be shown that $a$ exists with probability $1$. Find the probability that $a$ is even.
[i]Proposed by Justin Hseih[/i]
Determine, with proof, all possible values of $\gcd(a^2+b^2+c^2,abc)$ across all triples of positive integers $(a,b,c).$
Compute the number of ordered triples $(x,y,z)$ of integers satisfying
\[x^2+y^2+z^2=9.\]
[i]Proposed by Nathan Xiong[/i]
Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this?
[i]Proposed by Nathan Xiong[/i]
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
Suppose $a_1, a_2, a_3, \dots,$ is a sequence of real numbers such that $$a_n = \frac{a_{n-1}a_{n-2}}{3a_{n-2}-2a_{n-1}}$$ for all $n \ge 3$. If $a_1 = 1$ and $a_{10} = 10$, what is $a_{19}$?
[i]Proposed by Howard Halim[/i]
Let $N$ be the number of triples of positive integers $(a,b,c)$ with $a\leq b\leq c\leq 100$ such that the polynomial \[P(x)=x^2+(a^2+4b^2+c^2+1)x+(4ab+4bc-2ca)\] has integer roots in $x$. Find the last three digits of $N$.
Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\]
Let $T = TNYWR$. $T$ people each put a distinct marble into a bag; its contents are mixed randomly and one marble is distributed back to each person. Given that at least one person got their own marble back, what is the probability that everyone else also received their own marble?
Let $a$ and $b$ be positive integers such that $10< \gcd(a,b) < 20$ and $220 < \text{lcm}(a,b) < 230$. Find the difference between the smallest and largest possible values of $ab$.
The integers from $1$ to $9$ are arranged in a $3\times3$ grid. The rows and columns of the grid correspond to $6$ three-digit numbers, reading rows from left to right, and columns from top to bottom. Compute the least possible value of the largest of the $6$ numbers.
Let $\triangle ABC$ be equilateral. Let $D$ be the midpoint of side $AC,$ and let $DEFG$ be a square such that $D, F, B$ are collinear and $E,G$ lie on $AB,CB$ respectively. What fraction of the area of $\triangle ABC$ is covered by square $DEFG?$
[i]Proposed by Lohith Tummala[/i]
Suppose Pat and Rick are playing a game in which they take turns writing numbers from $\{1, 2, \dots, 97\}$ on a blackboard. In each round, Pat writes a number, then Rick writes a number; Rick wins if the sum of all the numbers written on the blackboard after $n$ rounds is divisible by 100. Find the minimum positive value of $n$ for which Rick has a winning strategy.
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
For how many permutations $\pi$ of $\{1,2,\ldots,9\}$ does there exist an integer $N$ such that \[N\equiv \pi(i)\pmod{i}\text{ for all integers }1\leq i\leq 9?\]
Let $\triangle ABC$ be a triangle with side lengths $a$, $b$, and $c$. Circle $\omega_A$ is the $A$-excircle of $\triangle ABC$, defined as the circle tangent to $BC$ and to the extensions of $AB$ and $AC$ past $B$ and $C$ respectively. Let $\mathcal{T}_A$ denote the triangle whose vertices are these three tangency points; denote $\mathcal{T}_B$ and $\mathcal{T}_C$ similarly. Suppose the areas of $\mathcal{T}_A$, $\mathcal{T}_B$, and $\mathcal{T}_C$ are $4$, $5$, and $6$ respectively. Find the ratio $a:b:c$.
For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots?