Found problems: 85335
2004 AMC 12/AHSME, 2
On the AMC 12, each correct answer is worth $ 6$ points, each incorrect answer is worth $ 0$ points, and each problem left unanswered is worth $ 2.5$ points. If Charlyn leaves $ 8$ of the $ 25$ problems unanswered, how many of the remaining problems must she answer correctly in order to score at least $ 100$?
$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$
PEN A Problems, 4
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
2012 Centers of Excellency of Suceava, 3
Let $ a,b,n $ be three natural numbers. Prove that there exists a natural number $ c $ satisfying:
$$ \left( \sqrt{a} +\sqrt{b} \right)^n =\sqrt{ c+(a-b)^n} +\sqrt{c} $$
[i]Dan Popescu[/i]
1998 Belarus Team Selection Test, 3
Let $1=d_1<d_2<d_3<...<d_k=n$ be all different divisors of positive integer $n$ written in ascending order. Determine all $n$ such that $$d_7^2+d_{10}^2=(n/d_{22})^2.$$
2010 Contests, 1
For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$.
(i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$.
(ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$.
(The number $|P|$ is the size of set $P$)
[i]Dan Schwarz, Romania[/i]
1930 Eotvos Mathematical Competition, 2
A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?
2003 Iran MO (3rd Round), 4
XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).
2021 Israel TST, 2
Find all functions $f:\mathbb{R}\to \mathbb{R}$ so that for any reals $x,y$ the following holds:
\[f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2\]
2008 German National Olympiad, 2
The triangle $ \triangle SFA$ has a right angle at $ F$. The points $ P$ and $ Q$ lie on the line $ SF$ such that the point $ P$ lies between $ S$ and $ F$ and the point $ F$ is the midpoint of the segment $ [PQ]$. The circle $ k_1$ is th incircle of the triangle $ \triangle SPA$. The circle $ k_2$ lies outside the triangle $ \triangle SQA$ and touches the segment $ [QA]$ and the lines $ SQ$ and $ SA$.
Prove that the sum of the radii of the circles $ k_1$ and $ k_2$ equals the length of $ [FA]$.
2011 Today's Calculation Of Integral, 748
Evaluate the following integrals.
(1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$
(2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$
2008 HMNT, 4
How many numbers between $1$ and $1,000,000$ are perfect squares but not perfect cubes?
1967 AMC 12/AHSME, 5
A triangle is circumscribed about a circle of radius $r$ inches. If the perimeter of the triangle is $P$ inches and the area is $K$ square inches, then $\frac{P}{K}$ is:
$ \text{(A)}\text{independent of the value of} \; r\qquad\text{(B)}\ \frac{\sqrt{2}}{r}\qquad\text{(C)}\ \frac{2}{\sqrt{r}}\qquad\text{(D)}\ \frac{2}{r}\qquad\text{(E)}\ \frac{r}{2} $
2001 Tournament Of Towns, 7
It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4?
2000 Stanford Mathematics Tournament, 23
What are the last two digits of ${7^{7^{7^7}}}$?
2002 IMO Shortlist, 6
Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.
1985 IMO Longlists, 20
Let $T$ be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points $(x, y, z)$ and $(u, v,w)$ are called [i]neighbors[/i] if $|x - u| + |y - v| + |z - w| = 1$. Show that there exists a subset $S$ of $T$ such that for each $p \in T$ , there is exactly one point of $S$ among $p$ and its [i]neighbors[/i].
1956 Polish MO Finals, 4
Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation
$$ a^2 + b^2 = c^2,$$
then:
1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $,
2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $,
3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.
1995 Argentina National Olympiad, 6
The $27$ points $(a,b,c)$ of the space are marked such that $a$, $b$ and $c$ take the values $0$, $1$ or $2$. We will call these points "junctures". Using $54$ rods of length $1$, all the joints that are at a distance of $1$ are joined together. A cubic structure of $2\times 2\times 2$ is thus formed. An ant starts from a juncture $A$ and moves along the rods; When it reaches a juncture it turns $90^\circ$ and changes rod. If the ant returns to $A$ and has not visited any juncture more than once except $A$, which it visited $2$ times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?
PEN G Problems, 11
Show that $\cos 1^{\circ}$ is irrational.
2008 Mathcenter Contest, 3
Set $ M= \{1,2,\cdots,2550\} $ and $\min A ,\ \max A $ represents the minimum and maximum values of the elements in the set $A$. For $ k \in \{1,2,\cdots 2006\} $define $$ x_k = \frac{1}{2008} \bigg (\sum_{A \subset M : n(A)= k} (\ min A + \max A) \, \bigg) $$. Find remainder from division $\sum_{i=1}^{2006} x_i^2$ with $2551$.
[i](passer-by)[/i]
1990 IMO Longlists, 10
Let $p, k$ and $x$ be positive integers such that $p \geq k$ and $x < \left[ \frac{p(p-k+1)}{2(k-1)} \right]$, where $[q]$ is the largest integer no larger than $q$. Prove that when $x$ balls are put into $p$ boxes arbitrarily, there exist $k$ boxes with the same number of balls.
2004 Purple Comet Problems, 5
Write the number $2004_{(5)}$ [ $2004$ base $5$ ] as a number in base $6$.
2009 Regional Olympiad of Mexico Center Zone, 5
Let $ABC$ be a triangle and let $D$ be the foot of the altitude from $A$. Let points $E$ and $F$ on a line through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, where $E$ and $F$ are points other than the point $D$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.
2018 PUMaC Live Round, 2.1
Compute the period (i.e. length of the repeating part) of the decimal expansion of $\tfrac{1}{729}$.
2019 Belarus Team Selection Test, 1.4
Let the sequence $(a_n)$ be constructed in the following way:
$$
a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots.
$$
Prove that $a_{180}>19$.
[i](Folklore)[/i]