Found problems: 85335
1999 Portugal MO, 2
How many positive integers are there such that $\frac{2n^2+4n+18}{3n+3}$ is an integer?
LMT Theme Rounds, 2023F 4C
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.
[i]Proposed by Christopher Cheng[/i]
[hide=Solution][i]Solution. [/i] $\boxed{6}$
Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]
2014 Cezar Ivănescu, 1
Let $ S $ be a nonempty subset of a finite group $ G, $ and $ \left( S^j \right)_{j\ge 1} $ be a sequence of sets defined as $ S^j=\left.\left\{\underbrace{xy\cdots z}_{\text{j terms}} \right| \underbrace{x,y,\cdots ,z}_{\text{j terms}} \in S \right\} . $ Prove that:
[b]a)[/b] $ \exists i_0\in\mathbb{N}^*\quad i\ge i_0\implies \left| S^i\right| =\left| S^{1+i}\right| $
[b]b)[/b] $ S^{|G|}\le G $
1996 May Olympiad, 5
In an electronic game of questions and answers, for each correct answer the player adds $5$ points on the screen, for each incorrect answer $2$ points are subtracted and when the player does not answer, no score is added or subtracted. Each game has $30$ questions. Francisco played $5$ games and in all of them he obtained the same number of points, greater than zero, but the number of correct answers, errors and unanswered questions in each game was different. Give all the possible scores that Francisco could obtain.
1953 Miklós Schweitzer, 9
[b]9.[/b] Let $w=f(x)$ be regular in $ \left | z \right |\leq 1$. For $0\leq r \leq 1$, denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$. Show that if the maximal length of the chords of $c_{1}$ is $1$, then for every $r$ such that $0\leq r \leq 1$, the maximal length of the chords of c, is not greater than $r$. [b](F. 1)[/b]
2023 CUBRMC, Individual
[b]p1.[/b] Find the largest $4$ digit integer that is divisible by $2$ and $5$, but not $3$.
[b]p2.[/b] The diagram below shows the eight vertices of a regular octagon of side length $2$. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure $270^o$. Compute the area of the shaded region.
[center][img]https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png[/img][/center]
[b]p3.[/b] Consider the numbers formed by writing full copies of $2023$ next to each other, like so: $$2023202320232023...$$
How many copies of $2023$ are next to each other in the smallest multiple of $11$ that can be written in this way?
[b]p4.[/b] A positive integer $n$ with base-$10$ representation $n = a_1a_2 ...a_k$ is called [i]powerful [/i] if the digits $a_i$ are nonzero for all $1 \le i \le k$ and
$$n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .$$
What is the unique four-digit positive integer that is [i]powerful[/i]?
[b]p5.[/b] Six $(6)$ chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation.
[b]p6.[/b] Given that the infinite sum $\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...$ is equal to $\frac{\pi^4}{90}$, compute the value of
$$\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}$$
[b]p7.[/b] Triangle $ABC$ is equilateral. There are $3$ distinct points, $X$, $Y$ , $Z$ inside $\vartriangle ABC$ that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio $1 : 1 : 2$ in some order. Find the ratio of the area of $\vartriangle ABC$ to that of $\vartriangle XY Z$.
[b]p8.[/b] For a fixed prime $p$, a finite non-empty set $S = \{s_1,..., s_k\}$ of integers is $p$-[i]admissible [/i] if there exists an integer $n$ for which the product $$(s_1 + n)(s_2 + n) ... (s_k + n)$$ is not divisible by $p$. For example, $\{4, 6, 8\}$ is $2$-[i]admissible[/i] since $(4+1)(6+1)(8+1) = 315$ is not divisible by $2$. Find the size of the largest subset of $\{1, 2,... , 360\}$ that is two-,three-, and five-[i]admissible[/i].
[b]p9.[/b] Kwu keeps score while repeatedly rolling a fair $6$-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a $1$, he stops rolling. For example, if the first roll is $1$, he gets a score of $1$, and if he rolls the sequence $(3, 4, 1)$, he gets a score of $3 + 3 \cdot 4 + 2 \cdot 1 = 17$. What is his expected score?
[b]p10.[/b] Let $\{a_1, a_2, a_3, ...\}$ be a geometric sequence with $a_1 = 4$ and $a_{2023} = \frac14$ . Let $f(x) = \frac{1}{7(1+x^2)}$. Find $$f(a_1) + f(a_2) + ... + f(a_{2023}).$$
[b]p11.[/b] Let $S$ be the set of quadratics $x^2 + ax + b$, with $a$ and $b$ real, that are factors of $x^{14} - 1$. Let $f(x)$ be the sum of the quadratics in $S$. Find $f(11)$.
[b]p12.[/b] Find the largest integer $0 < n < 100$ such that $n^2 + 2n$ divides $4(n- 1)! + n + 4$.
[b]p13.[/b] Let $\omega$ be a unit circle with center $O$ and radius $OQ$. Suppose $P$ is a point on the radius $OQ$ distinct from $Q$ such that there exists a unique chord of $\omega$ through $P$ whose midpoint when rotated $120^o$ counterclockwise about $Q$ lies on $\omega$. Find $OP$.
[b]p14.[/b] A sequence of real numbers $\{a_i\}$ satisfies
$$n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n$$
for each integer $n \ge 1$. Find the value of $a_{2023}$.
[b]p15.[/b] In $\vartriangle ABC$, let $\angle ABC = 90^o$ and let $I$ be its incenter. Let line $BI$ intersect $AC$ at point $D$, and let line $CI$ intersect $AB$ at point $E$. If $ID = IE = 1$, find $BI$.
[b]p16.[/b] For a positive integer $n$, let $S_n$ be the set of permutations of the first $n$ positive integers. If $p = (a_1, ..., a_n) \in S_n$, then define the bijective function $\sigma_p : \{1,..., n\} \to \{1, ..., n\}$ such that $\sigma_p (i) = a_i$ for all integers $1 \le i \le n$.
For any two permutations $p, q \in S_n$, we say $p$ and $q$ are friends if there exists a third permutation $r \in S_n$ such that for all integers $1 \le i \le n$, $$\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).$$
Find the number of friends, including itself, that the permutation $(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)$ has in $S_{10}$.
PS. You had better use hide for answers.
2004 France Team Selection Test, 2
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
1969 All Soviet Union Mathematical Olympiad, 120
Given natural $n$. Consider all the fractions $1/(pq)$, where $p$ and $q$ are relatively prime, $0<p<q\le n , p+q>n$. Prove that the sum of all such a fractions equals to $1/2$.
2024 Argentina Iberoamerican TST, 4
Find all natural numbers $n \geqslant 2$ with the property that there are two permutations $(a_1, a_2,\ldots, a_n) $ and $(b_1, b_2,\ldots, b_n)$ of the numbers $1, 2,\ldots, n$ such that $(a_1 + b_1, a_2 +b_2,\ldots, a_n + b_n)$ are consecutive natural numbers.
1970 IMO Longlists, 14
Let $\alpha + \beta +\gamma = \pi$. Prove that $\sum_{cyc}{\sin 2\alpha} = 2\cdot \left(\sum_{cyc}{\sin \alpha}\right)\cdot\left(\sum_{cyc}{\cos \alpha}\right)- 2\sum_{cyc}{\sin \alpha}$.
2017 BMT Spring, 9
Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b$ (mod $n$) for a fixed $n \in Z^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + ...$ Consider
$$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$
Then $P(n)$, $n \in Z^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.
2021 Regional Olympiad of Mexico Center Zone, 6
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions:
[list]
[*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$
[*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$
[/list]
Prove that $a_n=n$ for all positive integers $n$.
[i]Proposed by José Alejandro Reyes González[/i]
2023 Irish Math Olympiad, P3
Let $A, B, C, D, E$ be five points on a circle such that $|AB| = |CD|$ and $|BC| = |DE|$. The segments $AD$ and $BE$ intersect at $F$. Let $M$ denote the midpoint of segment $CD$. Prove that the circle of center $M$ and radius $ME$ passes through the midpoint of segment $AF$.
2022 Rioplatense Mathematical Olympiad, 4
Let $ABCD$ be a parallelogram and $M$ is the intersection of $AC$ and $BD$. The point $N$ is inside of the $\triangle AMB$ such that $\angle AND=\angle BNC$. Prove that $\angle MNC=\angle NDA$ and $\angle MND=\angle NCB$.
2006 Regional Competition For Advanced Students, 3
In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$.
Given are the points $ A$ and $ B$. Determine the set of points $ P\equal{}P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle.
2002 AMC 10, 6
Cindy was asked by her teacher to subtract $ 3$ from a certain number and then divide the result by $ 9$. Instead, she subtracted $ 9$ and then divided the result by $ 3$, giving an answer of $ 43$. What would her answer have been had she worked the problem correctly?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 51 \qquad \textbf{(E)}\ 138$
2020 Balkan MO Shortlist, A4
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.
[i]Brazitikos Silouanos, Greece[/i]
2010 Stanford Mathematics Tournament, 9
For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$.
Fi nd the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.
1989 Tournament Of Towns, (236) 4
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin)
2021 Durer Math Competition Finals, 3
Let $A$ and $B$ different points of a circle $k$ centered at $O$ in such a way such that $AB$ is not a diagonal of $k$. Furthermore, let $X$ be an arbitrary inner point of the segment $AB$. Let $k_1$ be the circle that passes through the points $A$ and $X$, and $A$ is the only common point of $k$ and $k_1$. Similarly, let $k_2$ be the circle that passes through the points $B$ and $X$, and $B$ is the only common point of $k$ and $k_2$. Let $M$ be the second intersection point of $k_1$ and $k_2$. Let $Q$ denote the center of circumscribed circle of the triangle $AOB$. Let $O_1$ and $O_2$ be the centers of $k_1$ and $k_2$. Show that the points $M,O,O_1,O_2,Q$ are on a circle.
2020 LMT Spring, 25
Let $\triangle ABC$ be a triangle such that $AB=5,AC=8,$ and $\angle BAC=60^{\circ}$. Let $\Gamma$ denote the circumcircle of $ABC$, and let $I$ and $O$ denote the incenter and circumcenter of $\triangle ABC$, respectively. Let $P$ be the intersection of ray $IO$ with $\Gamma$, and let $X$ be the intersection of ray $BI$ with $\Gamma$. If the area of quadrilateral $XICP$ can be expressed as $\frac{a\sqrt{b}+c\sqrt{d}}{e}$, where $a$ and $d$ are squarefree positive integers and $\gcd(a,c,e)=1$, compute $a+b+c+d+e$.
1978 IMO Longlists, 11
Find all natural numbers $n < 1978$ with the following property: If $m$ is a natural number, $1 < m < n$, and $(m, n) = 1$ (i.e., $m$ and $n$ are relatively prime), then $m$ is a prime number.
2011 All-Russian Olympiad, 2
Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.
2019 Durer Math Competition Finals, 8
A chess piece is placed on one of the squares of an $8\times 8$ chessboard where it begins a tour of the board: it moves from square to square, only moving horizontally or vertically. It visits every square precisely once, and ends up exactly where it started. What is the maximum number of times the piece can change direction along its tour?
2016 Purple Comet Problems, 1
Two integers have a sum of 2016 and a difference of 500. Find the larger of the two integers.