Found problems: 85335
2016 PUMaC Algebra Individual A, A3
For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2y}$. The sum of the solutions to the equation
\[4096f(f(x, x), x) = x^{13}\]
can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.
2008 National Olympiad First Round, 16
A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$, at least $2$ of any $40$ gave exactly $2$, and at least $3$ of any $40$ gave exactly $1$ correct answers. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers?
$
\textbf{(A)}\ 18
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 26
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ \text{None of the above}
$
2020 Jozsef Wildt International Math Competition, W52
If $f\in C^{(3)}([0,1])$ such that $f(0)=f(1)=f'(0)=0$ and $|f'''(x)|\le1,(\forall)x\in[0,1]$, show that:
a)
$$|f(x)|\le\frac{x(1-x)}{\sqrt3}\cdot\left(\int^x_0\frac{f(t)}{t(1-t)}dt\right)^{1/2},(\forall)x\in[0,1]$$
b)
$$|f'(x)|\le\frac{1-2x}{\sqrt3}\cdot\left(\int^x_0\frac{|f(t)|}{t(1-t)}dt\right)^{1/2},(\forall)x\in\left[0,\frac12\right]$$
c)
$$\int^1_0(1-x)^2\cdot\frac{|f(x)|}xdx\ge9\int^1_0\left(\frac{f(x)}x\right)^2dx$$
[i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]
MBMT Team Rounds, 2020.7
Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
[i]Proposed by Gabriel Wu[/i]
2012 Tournament of Towns, 1
Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each positive integer $k$ there exists a positive integer $t = t(k)$ such that $a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each positive integer k?
2000 Saint Petersburg Mathematical Olympiad, 11.7
It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true:
$$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$
Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).
1979 Chisinau City MO, 182
Prove that a section of a cube by a plane cannot be a regular pentagon.
1989 French Mathematical Olympiad, Problem 5
Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Denote
$$s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}.$$
(a) Let $\lambda>1$ be a real number. Show that $s'<\lambda s+\frac\lambda{\lambda-1}$.
(b) Deduce that $\sqrt{s'}<\sqrt s+1$.
2012 India PRMO, 7
In $\vartriangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$?
1999 AIME Problems, 14
Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
1991 Austrian-Polish Competition, 4
Let $P(x)$ be a real polynomial with $P(x) \ge 0$ for $0 \le x \le 1$. Show that there exist polynomials $P_i (x) (i = 0, 1,2)$ with $P_i (x) \ge 0$ for all real x such that $P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)$.
1998 India Regional Mathematical Olympiad, 5
Find the minimum possible least common multiple of twenty natural numbers whose sum is $801$.
2018 Chile National Olympiad, 4
Find all postitive integers n such that
$$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$
where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.
1996 Romania National Olympiad, 3
Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.
2008 Hungary-Israel Binational, 3
A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary.
[i]Author: Kei Irie, Japan[/i]
2012 CHKMO, 2
Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen.
2011 ISI B.Math Entrance Exam, 3
For $n\in\mathbb{N}$ prove that
\[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]
2009 Bosnia Herzegovina Team Selection Test, 2
Find all pairs $\left(a,b\right)$ of posive integers such that $\frac{a^{2}\left(b-a\right)}{b+a}$ is square of prime.
1950 Moscow Mathematical Olympiad, 182
Prove that $\frac{1}{2} \frac{3}{4} \frac{5}{6} \frac{7}{8} ... \frac{99}{100 } <\frac{1}{10}$.
2022 AMC 12/AHSME, 17
Suppose $a$ is a real number such that the equation
$$a\cdot(\sin x+\sin(2x))=\sin(3x)$$
has more than one solution in the interval $(0,\pi)$. The set of all such $a$ can be written in the form $(p,q)\cup(q,r)$, where $p$, $q$, and $r$ are real numbers with $p<q<r$. What is $p+q+r$?
$\textbf{(A) }-4\qquad\textbf{(B) }-1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }4$
2016 Latvia National Olympiad, 4
Find the least prime factor of the number $\frac{2016^{2016}-3}{3}$.
2022 Silk Road, 2
Distinct positive integers $A$ and $B$ are given$.$ Prove that there exist infinitely many positive integers that can be represented both as $x_{1}^2+Ay_{1}^2$ for some positive coprime integers $x_{1}$ and $y_{1},$ and as $x_{2}^2+By_{2}^2$ for some positive coprime integers $x_{2}$ and $y_{2}.$
[i](Golovanov A.S.)[/i]
2021 AMC 10 Spring, 7
Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that
$\bullet$ all of his happy snakes can add
$\bullet$ none of his purple snakes can subtract, and
$\bullet$ all of his snakes that can’t subtract also can’t add
Which of these conclusions can be drawn about Tom’s snakes?
$\textbf{(A)}$ Purple snakes can add.
$\textbf{(B)}$ Purple snakes are happy.
$\textbf{(C)}$ Snakes that can add are purple.
$\textbf{(D)}$ Happy snakes are not purple.
$\textbf{(E)}$ Happy snakes can't subtract.
2005 Baltic Way, 15
Let the lines $e$ and $f$ be perpendicular and intersect each other at $H$. Let $A$ and $B$ lie on $e$ and $C$ and $D$ lie on $f$, such that all five points $A,B,C,D$ and $H$ are distinct. Let the lines $b$ and $d$ pass through $B$ and $D$ respectively, perpendicularly to $AC$; let the lines $a$ and $c$ pass through $A$ and $C$ respectively, perpendicularly to $BD$. Let $a$ and $b$ intersect at $X$ and $c$ and $d$ intersect at $Y$. Prove that $XY$ passes through $H$.
2019 ASDAN Math Tournament, 6
Consider a triangle $\vartriangle ACE$ with $\angle ACE = 45^o$ and $\angle CEA = 75^o$. Define points $Q, R$, and $P$ such that $AQ$, $CR$, and $EP$ are the altitudes of $\vartriangle ACE$. Let $H$ be the intersection of $AQ$, $CR$, and $EP$.
Next define points $B, D$, and $F$ as follows. Extend $EP$ to point $B$ such that $BP = HP$, extend $AQ$ to point $D$ such that $DQ = HQ$, and extend $CR$ to point $F$ such that $F R = HR$. Finally, lengths $CH = 2$, $AH =\sqrt2$, and $EH =\sqrt3 - 1$. Compute the area of hexagon $ABCDEF$.