Found problems: 85335
A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising?
The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to
$ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$
Let $ABC$ be a triagle inscribed in a circle $(O)$. A variable line through the orthocenter $H$ of the triangle meets the circle $(O)$ at two points $P , Q$. Two lines through $P, Q$ that are perpendicular to $AP , AQ$ respectively meet $BC$ at $M, N$ respectively. Prove that the line through $P$ perpendicular to $OM$ and the line through $Q$ perpendicular to $ON$ meet each other at a point on the circle $(O)$.
Nguyễn Văn Linh
Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.
Find the number of ordered pairs of integers $(x, y)$ with $0 \le x, y \le 40$ where $$\frac{x^2-xy^2+1}{41}$$ is an integer.
Let $n$ be a positive integer and $a_1\le a_2\le\ldots\le a_{n+1}$ and $b_1\le b_2\le\ldots\le b_n$ be real numbers such that for all $k\le n$, \[\binom nk\sum_{\substack{1\le i_1<i_2<\ldots<i_k\le n+1,\\i_1,i_2,\ldots,i_k\in\mathbb N}}a_{i_1}a_{i_2}\ldots a_{i_k} = \binom{n+1}k\sum_{\substack{1\le j_1<j_2<\ldots<j_k\le n,\\j_1,j_2,\ldots,j_k\in\mathbb N}}b_{j_1}b_{j_2}\ldots b_{j_k}.\] Show that \[a_1\le b_1\le a_2\le b_2\le \ldots \le a_n\le b_n\le a_{n+1}.\]
[i](Proposed by Ivan Chan Guan Yu)[/i]
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
Do there exist $2024$ non-zero reals $a_1, a_2, \ldots, a_{2024}$, such that $$\sum_{i=1}^{2024}(a_i^2+\frac{1}{a_i^2})+2\sum_{i=1}^{2024} \frac{a_i} {a_{i+1}}+2024=2\sum_{i=1}^{2024}(a_i+\frac{1}{a_i})?$$
Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.
Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.
The last two digits of number of $\left[\frac{10^{93}}{10^{31}+1}\right]$ is________.
The subsets $A_1,A_2,\ldots ,A_{2000}$ of a finite set $M$ satisfy $|A_i|>\frac{2}{3}|M|$ for each $i=1,2,\ldots ,2000$. Prove that there exists $m\in M$ which belongs to at least $1334$ of the subsets $A_i$.
Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*}
\log_2(xyz-3+\log_5 x) &= 5 \\
\log_3(xyz-3+\log_5 y) &= 4 \\
\log_4(xyz-3+\log_5 z) &= 4.
\end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.
For which values of $ k$ and $ d$ has the system $ x^3\plus{}y^3\equal{}2$ and $ y\equal{}kx\plus{}d$ no real solutions $ (x,y)$?
Show that, for any sequence $a_1,a_2,\ldots$ of real numbers, the two conditions
\[
\lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha
\]
and
\[
\lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha
\]
are equivalent.
The incircle of triangle $ABC$ touches $AC$ and $BC$ respectively $P$ and $Q$. Let $N$ and $M$ be the midpoints of the sides $AC$ and $BC$ respectively.$AM$ and $BP$,$BN$ and $AQ$ intersects at the points $X$ and $Y$ respectively. If the points $C,X$ and $Y$ are collinear , then prove that $CX$ is the angle bisector of $\angle ACB$.
A tower of height $h$ is a stack of contiguous rows of squares of height $h$ such that
[list]
[*] the bottom row of the tower has $h$ squares,
[*] each row above the bottom row has one fewer square than the row below it, and within each row the squares are contiguous,
[*] the squares in any given row all lie directly above a square in the row below.
[/list]
A tower is called balanced if when the squares of the tower are colored black and white in a checkerboard fashion, the number of black squares is equal to the number of white squares. For example, the figure above shows a tower of height 5 that is not balanced, since there are 7 white squares and 8 black squares.
How many balanced towers are there of height 2016?
[u]Round 1[/u]
[b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ?
[b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$?
[b]p3.[/b] Find the number of three-by-three grids such that
$\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and
$\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square.
[u]Round 2 [/u]
[b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$.
[b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$.
[b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$.
[u]Round 3 [/u]
[b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$?
[b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on?
[b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that
$\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and
$\bullet$ the only time she visits town $10$ is at the very end?
One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$.
[u]Round 4[/u]
[b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$?
[b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if
1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and
2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers.
Find the sum of all primed integers.
[b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$.
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.
In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.
Given an acute scalene triangle $ABC$ inscribed in circle $(O)$. Let $H$ be its orthocenter and $M$ be the midpoint of $BC$. Let $D$ lie on the opposite rays of $HA$ so that $BC=2DM$. Let $D'$ be the reflection of $D$ through line $BC$ and $X$ be the intersection of $AO$ and $MD$.
a) Show that $AM$ bisects $D'X$.
b) Similarly, we define the points $E,F$ like $D$ and $Y,Z$ like $X$. Let $S$ be the intersection of tangent lines from $B,C$ with respect to $(O)$. Let $G$ be the projection of the midpoint of $AS$ to the line $AO$. Show that there exists a point with the same power to all the circles $(BEY),(CFZ),(SGO)$ and $(O)$.
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
$A,B,C,D,E$, and $F$ are points of a convex hexagon, and there is a circle such that $A,B,C,D,E$, and $F$ are all on the circle. If $\angle ABC = 72^o$, $\angle BCD = 96^o$, $\angle CDE = 118^o$, and $\angle DEF = 104^o$, what is $\angle EFA$?
Let $M, N, P$ be arbitrary points on the sides $BC, CA, AB$ respectively of an acute-angled triangle $ABC$. Prove that at least one of the following inequalities is satisfied:
$NP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB$
Michelle is at the bottom-left corner of a $6\times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$; the first time Michelle moves to one of these points she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?