Found problems: 85335
If $b>1$, $\sin x>0$, $\cos x>0$, and $\log_b \sin x = a$, then $\log_b \cos x$ equals
$\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}$
Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality
\[f(xf(y))=f(xy)+x\]
Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table:
$\begin{array} {|c|c|c|} \cline{1-3}
10 & & \\ \cline{1-3}
& & 9 \\ \cline{1-3}
& 3 & \\ \cline{1-3}
11 & & 17 \\ \cline{1-3}
& 20 & \\ \cline{1-3}
\end{array}$
such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.
Suppose $a$ and $b$ are positive integers such that $a^b=2^{2023}.$ Compute the smallest possible value of $b^a.$
In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively. $O_1I_1$ intersects $O_2I_2$ at $P$. Find the locus of point $P$ as $D$ moves along $BC$.
A car and a motorcyclist left point $A$ in the direction of point $B$ at $10$ o'clock, and half an hour later a cyclist left point $B$ (in the direction of point A) and a pedestrian left (in the direction of point $A$) The car met the cyclist at $11$ o'clock hour and half an hour later overtook the pedestrian, and the motorcyclist overtook the pedestrian at $12:30$ p.m. At what time did the motorcyclist and the cyclist meet? (Speeds and directions of movement of ALL participants)
Consider digits $\underline{A}, \underline{B}, \underline{C}, \underline{D}$, with $\underline{A} \neq 0,$ such that $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$ Compute the sum of all distinct possible values of $\underline{A} + \underline{B} + \underline{C} + \underline{D}$.
[i]Proposed by Kyle Lee[/i]
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
Given a set of $2^{2016}$ cards with the numbers $1,2, ..., 2^{2016}$ written on them. We divide the set of cards into pairs arbitrarily, from each pair, we keep the card with larger number and discard the other. We now again divide the $2^{2015}$ remaining cards into pairs arbitrarily, from each pair, we keep the card with smaller number and discard the other. We now have $2^{2014}$ cards, and again divide these cards into pairs and keep the larger one in each pair. We keep doing this way, alternating between keeping the larger number and keeping the smaller number in each pair, until we have just one card left. Find all possible values of this final card.
To the nearest degree, find the measure of the largest angle in a triangle with side lengths $3$, $5$, and $7$.
$ABC$ is an isosceles triangle with $AB = AC$. Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$, $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$
Two square sheets have areas equal to $ 2003$. Each of the sheets is arbitrarily divided into $ 2003$ nonoverlapping polygons, besides, each of the polygons has an unitary area. Afterward, one overlays two sheets, and it is asked to prove that the obtained double layer can be punctured $ 2003$ times, so that each of the $ 4006$ polygons gets punctured precisely once.
Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.
Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.
Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$.
[i] M. Ivanov [/i]
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]
[/list]
Let $n \geq 2$ be a positive integer. Determine the number of $n$-tuples $(x_1, x_2, \ldots, x_n)$ such that $x_k \in \{0, 1, 2\}$ for $1 \leq k \leq n$ and $\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k$ is divisible by $3$.
On the board, the numbers from $1$ to $n$ are written. Achka (A) and Bavachka (B) play the following game. First, A erases one number, then B erases two consecutive numbers, then A erases three consecutive numbers, and finally B erases four consecutive numbers. What is the smallest $n$ such that B can definitely make her moves, no matter how A plays?
Let $ S$ be a set of all positive integers which can be represented as $ a^2 \plus{} 5b^2$ for some integers $ a,b$ such that $ a\bot b$. Let $ p$ be a prime number such that $ p \equal{} 4n \plus{} 3$ for some integer $ n$. Show that if for some positive integer $ k$ the number $ kp$ is in $ S$, then $ 2p$ is in $ S$ as well.
Here, the notation $ a\bot b$ means that the integers $ a$ and $ b$ are coprime.
In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.
Let $d \geq 1$ be an integer that is not the square of an integer. Prove that for every integer $n \geq 1,$
\[(n \sqrt d +1) \cdot | \sin(n \pi \sqrt d )| \geq 1\]
King Middle School has $ 1200$ students. Each student takes $ 5$ classes a day. Each teacher teaches $ 4$ classes. Each class has $ 30$ students and $ 1$ teacher. How many teachers are there at King Middle School?
\[ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 45 \qquad
\textbf{(E)}\ 50 \qquad
\]
Elax creates a partially filled $4 \times 4$ grid, and is trying to write in positive integers such that any four cells that share no rows and columns always sum to a number $S$. Given that the sum of the numbers in the top row is also $S$, what is the missing cell number?
[asy]
size(100);
add(grid(4,4));
label("$11$", (0.5,1.5));
label("$10$", (0.5,2.5));
label("?", (0.5,3.5));
label("$8$", (1.5,3.5));
label("$7$", (2.5,2.5));
label("$4$", (3.5,0.5));
label("$9$", (3.5,1.5));
[/asy]
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }12$