Found problems: 85335
2021 JHMT HS, 4
For a natural number $n,$ let $a_n$ be the sum of all products $xy$ over all integers $x$ and $y$ with $1 \leq x < y \leq n.$ For example, $a_3 = 1\cdot2 + 2\cdot3 + 1\cdot3 = 11.$ Determine the smallest $n \in \mathbb{N}$ such that $n > 1$ and $a_n$ is a multiple of $2020.$
2015 CCA Math Bonanza, I9
There is $1$ integer in between $300$ and $400$ (base $10$) inclusive such that its last digit is $7$ when written in bases $8$, $10$, and $12$. Find this integer, in base $10$.
[i]2015 CCA Math Bonanza Individual Round #9[/i]
2024 Singapore MO Open, Q5
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$.
[i]Proposed by oneplusone[/i]
1985 AMC 12/AHSME, 7
In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, $ a \times b \minus{} c$ in such languages means the same as $ a(b\minus{}c)$ in ordinary algebraic notation. If $ a \div b \minus{} c \plus{} d$ is evaluated in such a language, the result in ordinary algebraic notation would be
$ \textbf{(A)}\ \frac{a}{b} \minus{} c \plus{} d \qquad \textbf{(B)}\ \frac{a}{b} \minus{} c \minus{} d \qquad \textbf{(C)}\ \frac{d \plus{} c \minus{} b}{a} \qquad \textbf{(D)}\ \frac{a}{b \minus{} c \plus{} d} \qquad \textbf{(E)}\ \frac{a}{b\minus{}c\minus{}d}$
2016 Harvard-MIT Mathematics Tournament, 6
Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M$, $N$, $P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E$, $F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U$, $V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $\widehat{BAC}$ of $\Gamma$.
Given that $AB = 5$, $AC = 8$, and $\angle A = 60^{\circ}$, compute the area of triangle $XUV$.
1971 IMO Longlists, 40
Consider the set of grid points $(m,n)$ in the plane, $m,n$ integers. Let $\sigma$ be a finite subset and define
\[S(\sigma)=\sum_{(m,n)\in\sigma}(100-|m|-|n|) \]
Find the maximum of $S$, taken over the set of all such subsets $\sigma$.
2016 ASDAN Math Tournament, 10
Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.
1989 IMO Longlists, 9
Do there exist two sequences of real numbers $ \{a_i\}, \{b_i\},$ $ i \in \mathbb{N},$ satisfying the following conditions:
\[ \frac{3 \cdot \pi}{2} \leq a_i \leq b_i\]
and
\[ \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i}\]
$ \forall i \in \mathbb{N}$ and all $ x,$ with $ 0 < x < 1?$
2007 China Northern MO, 2
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.
a) Solve the equation $ f(x) = 0$.
b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]
2010 Iran Team Selection Test, 2
Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$
\[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]
2005 AMC 10, 2
For each pair of real numbers $ a\not\equal{} b$, define the operation $ \star$ as \[(a \star b) \equal{} \frac{a \plus{} b}{a \minus{} b}.\] What is the value of $ ((1 \star 2) \star 3)$?
$ \textbf{(A)}\ \minus{}\frac{2}{3}\qquad
\textbf{(B)}\ \minus{}\frac{1}{5}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac{1}{2}\qquad
\textbf{(E)}\ \text{This value is not defined.}$
2016 VJIMC, 4
Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying
$$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$
for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e.
$$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$
2001 All-Russian Olympiad, 3
The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set $D$ of towns [i]dominant[/i] if every town not in $D$ is connected by a road to a town in $D$. Suppose that each dominant set consists of at least $k$ towns. Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road.
2020 Jozsef Wildt International Math Competition, W49
Let $a,b,c>0$ so that $a+b+c=1$. Then prove that
$$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$
[i]Proposed by Marius Drăgan[/i]
2001 Estonia National Olympiad, 3
A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.
2012 Kosovo National Mathematical Olympiad, 1
Find the two last digits of $2012^{2012}$.
2016 NIMO Problems, 3
Let $f$ be the quadratic function with leading coefficient $1$ whose graph is tangent to that of the lines $y=-5x+6$ and $y=x-1$. The sum of the coefficients of $f$ is $\tfrac pq$, where $p$ and $q$ are positive relatively prime integers. Find $100p + q$.
[i]Proposed by David Altizio[/i]
1981 Canada National Olympiad, 3
Given a finite collection of lines in a plane $P$, show that it is possible to draw an arbitrarily large circle in $P$ which does not meet any of them. On the other hand, show that it is possible to arrange a countable infinite sequence of lines (first line, second line, third line, etc.) in $P$ so that every circle in $P$ meets at least one of the lines. (A point is not considered to be a circle.)
2005 Bulgaria National Olympiad, 1
Determine all triples $\left( x,y,z\right)$ of positive integers for which the number \[ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} \] is an integer .
2013 Iran Team Selection Test, 5
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
2019 Teodor Topan, 4
Let $ S $ be a finite [url=https://en.wikipedia.org/wiki/Cancellation_property]cancellative semigroup.[/url]
[b]a)[/b] Prove that $ S $ contains an idempotent element.
[b]b)[/b] Prove that $ S $ is a group.
[b]c)[/b] Disprove subpoint [b]b)[/b] in the case that $ S $ would not be finite.
[i]Vlad Mihaly[/i]
2021 Princeton University Math Competition, 5
Given a real number $t$ with $0 < t < 1$, define the real-valued function $f(t, \theta) = \sum^{\infty}_{n=-\infty} t^{|n|}\omega^n$, where $\omega = e^{i \theta} = \cos \theta + i\sin \theta$. For $\theta \in [0, 2\pi)$, the polar curve $r(\theta) = f(t, \theta)$ traces out an ellipse $E_t$ with a horizontal major axis whose left focus is at the origin. Let $A(t)$ be the area of the ellipse $E_t$. Let $A\left( \frac12 \right) = \frac{a\pi}{b}$ , where $a, b$ are relatively prime positive integers. Find $100a +b$ .
2013 IMC, 5
Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime?
[i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]
2021 Oral Moscow Geometry Olympiad, 1
Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$.
[img]https://cdn.artofproblemsolving.com/attachments/6/1/eeb98ccdee801361f9f66b8f6b2da4714e659f.png[/img]
2010 Serbia National Math Olympiad, 1
Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$.
[i]Proposed by Dusan Djukic[/i]