Found problems: 85335
2018 IFYM, Sozopol, 3
The number 1 is a solution of the equation
$(x + a)(x + b)(x + c)(x + d) = 16$,
where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.
2018 Ramnicean Hope, 1
Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation
$$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$
Calculate $ \int_{-2019}^{2019}f(x)dx . $
[i]Constantin Rusu[/i]
2002 Irish Math Olympiad, 2
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$.
Determine $ n$.
2014 PUMaC Geometry A, 5
There is a point $D$ on side $AC$ of acute triangle $\triangle ABC$. Let $AM$ be the median drawn from $A$ (so $M$ is on $BC$) and $CH$ be the altitude drawn from $C$ (so $H$ is on $AB$). Let $I$ be the intersection of $AM$ and $CH$, and let $K$ be the intersection of $AM$ and line segment $BD$. We know that $AK=8$, $BK=8$, and $MK=6$. Find the length of $AI$.
1949-56 Chisinau City MO, 49
Prove the identity: $$\cos \frac{\pi}{7} \cdot \cos \frac{4\pi}{7} \cdot \cos \frac{5\pi}{7} = \frac{1}{8}$$
2024 CCA Math Bonanza, I14
Larry initially has a one character string that is either `a', `b', `c', or `d'. Every minute, he chooses a character in the string and:
[list]
[*] if it's an `a' he can replace it with `ac' or `da',
[*] if it's a `b' he can replace it with `cb' or `bd',
[*] if it's a `c' he can replace it with `cc' or `ba',
[*] if it's a `d' he can replace it with `dd' or `ab'.
[/list]
Larry does the above process for $10$ minutes. Find the number of possible strings he can end up with that are a permutation of `aabbccccddd'.
[i]Individual #14[/i]
2025 CMIMC Geometry, 5
Let $\triangle{ABC}$ be an equilateral triangle. Let $E_{AB}$ be the ellipse with foci $A, B$ passing through $C,$ and in the parallel manner define $E_{BC}, E_{AC}.$ Let $\triangle{GHI}$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges of $\triangle{GHI}$ do not intersect those of $\triangle{ABC}.$ Compute the ratio of the largest sides of $\triangle{GHI}$ and $\triangle{ABC}.$
2010 Contests, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
2006 JBMO ShortLists, 10
Let $ ABCD$ be a trapezoid inscribed in a circle $ \mathcal{C}$ with $ AB\parallel CD$, $ AB\equal{}2CD$. Let $ \{Q\}\equal{}AD\cap BC$ and let $ P$ be the intersection of tangents to $ \mathcal{C}$ at $ B$ and $ D$. Calculate the area of the quadrilateral $ ABPQ$ in terms of the area of the triangle $ PDQ$.
2018 Turkey MO (2nd Round), 5
Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which
$$
({\rm gcd}(a_i,a_j))^2|a_k.
$$
2020 Argentina National Olympiad Level 2, 3
Let $ABCD$ be a parallelogram with $\angle ABC = 105^\circ$. Inside the parallelogram, there is a point $E$ such that triangle $BEC$ is equilateral and $\angle CED = 135^\circ$. Let $K$ be the midpoint of side $AB$. Determine the measure of angle $\angle BKC$.
2017 BMT Spring, 4
What is the greatest multiple of $9$ that can be formed by using each of the digits in the set $\{1, 3,5, 7, 9\}$ at most once.
2014 BMT Spring, 11
Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$. Find $a_5$.
2010 Princeton University Math Competition, 8
Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers $\{1, 2, 3\ldots , 10\}$ across his paper such that:
[list]
[*]The leftmost number is 1.
[*]The rightmost number is 10.
[*]Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list]
How many such permutations are there?
2019 PUMaC Individual Finals A, B, A2
Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$,
$p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$
2001 Brazil Team Selection Test, Problem 4
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP : PQ : QC = AC : CB : BA$.
Prove that the points $A$, $P$, $Q$ and $O$ lie on one circle.
[i]Alternative formulation.[/i] Let $O$ be the center of the circumcircle of a triangle $ABC$. If $P$ and $Q$ are points on the sides $AB$ and $AC$, respectively, satisfying $\frac{BP}{PQ}=\frac{CA}{BC}$ and $\frac{CQ}{PQ}=\frac{AB}{BC}$, then show that the points $A$, $P$, $Q$ and $O$ lie on one circle.
2013 Finnish National High School Mathematics Competition, 2
In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?
1986 IMO Shortlist, 5
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
LMT Guts Rounds, 2020 F10
$2020$ magicians are divided into groups of $2$ for the Lexington Magic Tournament. After every $5$ days, which is the duration of one match, teams are rearranged so no $2$ people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n?$
[i]Proposed by Ephram Chun[/i]
2014 Purple Comet Problems, 10
Given that $x$ and $y$ satisfy the two equations
\begin{align*}\frac1x+\frac1y&=4\\\\\frac2x+\frac3y&=7\end{align*}
evaluate $\dfrac{7-4y}x$.
2015 HMMT Geometry, 8
Let $S$ be the set of [b]discs[/b] $D$ contained completely in the set $\{ (x,y) : y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^2-\frac{3}{4}$. What is the area of the union of the elements of $S$?
2025 Korea Winter Program Practice Test, P5
In a convex quadrilateral $ABCD$, $\angle ABC = \angle CDA$. Let $X \neq C$ be the intersection of the circumcircle of $\triangle BCD$ and circle with diameter $AC$. Prove that the tangent to the circumcircle of $\triangle BCD$ at $X$, the tangent to the circumcircle of $\triangle ABD$ at $A$ concur on $BD$.
2012 National Olympiad First Round, 10
How many positive integers $n$ are there such that there are $20$ positive integers that are less than $n$ and relatively prime with $n$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{None}$
2013 USAMO, 2
For a positive integer $n\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move. Prove that $a_{n-1}+a_n=2^n$ for all $n\geq 4$.
1992 Dutch Mathematical Olympiad, 5
We consider regular $ n$-gons with a fixed circumference $ 4$. Let $ r_n$ and $ a_n$ respectively be the distances from the center of such an $ n$-gon to a vertex and to an edge.
$ (a)$ Determine $ a_4,r_4,a_8,r_8$.
$ (b)$ Give an appropriate interpretation for $ a_2$ and $ r_2$
$ (c)$ Prove that $ a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n)$ and $ r_{2n}\equal{}\sqrt{a_2n r_n}.$
$ (d)$ Define $ u_0\equal{}0, u_1\equal{}1$ and $ u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1})$ for $ n$ even or $ u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}}$ for $ n$ odd. Determine $ \displaystyle\lim_{n\to\infty}u_n$.