Found problems: 85335
LMT Team Rounds 2021+, 3
Beter Pai wants to tell you his fastest $40$-line clear time in Tetris, but since he does not want Qep to realize she is
better at Tetris than he is, he does not tell you the time directly. Instead, he gives you the following requirements,
given that the correct time is t seconds:
$\bullet$ $t < 100$.
$\bullet$ $t$ is prime.
$\bullet$ $t -1$ has 5 proper factors.
$\bullet$ all prime factors of $t +1$ are single digits.
$\bullet$ $t -2$ is a multiple of $3$.
$\bullet$ $t +2$ has $2$ factors.
Find t.
2008 Paraguay Mathematical Olympiad, 1
How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?
1955 AMC 12/AHSME, 13
The fraction $ \frac{a^{\minus{}4}\minus{}b^{\minus{}4}}{a^{\minus{}2}\minus{}b^{\minus{}2}}$ is equal to:
$ \textbf{(A)}\ a^{\minus{}6}\minus{}b^{\minus{}6} \qquad
\textbf{(B)}\ a^{\minus{}2}\minus{}b^{\minus{}2} \qquad
\textbf{(C)}\ a^{\minus{}2}\plus{}b^{\minus{}2} \\
\textbf{(D)}\ a^2\plus{}b^2 \qquad
\textbf{(E)}\ a^2\minus{}b^2$
1991 Tournament Of Towns, (282) 2
Each of three given circles with radii $1$, $r$ and $r$ touches the others from the outside. For what values of $r$ does there exist a triangle “circumscribed” to these circles? (This means the circles lie inside the triangle, each circle touching two sides of the triangle and each side of the triangle touching two circles.)
(N.B. Vasiliev, Moscow)
1946 Putnam, B2
Let $A, B$ be two variable points on a parabola $P_{0}$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A,B$ and the vertex of $P_0$ is a parabola $P_1 .$ Apply the same process to $P_1$ and repeat the process, obtaining the sequence of parabolas $P_1, P_2 , \ldots, P_n$. If the equation of $P_0$ is $y=m x^2$, find the equation of $P_n .$
1998 AMC 8, 7
$ 100\times 19.98\times 1.998\times 1000\equal{} $
$ \text{(A)}\ (1.998)^{2}\qquad\text{(B)}\ (19.98)^{2}\qquad\text{(C)}\ (199.8)^{2}\qquad\text{(D)}\ (1998)^{2}\qquad\text{(E)}\ (19980)^{2} $
1975 Canada National Olympiad, 8
Let $ k$ be a positive integer. Find all polynomials
\[ P(x) \equal{} a_0 \plus{} a_1 x \plus{} \cdots \plus{} a_n x^n,\]
where the $ a_i$ are real, which satisfy the equation
\[ P(P(x)) \equal{} \{ P(x) \}^k\]
2013 IMO Shortlist, G4
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.
1996 Taiwan National Olympiad, 4
Show that for any real numbers $a_{3},a_{4},...,a_{85}$, not all the roots of the equation $a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0$ are real.
2015 Brazil National Olympiad, 4
Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors.
a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$ iff $n$ is prime or $n=4$.
b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$
1998 Korea - Final Round, 1
Let $ x,y,z$ be positive real numbers satisfying $ x\plus{}y\plus{}z\equal{}xyz$. Prove that:
\[\frac1{\sqrt{1+x^2}}+\frac1{\sqrt{1+y^2}}+\frac1{\sqrt{1+z^2}}\leq\frac{3}{2}\]
2023 AMC 10, 16
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
2000 Moldova National Olympiad, Problem 3
Consider the sets $A_1=\{1\}$, $A_2=\{2,3,4\}$, $A_3=\{5,6,7,8,9\}$, etc. Let $b_n$ be the arithmetic mean of the smallest and the greatest element in $A_n$. Show that the number $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$ is a prime integer.
2010 BAMO, 3
All vertices of a polygon $P$ lie at points with integer coordinates in the plane, and all sides of $P$ have integer lengths. Prove that the perimeter of $P$ must be an even number.
2013 National Chemistry Olympiad, 60
Which vitamin is the most soluble in water?
${ \textbf{(A)}\ \text{A} \qquad\textbf{(B)}\ \text{K} \qquad\textbf{(C)}\ \text{C} \qquad\textbf{(D)}}\ \text{D} \qquad $
1997 Romania Team Selection Test, 2
Let $P$ be the set of points in the plane and $D$ the set of lines in the plane. Determine whether there exists a bijective function $f: P \rightarrow D$ such that for any three collinear points $A$, $B$, $C$, the lines $f(A)$, $f(B)$, $f(C)$ are either parallel or concurrent.
[i]Gefry Barad[/i]
1949 Putnam, A4
Given that $P$ is a point inside a tetrahedron with vertices at $A, B, C$ and $D$, such that the sum of the distances $PA+PB+PC+PD$ is a minimum, show that the two angles $\angle APB$ and $\angle CPD$ are equal and are bisected by the same straight line. What other pair of angles must be equal?
2015 HMNT, 1
Triangle $ABC$ is isosceles, and $\angle ABC=x^{\circ}$. If the sum of the possible measures of $\angle BAC$ is $240^{\circ}$, find $x$.
2010 IMC, 2
Let $a_0,a_1,\dots,a_n$ be positive real numbers such that $a_{k+1}-a_k \geq 1$ for all $k=0,1,\dots,n-1.$ Prove that
\[1+\frac{1}{a_0} \left( 1+\frac1{a_1-a_0}\right)\cdots\left(1+\frac1{a_n-a_0}\right)\leq \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right)\cdots \left(1+\frac1{a_n}\right).\]
2018 Moldova EGMO TST, 8
Let $ABC$ be a triangle with $AB=c$ , $BC=a$ and $AC=b$. If $ x,y\in\mathbb{R}$ satisfy $ \frac{1}{x} +\frac{1}{y+z} = \frac{1}{a} $ , $ \frac{1}{y} +\frac{1}{x+z} = \frac{1}{b} $ , $ \frac{1}{z} +\frac{1}{y+x} = \frac{1}{c} $ . Prove that the following equality holds $ x(p-a) + y(p-b) + z(p-c) = 3r^2 + 12R*r , $ Where $p$ is semi-perimeter, $R$ is the circumradius and $r$ is the inradius.
2014 Contests, 2
$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.
2007 Thailand Mathematical Olympiad, 16
What is the smallest positive integer with $24$ positive divisors?
2020 Brazil Team Selection Test, 1
Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form:
[asy]
size(1.5cm);
draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0));
[/asy]
The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible?
[i]Proposed by Daniel Kohen[/i]
2007 Tournament Of Towns, 3
$D$ is the midpoint of the side $BC$ of triangle $ABC$. $E$ and $F$ are points on $CA$ and $AB$ respectively, such that $BE$ is perpendicular to $CA$ and $CF$ is perpendicular to $AB$. If $DEF$ is an equilateral triangle, does it follow that $ABC$ is also equilateral?
2005 Moldova National Olympiad, 10.7
Determine all strictly increasing functions $ f: R\rightarrow R$ satisfying relationship $ f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005$
for any real values of x and y.