Found problems: 229
Let $f(x)=x^4-4x^2+2.$ Find the smallest natural $n \in \mathbb{N}$ such that there exists $k,c \in \mathbb{N}$ with $$\left|f^k\left(\frac{n^2+1}{n}\right)-c^{144}\right| < \frac{1}{100}.$$
A positive integer $N$ is a [i]triple-double[/i] if there exists non-negative integers $a$, $b$, $c$ such that $2^a + 2^b + 2^c = N$. How many three-digit numbers are triple-doubles?
[i]Proposed by Giacomo Rizzo[/i]
Find the sum of all positive integers $n\le 2024$ such that all pairs of distinct positive integers $(a,b)$ that satisfy $ab=n$ have a sum that is a perfect square.
Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$.
[i]Proposed by Nathan Xiong[/i]
There are distinct quadratics $e(x)$, $p(x)$, $h(x)$, $r(x)$, $a(x)$, and $m(x)$ with leading coefficient $1$, such that their roots are $2$ distinct values from the set $\{3, 4, 5, 6\}$. James takes three of these quadratics, sums two, and subtracts the last. Given that this new quadratic has a root at $0$, find its other root.
Square $ABCD$ has side length $2$. For each $0 \leq r \leq 2$, point $P_r$ is on side $\overline{AB}$ with $AP_r = r$, and square $\Sigma_r$ is constructed with diagonal $\overline{DP_r}$. Let region $\mathcal{R}$ be the set of all points that are in both $\Sigma_0$ and $\Sigma_2$, but not in $\Sigma_r$ for at least one value of $r$. Find the area of the convex hull of $\mathcal{R}$.
[i]Proposed by Justin Hsieh[/i]
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
Let $U$ be the set of all complex numbers $m$ such that the $4$ roots of $(x^2+2x+5)(x^2-2mx+25)=0$ are concyclic in the complex plane. One can show that when the points of $U$ are plotted on the complex plane, it is visualized as the finite union of some curves. Find the sum of the lengths of those curves (i.e. the perimeter of $U$).
Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook [i]attacks[/i] another rook if the two rooks are in the same row or column and there are no other rooks between them.
Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks $4$ other rooks.
Determine the maximum possible value of $$\sqrt{x}(2\sqrt{x}+\sqrt{1-x})(3\sqrt{x}+4\sqrt{1-x})$$ over all $x\in [0,1]$.
Let a tree on $mn + 1$ vertices be $(m,n)$-nice if the following conditions hold:
[list]
[*] $m + 1$ colors are assigned to the nodes of the tree
[*] for the first $m$ colors, there will be exactly $n$ nodes of color $i$ $(1\le i \le m)$
[*] the root node of the tree will be the unique node of color $m+1$. \item the $(m,n)$-nice trees must also satisfy the condition that for any two non-root nodes $i, j$, if the color of $i$ equals the color of $j$, then the color of the parent of $i$ equals the color of the parent of $j$.
[*] Nodes of the same color are considered indistinguishable (swapping any two of them results in the same tree).
[/list]
Let $N(u,v,l)$ denote the number of $(u,v)$-nice trees with $l$ leaves. Note that $N(2,2,2) = 2, N(2,2,3) = 4, N(2,2,4) = 6$. Compute the remainder when $\sum_{l = 123}^{789} N(8,101,l)$ is divided by $101$.
Definition: Any rooted, ordered tree consists of some set of nodes, each of which has a (possibly empty) ordered list of children. Each node is the child of exactly one other node, with the exception of the root, which has not parent. There also cannot be any cycles of nodes which are all linearly children of each other.
[i]Proposed by Advait Nene[/i]
Evaluate
\[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\]
[i]Proposed by Nathan Xiong[/i]
Amy has a six-sided die which always rolls values greater than or equal to the previous roll. She rolls the die repeatedly until she rolls a $6$. Find the expected value of the sum of all distinct values she has rolled when she finishes.
The value of
\[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
Currently, Selena’s analog clock says $4{:}00$. Suddenly her clock breaks, so the hour hand moves $12$ times as fast as it normally does, but the minute hand stays the same speed. Find the degree measure of the smaller angle formed by the minute and the hour hand $2024$ minutes from now.
Find the sum of all two-digit prime numbers whose digits are also both prime numbers.
[i]Proposed by Nathan Xiong[/i]
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ onto sides $AC$ and $AB$, respectively. Lines $BD$ and $CE$ intersect $\omega$ again at points $P \neq B$ and $Q \neq C$. Suppose that $PD=3$, $QE=2$, and $AP \parallel BC$. Compute $DE$.
[i]Proposed by Kyle Lee[/i]
Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown:
[asy]
size(5cm);
real n = 5;
for (int i = 0; i < n; ++i) {
draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));
}
for (int i = 0; i < n; ++i) {
draw((n-i,0)--((n-i)/2,(n-i)*0.866));
}
for (int i = 0; i < n; ++i) {
draw((i,0)--((n+i)/2,(n-i)*0.866));
}
[/asy]
Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Andrew Wen[/i]
For two real numbers $x$ and $y$, let $x\circ y=\frac{xy}{x+y}$. The value of
\[1 \circ (2 \circ (3 \circ (4 \circ 5)))\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
Define an integer $n \ge 0$ to be \textit{two-far} if there exist integers $a$ and $b$ such that $a$, $b$, and $n + a + b$ are all powers of two. If $N$ is the number of two-far integers less than 2048, find the remainder when $N$ is divided by 100.
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
George is taking a ten-question true-false exam, where the answer key has been selected uniformly at random; however, he doesn't know any of the answers! Luckily, a friend has helpfully hinted that no two consecutive questions have true as the correct answer. If George takes the exam and maximizes the expected number of questions he gets correct, how many of his answers are expected to be right?
Find the number of ordered triples of positive integers $(a,b,c),$ where $1 \leq a,b,c \leq 10,$ with the property that $\gcd(a,b), \gcd(a,c),$ and $\gcd(b,c)$ are all pairwise relatively prime.
[i]Proposed by Kyle Lee[/i]
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$, and let $T = 10X + Y$ for an integer $X$ and a digit $Y$. Suppose that $a$ and $b$ are real numbers satisfying $a+\frac1b=Y$ and $\frac{b}a=X$. Compute $(ab)^4+\frac1{(ab)^4}$.
Points $A$, $B$, and $C$ lie in the plane such that $AB=13$, $BC=14$, and $CA=15$. A peculiar laser is fired from $A$ perpendicular to $\overline{BC}$. After bouncing off $BC$, it travels in a direction perpendicular to $CA$. When it hits $CA$, it travels in a direction perpendicular to $AB$, and after hitting $AB$ its new direction is perpendicular to $BC$ again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape $T_\infty$. What is the ratio of the perimeter of $T_\infty$ to the perimeter of $\triangle ABC$?