Found problems: 85335
2007 ITest, 22
Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }223&\textbf{(U) }678\\\\
\textbf{(V) }2007 & &\end{array}$
2022 BMT, 16
A street on Stanford can be modeled by a number line. Four Stanford students, located at positions $1$, $9$, $25$ and $49$ along the line, want to take an UberXL to Berkeley, but are not sure where to meet the driver. Find the smallest possible total distance walked by the students to a single position on the street. (For example, if they were to meet at position $46$, then the total distance walked by the students would be $45 + 37 + 21 + 3 = 106$, where the distances walked by the students at positions $1$, $9$, $25$ and $49$ are summed in that order.)
2014-2015 SDML (High School), 8
What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
$\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$
2022 AMC 12/AHSME, 16
Suppose $x$ and $y$ are positive real numbers such that
$x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$.
What is the greatest possible value of $\log_2{y}$?
$\textbf{(A)}3~\textbf{(B)}4~\textbf{(C)}3+\sqrt{2}~\textbf{(D)}4+\sqrt{3}~\textbf{(E)}7$
1985 Brazil National Olympiad, 4
$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.
2000 AIME Problems, 9
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$
2019 PUMaC Algebra B, 4
Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.
2006 Taiwan National Olympiad, 2
Find all reals $x$ satisfying $0 \le x \le 5$ and
$\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.
2020 Iran Team Selection Test, 5
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:
$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$
[i]Proposed by Mohammad Amin Sharifi[/i]
2017 HMIC, 3
Let $v_1, v_2, \ldots, v_m$ be vectors in $\mathbb{R}^n$, such that each has a strictly positive first coordinate. Consider the following process. Start with the zero vector $w = (0, 0, \ldots, 0) \in \mathbb{R}^n$. Every round, choose an $i$ such that $1 \le i \le m$ and $w \cdot v_i \le 0$, and then replace $w$ with $w + v_i$.
Show that there exists a constant $C$ such that regardless of your choice of $i$ at each step, the process is guaranteed to terminate in (at most) $C$ rounds. The constant $C$ may depend on the vectors $v_1, \ldots, v_m$.
1996 Vietnam National Olympiad, 1
Find all $ f: \mathbb{N}\to\mathbb{N}$ so that :
$ f(n) \plus{} f(n \plus{} 1) \equal{} f(n \plus{} 2)f(n \plus{} 3) \minus{} 1996$
2023/2024 Tournament of Towns, 2
2. A unit square paper has a triangle-shaped hole (vertices of the hole are not on the border of the paper). Prove that a triangle with area of $1 / 6$ can be cut from the remaining paper.
Alexandr Yuran
2017 AMC 10, 14
An integer $N$ is selected at random in the range $1\le N \le 2020.$ What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$
2013 AIME Problems, 2
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5$,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5$.
1990 Romania Team Selection Test, 10
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
2023 LMT Spring, 8
Ephramis taking his final exams. He has $7$ exams and his school holds finals over $3$ days. For a certain arrangement of finals, let $f$ be the maximum number of finals Ephram takes on any given day. Find the expected value of $f$ .
1992 Tournament Of Towns, (325) 2
Consider a right triangle $ABC$, where $A$ is the right angle, and $AC > AB$. Points $E$ on $AC$ and $D$ on $BC$ are chosen so that$ AB = AE = BD$. Prove that the triangle $ADE$ is right if and only if the ratio $AB : AC : BC$ of sides of the triangle $ABC$ is $3 : 4 : 5$.
(A. Parovan)
1960 Poland - Second Round, 4
Prove that if $ n $ is a non-negative integer, then number $$
2^{n+2} + 3^{2n+1}$$
is divisible by $7$.
2016 NIMO Problems, 5
The equation $x^3 - 3x^2 - 7x - 1 = 0$ has three distinct real roots $a$, $b$, and $c$. If \[\left( \dfrac{1}{\sqrt[3]{a}-\sqrt[3]{b}} + \dfrac{1}{\sqrt[3]{b}-\sqrt[3]{c}} + \dfrac{1}{\sqrt[3]{c}-\sqrt[3]{a}} \right)^2 = \dfrac{p\sqrt[3]{q}}{r}\] where $p$, $q$, $r$ are positive integers such that $\gcd(p, r) = 1$ and $q$ is not divisible by the cube of a prime, find $100p + 10q + r$.
[i]Proposed by Michael Tang and David Altizio[/i]
2015 Tuymaada Olympiad, 1
On the football training there was $n$ footballers - forwards and goalkeepers. They made $k$ goals. Prove that main trainer can give for every footballer squad number from $1$ to $n$ such, that for every goal the difference between squad number of forward and squad number of goalkeeper is more than $n-k$.
[i](S. Berlov)[/i]
1991 All Soviet Union Mathematical Olympiad, 544
Does there exist a triangle in which two sides are integer multiples of the median to that side?
Does there exist a triangle in which every side is an integer multiple of the median to that side?
2023 Brazil Cono Sur TST, 1
A $2022 \times 2022$ squareboard was divided into $L$ and $Z$ tetrominoes. Each tetromino consists of four squares, which can be rotated or flipped. Determine the least number of $Z$-tetrominoes necessary to cover the $2022 \times 2022$ squareboard.
2001 Swedish Mathematical Competition, 5
Find all polynomials $p(x)$ such that $p'(x)^2 = c p(x) p''(x)$ for some constant $c$.
2011 Switzerland - Final Round, 2
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]
Brazil L2 Finals (OBM) - geometry, 2022.3
Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.