Found problems: 85335
2020/2021 Tournament of Towns, P2
Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.)
[i]Boris Frenkin[/i]
2020 IMO Shortlist, C2
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
1999 Romania National Olympiad, 1
Let $P(x) = 2x^3-3x^2+2$, and the sets:
$$A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}$$ Prove that the sets $A \cap B$ and $A\cap C$ have the same number of elements
1985 Putnam, A1
Determine, with proof, the number of ordered triples $\left(A_{1}, A_{2}, A_{3}\right)$ of sets which have the property that
(i) $A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},$ and
(ii) $A_{1} \cap A_{2} \cap A_{3}=\emptyset.$
Express your answer in the form $2^{a} 3^{b} 5^{c} 7^{d},$ where $a, b, c, d$ are nonnegative integers.
2018 AMC 10, 15
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[asy]
draw(circle((0,0),13));
draw(circle((5,-6.2),5));
draw(circle((-5,-6.2),5));
label("$B$", (9.5,-9.5), S);
label("$A$", (-9.5,-9.5), S);
[/asy]
$\textbf{(A) } 21 \qquad \textbf{(B) } 29 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 69 \qquad \textbf{(E) } 93 $
2024 LMT Fall, 12
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
2015 CentroAmerican, Problem 5
Let $ABC$ be a triangle such that $AC=2AB$. Let $D$ be the point of intersection of the angle bisector of the angle $CAB$ with $BC$. Let $F$ be the point of intersection of the line parallel to $AB$ passing through $C$ with the perpendicular line to $AD$ passing through $A$. Prove that $FD$ passes through the midpoint of $AC$.
2011 Baltic Way, 5
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
\[f(f(x))=x^2-x+1\]
for all real numbers $x$. Determine $f(0)$.
Kvant 2021, M2668
Two circles are given for which there is a family of quadrilaterals circumscribed around the first circle and inscribed in the second. Let's denote by $a, b, c$ and $d{}$ the consecutive lengths of the sides of one of these quadrilaterals. Prove that the sum \[\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}\]does not depend on the choice of the quadrilateral.
[i]Proposed by I. Weinstein[/i]
2014 HMNT, 7
Consider the set of $5$-tuples of positive integers at most $5$. We say the tuple ($a_1$, $a_2$, $a_3$, $a_4$, $a_5$) is [i]perfect[/i] if for any distinct indices $i$, $j$, $k$, the three numbers $a_i$, $a_j$ , $a_k$ do not form an arithmetic progression (in any order). Find the number of perfect $5$-tuples.
1992 IMTS, 4
An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm?
1975 All Soviet Union Mathematical Olympiad, 212
Prove that for all the positive numbers $a,b,c$ the following inequality is valid:
$$a^3+b^3+c^3+3abc>ab(a+b)+bc(b+c)+ac(a+c)$$
1992 AMC 12/AHSME, 28
Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is
$ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $
2014 Iran Team Selection Test, 6
$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively .
Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$.
Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$.
Prove that the circumcenter of $AIT$ is on the $BC$.
2019 Hanoi Open Mathematics Competitions, 5
Let $ABC$ be a triangle and $AD$ be the bisector of the triangle ($D \in (BC)$) Assume that $AB =14$ cm,
$AC = 35$ cm and $AD = 12$ cm; which of the following is the area of triangle $ABC$ in cm$^2$?
[b]A.[/b] $\frac{1176}{5}$ [b]B.[/b] $\frac{1167}{5}$ [b]C.[/b] $234$ [b]D.[/b] $\frac{1176}{7}$ [b]E.[/b] $236$
1999 Tournament Of Towns, 7
Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides.
(A Kanel)
2003 Paraguay Mathematical Olympiad, 3
Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in $33$ years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.
2002 India IMO Training Camp, 3
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?
2019 Azerbaijan Senior NMO, 5
Prove that for any $a;b;c\in\mathbb{R^+}$, we have $$(a+b)^2+(a+b+4c)^2\geq \frac{100abc}{a+b+c}$$ When does the equality hold?
1985 IMO Longlists, 52
In the triangle $ABC$, let $B_1$ be on $AC, E$ on $AB, G$ on $BC$, and let $EG$ be parallel to $AC$. Furthermore, let $EG$ be tangent to the inscribed circle of the triangle $ABB_1$ and intersect $BB_1$ at $F$. Let $r, r_1$, and $r_2$ be the inradii of the triangles $ABC, ABB_1$, and $BFG$, respectively. Prove that $r = r_1 + r_2.$
2012 Today's Calculation Of Integral, 842
Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$
1955 AMC 12/AHSME, 2
The smaller angle between the hands of a clock at $ 12: 25$ p.m. is:
$ \textbf{(A)}\ 132^\circ 30' \qquad
\textbf{(B)}\ 137^\circ 30' \qquad
\textbf{(C)}\ 150^\circ \qquad
\textbf{(D)}\ 137^\circ 32' \qquad
\textbf{(E)}\ 137^\circ$
2021 Malaysia IMONST 2, 5
There are $n$ guests at a gathering. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. Determine all possible values of $n$.
2019 BMT Spring, 1
Let $p$ be a polynomial with degree less than $4$ such that $p(x)$ attains a maximum at $x = 1$. If $p(1) = p(2) = 5$, find $p(10)$.
Geometry Mathley 2011-12, 8.1
Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$.
Kostas Vittas