Found problems: 85335
2016 CentroAmerican, 3
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.
2017 Regional Olympiad of Mexico Northeast, 4
Let $\Gamma$ be the circumcircle of the triangle $ABC$ and let $M$ be the midpoint of the arc $\Gamma$ containing $A$ and bounded by $B$ and $C$. Let $P$ and $Q$ be points on the segments $AB$ and $AC$, respectively, such that $BP = CQ$. Prove that $APQM$ is a cyclic quadrilateral.
2007 Today's Calculation Of Integral, 188
Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.
2010 Ukraine Team Selection Test, 6
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.
2006 Austrian-Polish Competition, 7
Find all nonnegative integers $m,n$ so that \[\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}\]
2012 Olympic Revenge, 5
Let $x_1,x_2,\ldots ,x_n$ positive real numbers. Prove that:
\[\sum_{cyc} \frac{1}{x_i^3+x_{i-1}x_ix_{i+1}} \le \sum_{cyc} \frac{1}{x_ix_{i+1}(x_i+x_{i+1})}\]
2011 NZMOC Camp Selection Problems, 4
Let a point $P$ inside a parallelogram $ABCD$ be given such that $\angle APB +\angle CPD = 180^o$. Prove that $AB \cdot AD = BP \cdot DP + AP \cdot CP$.
2015 BMT Spring, Tie 2
The unit square $ABCD$ has $E$ as midpoint of $AD$ and a circle of radius $r$ tangent to $AB$, $BC$, and $CE$. Determine $r$.
2014 BMT Spring, 3
Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $\$ 37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.
1985 AMC 8, 23
King Middle School has $ 1200$ students. Each student takes $ 5$ classes a day. Each teacher teaches $ 4$ classes. Each class has $ 30$ students and $ 1$ teacher. How many teachers are there at King Middle School?
\[ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 45 \qquad
\textbf{(E)}\ 50 \qquad
\]
LMT Team Rounds 2010-20, 2020.S12
In the figure above, the large triangle and all four shaded triangles are equilateral. If the areas of triangles $A, B,$ and $C$ are $1, 2,$ and $3,$ respectively, compute the smallest possible integer ratio between the area of the entire triangle to the area of triangle $D.$
[Insert Diagram]
[i]Proposed by Alex Li[/i]
2012 Stanford Mathematics Tournament, 3
Let $ABC$ be an equilateral triangle of side 1. Draw three circles $O_a$, $O_b$, $O_c$ with diameters $BC$, $CA$, and $AB$, respectively. Let $S_a$ denote the area of the region inside $O_a$ and outside of $O_b$ and $O_c$. Define $S_b$ and $S_c$ similarly, and let $S$ be the area of intersection between the three circles. Find $S_a+S_b+S_c-S$.
2005 Purple Comet Problems, 2
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?
2014 IMO Shortlist, C1
Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles.
[i]Proposed by Serbia[/i]
1999 Slovenia National Olympiad, Problem 3
Let $O$ be the circumcenter of a triangle $ABC$, $P$ be the midpoint of $AO$, and $Q$ be the midpoint of $BC$. If $\angle ABC=4\angle OPQ$ and $\angle ACB=6\angle OPQ$, compute $\angle OPQ$.
2023 MOAA, 16
Compute the sum $$\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}$$ where $\varphi(n)$ returns the number of positive integers less than $n$ that are relatively prime to $n$.
[i]Proposed by Andy Xu[/i]
2018 District Olympiad, 4
Let $ABC$ be a triangle with $\angle A = 80^o$ and $\angle C = 30^o$. Consider the point $M$ inside the triangle $ABC$ so that $\angle MAC= 60^o$ and $\angle MCA = 20^o$. If $N$ is the intersection of the lines $BM$ and $AC$ to show that a $MN$ is the bisector of the angle $\angle AMC$.
Kvant 2022, M2687
We have a regular $n{}$-gon, with $n\geqslant 4$. We consider the arrangements of $n{}$ numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement $K{}$, we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by $\alpha(K)$.
[list=a]
[*]Find the largest possible value of $\alpha(K)$.
[*]Find the number of arrangements for which $\alpha(K)$ takes this largest possible value.
[/list]
[i]Proposed by P. Kozhevnikov[/i]
2005 AIME Problems, 8
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.
2018 Polish MO Finals, 3
Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that
$$f(f(x)+f(y))+cxy=f(x+y).$$
Russian TST 2018, P1
The natural numbers $a > b$ are such that $a-b=5b^2-4a^2$. Prove that the number $8b + 1$ is composite.
2011 Rioplatense Mathematical Olympiad, Level 3, 4
We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ . Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$. Denote $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.
2021 DIME, 13
Let $\triangle ABC$ have side lengths $AB=7$, $BC=8$, and $CA=9$. Let $D$ be the projection from $A$ to $\overline{BC}$ and $D'$ be the reflection of $D$ over the perpendicular bisector of $\overline{BC}$. Let $P$ and $Q$ be distinct points on the line through $D'$ parallel to $\overline{AC}$ such that $\angle APB = \angle AQB = 90^{\circ}$. The value of $AP+AQ$ can be written as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
[i]Proposed by i3435[/i]
2020 DMO Stage 1, 5.
[b]Q.[/b] Let $ABC$ be a triangle, where $L_A, L_B, L_C$ denote the internal angle bisectors of $\angle BAC, \angle ABC, \angle ACB$ respectively and $\ell_A, \ell_B, \ell_C$, the altitudes from the corresponding vertices. Suppose $ L_A\cap \overline{BC} = \{A_1\}$, $\ell_A \cap \overline{BC} = \{A_2\}$ and the circumcircle of $\triangle AA_1A_2$ meets $AB$ and $AC$ at $S$ and $T$ respectively. If $\overline{ST} \cap \overline{BC} = \{A'\}$, prove that $A',B',C'$ are collinear, where $B'$ and $C'$ are defined in a similar manner.
[i]Proposed by Functional_equation[/i]
2014 Purple Comet Problems, 23
Suppose $x$ is a real number satisfying $x^2-990x+1=(x+1)\sqrt x$. Find $\sqrt x+\tfrac1{\sqrt x}$.