Found problems: 85335
2017 CCA Math Bonanza, I10
Find the sum of the two smallest possible values of $x^\circ$ (in degrees) that satisfy the following equation if $x$ is greater than $2017^\circ$: $$\cos^59x+\cos^5x=32\cos^55x\cos^54x+5\cos^29x\cos^2x\left(\cos9x+\cos x\right).$$
[i]2017 CCA Math Bonanza Individual Round #10[/i]
LMT Speed Rounds, 2010.13
Let $ABC$ be a non-degenerate triangle inscribed in a circle, such that $AB$ is the diameter of the circle. Let the angle bisectors of the angles at $A$ and $B$ meet at $P.$ Determine the maximum possible value of $\angle APB,$ in degrees.
2021 HMNT, 3
Suppose $m$ and $n$ are positive integers for which
$\bullet$ the sum of the first $m$ multiples of $n$ is $120$, and
$\bullet$ the sum of the first $m^3$ multiples of$ n^3$ is $4032000$.
Determine the sum of the first $m^2$ multiples of $n^2$
2018 Turkey EGMO TST, 4
There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which has the heavier one becomes the lighter one. Determine the maximal possible value of $n$.
IV Soros Olympiad 1997 - 98 (Russia), 9.10
On the plane there is an image of a circle with a marked center. Let an arbitrary angle be drawn on this plane. Using one ruler, construct the bisector of this angle.
2014 Danube Mathematical Competition, 4
Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.
2021 CMIMC, 2.5 1.2
Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2\text{ mod }n)$th hole, where $k\text{ mod }n$ is the remainder when $k$ is divided by $n$. What is minimum $n$ such that there is at most one pigeon per hole?
[i]Proposed by Christina Yao[/i]
1995 Miklós Schweitzer, 2
Given $f,g\in L^1[0,1]$ and $\int_0^1 f = \int_0^1 g=1$, prove that there exists an interval I st $\int_I f = \int_I g=\frac12$.
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 :-
$1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0$.