Found problems: 85335
2009 Today's Calculation Of Integral, 449
Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$
ICMC 4, 4
Does there exist a set $\mathcal{R}$ of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of $\mathcal{R}$?
[i]Proposed by Tony Wang[/i]
2016 Kosovo National Mathematical Olympiad, 5
If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .
2008 ITest, 10
Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room three feet above the floor. Over the next few mornings, Tony moves the spider up three feet from the point where he finds it. If the wall in the living room is $18$ feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?
2025 Turkey EGMO TST, 2
Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying:
\[
\sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n
\]
for all positive integers $n$.
2018 PUMaC Combinatorics B, 8
Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of
$$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$
then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?
2002 Mexico National Olympiad, 1
The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that
$AB$ goes to $DA$
$DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$
then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?
1990 ITAMO, 3
Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.
II Soros Olympiad 1995 - 96 (Russia), 11.5
The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?
Russian TST 2015, P2
Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.
1994 Tournament Of Towns, (400) 2
$60$ children participate in a summer camp. Among any $10$ of the children there are three or more who live in the same block. Prove that there must be $15$ or more children from the same block.
(Folklore)
2009 Dutch IMO TST, 1
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.
2017 Rioplatense Mathematical Olympiad, Level 3, 4
Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?
1986 AIME Problems, 4
Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.
\[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]
2021 Indonesia TST, N
For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.
2009 Stanford Mathematics Tournament, 12
A number $N$ has 2009 positive factors. What is the maximum number of positive factors that $N^2$ could have?
2024 BMT, 4
Two circles, $\omega_1$ and $\omega_2$, are internally tangent at $A.$ Let $B$ be the point on $\omega_2$ opposite of $A.$ The radius of $\omega_1$ is $4$ times the radius of $\omega_2.$ Point $P$ is chosen on the circumference of $\omega_1$ such that the ratio $\tfrac{AP}{BP}=\tfrac{2\sqrt{3}}{\sqrt{7}}.$ Let $O$ denote the center of $\omega_2.$ Determine $\tfrac{OP}{AO}.$
1955 AMC 12/AHSME, 34
A $ 6$-inch and $ 18$-inch diameter pole are placed together and bound together with wire. The length of the shortest wire that will go around them is:
$ \textbf{(A)}\ 12\sqrt{3}\plus{}16\pi \qquad
\textbf{(B)}\ 12\sqrt{3}\plus{}7\pi \qquad
\textbf{(C)}\ 12\sqrt{3}\plus{}14\pi \\
\textbf{(D)}\ 12\plus{}15\pi \qquad
\textbf{(E)}\ 24\pi$
1966 Poland - Second Round, 4
Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.
2022 China Northern MO, 1
As shown in the figure, given $\vartriangle ABC$ with $AB \perp AC$, $AB=BC$, $D$ is the midpoint of the side $AB$, $DF\perp DE$, $DE=DF$ and $BE \perp EC$. Prove that $\angle AFD= \angle CEF$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/f16a8c8c463874f3ccb333d91cdef913c34189.png[/img]
2000 National Olympiad First Round, 20
For every real $x$, the polynomial $p(x)$ whose roots are all real satisfies $p(x^2-1)=p(x)p(-x)$. What can the degree of $p(x)$ be at most?
$ \textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{There is no upper bound for the degree of } p(x)
\qquad\textbf{(E)}\ \text{None}
$
2011 National Olympiad First Round, 2
How many of the coefficients of $(x+1)^{65}$ cannot be divisible by $65$?
$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{None}$
2022 Princeton University Math Competition, A8
The function $f$ sends sequences to sequences in the following way: given a sequence $\{a_n\}_{n=0}^{\infty}$ of real numbers, $f$ sends $\{a_n\}_{n=0}^{\infty}$ to the sequence $\{b_n\}_{n=0}^{\infty},$ where $b_n=\sum_{k=0}^n a_k \tbinom{n}{k}$ for all $n \ge 0.$ Let $\{F_n\}_{n=0}^{\infty}$ be the Fibonacci sequence, defined by $F_0=0, F_1=1,$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \ge 0.$ Let $\{c_n\}_{n=0}^{\infty}$ denote the sequence obtained by applying the function $f$ to the sequence $\{F_n\}_{n=0}^{\infty}$ $2022$ times. Find $c_5 \pmod{1000}.$
2014 Sharygin Geometry Olympiad, 5
In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$.
1994 AMC 8, 3
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at
$\text{(A)}\ \text{3:40 P.M.} \qquad \text{(B)}\ \text{3:55 P.M.} \qquad \text{(C)}\ \text{4:10 P.M.} \qquad \text{(D)}\ \text{4:25 P.M.} \qquad \text{(E)}\ \text{4:40 P.M.}$