Found problems: 85335
2015 Purple Comet Problems, 15
On the inside of a square with side length 60, construct four congruent isosceles triangles each with base
60 and height 50, and each having one side coinciding with a different side of the square. Find the area of
the octagonal region common to the interiors of all four triangles.
1972 AMC 12/AHSME, 15
A contractor estimated that one of his two bricklayers would take $9$ hours to build a certain wall and the other $10$ hours. However, he knew from experience that when they worked together, their combined output fell by $10$ bricks per hour. Being in a hurry, he put both men on the job and found that it took exactly 5 hours to build the wall. The number of bricks in the wall was
$\textbf{(A) }500\qquad\textbf{(B) }550\qquad\textbf{(C) }900\qquad\textbf{(D) }950\qquad \textbf{(E) }960$
2016 VJIMC, 3
For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix
$$\begin{bmatrix}
1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\
0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\
1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\
0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\
0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1
\end{bmatrix}$$
2020 Taiwan TST Round 2, 1
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
2020 BMT Fall, 25
Submit an integer between $1$ and $50$, inclusive. You will receive a score as follows:
If some number is submitted exactly once: If $E$ is your number, $A$ is the closest number to $E$ which received exactly one submission, and $M$ is the largest unique submission, you will receive $\frac{25}{M} (A - |E - A|)$ points, rounded to the nearest integer.
If no number was submitted exactly once: If $E$ is your number, $K$ is the number of people who submitted $E$, and $M$ is the number of people who submitted the most commonly submitted number, then you will receive $\frac{25(M-K)}{M}$ points, rounded to the nearest integer.
2025 Euler Olympiad, Round 2, 2
Points $A$, $B$, $C$, and $D$ lie on a line in that order, and points $E$ and $F$ are located outside the line such that $EA=EB$, $FC=FD$ and $EF \parallel AD$. Let the circumcircles of triangles $ABF$ and $CDE$ intersect at points $P$ and $Q$, and the circumcircles of triangles $ACF$ and $BDE$ intersect at points $M$ and $N$. Prove that the lines $PQ$ and $MN$ pass through the midpoint of segment $EF$.
[i]
Proposed by Giorgi Arabidze, Georgia[/i]
2020 MMATHS, I8
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_ib_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?
[i]Proposed by Andrew Wu[/i]
2012 Dutch IMO TST, 1
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
2015 HMNT, 3
Let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$. For example, $\lfloor \pi \rfloor=3$, and $\{\pi\}=0.14159\dots$, while $\lfloor 100 \rfloor=100$ and $\{100\}=0$. If $n$ is the largest solution to the equation $\frac{\lfloor n \rfloor}{n}=\frac{2015}{2016}$, compute $\{n\}$.
2011 F = Ma, 8
When a block of wood with a weight of $\text{30 N}$ is completely submerged under water the buoyant force on the block of wood from the water is $\text{50 N}$. When the block is released it floats at the surface. What fraction of the block will then be visible above the surface of the water when the block is floating?
(A) $1/15$
(B) $1/5$
(C) $1/3$
(D) $2/5$
(E) $3/5$
2023 Turkey Olympic Revenge, 4
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$ is a perfect square.
[i]Proposed by Barış Koyuncu[/i]
2010 ELMO Shortlist, 2
For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$,
\[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\]
[i]Alex Zhu.[/i]
2018 Saint Petersburg Mathematical Olympiad, 6
$\alpha,\beta$ are positive irrational numbers and $[\alpha[\beta x]]=[\beta[\alpha x]]$ for every positive $x$. Prove that $\alpha=\beta$
2010 India National Olympiad, 2
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.
2021 Romania Team Selection Test, 3
Let $\mathcal{P}$ be a convex quadrilateral. Consider a point $X$ inside $\mathcal{P}.$ Let $M,N,P,Q$ be the projections of $X$ on the sides of $\mathcal{P}.$ We know that $M,N,P,Q$ all sit on a circle of center $L.$ Let $J$ and $K$ be the midpoints of the diagonals of $\mathcal{P}.$ Prove that $J,K$ and $L$ lie on a line.
2006 ISI B.Math Entrance Exam, 3
Find all roots of the equation :-
$1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0$.
1982 IMO Shortlist, 12
Four distinct circles $C,C_1, C_2$, C3 and a line L are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_1, C_2, C_3$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be $1$, determine the distance between its center and $L.$
2023 Chile Junior Math Olympiad, 1
Determine the number of three-digit numbers with the following property:
The number formed by the first two digits is prime and the number formed by the last two digits is prime.
1964 Poland - Second Round, 4
Find the real numbers $ x, y, z $ satisfying the system of equations
$$(z - x)(x - y) = a $$
$$(x - y)(y - z) = b$$
$$(y - z)(z - x) = c$$
where $ a, b, c $ are given real numbers.
III Soros Olympiad 1996 - 97 (Russia), 10.2
On a side of the triangle, take four points $K$, $P$, $H$ and $M$, which are respectively the midpoint of this side, the foot of the bisector with the opposite angle of the triangle, the touchpoint of this side of the circle inscribed in the triangle and the foot of the corresponding altitude. Find $KH$ if $KP = a$, $KM =b$.
I Soros Olympiad 1994-95 (Rus + Ukr), 10.1
The equation $x^2 + bx + c = 0$ has two different roots $x_1$ and $x_2$. It is also known that the numbers $b$, $x_1$, $c$, $x_2$ in the indicated order form an arithmetic progression. Find the difference of this progression.
2000 Croatia National Olympiad, Problem 1
Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.
2003 South africa National Olympiad, 6
In $\Delta ABC$, the sum of the sides is $2s$ and the radius of the incircle is $r$. Three semicircles with diameters $AB$, $BC$ and $CA$ are drawn on the outside of $ABC$. A circle with radius $t$ touches all three semicircles. Prove that \[ \frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r. \]
2009 Today's Calculation Of Integral, 416
Answer the following questions.
(1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$.
(2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$.
Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$.
(3) Find $ f_n(x)$ for all $ n$.
(4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.