Found problems: 85335
1995 Austrian-Polish Competition, 3
Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .
2015 Junior Regional Olympiad - FBH, 5
It is given $2015$ numbers such that every one of them when gets replaced with sum of the rest, we get same $2015$ same numbers. Prove that product of all numbers is $0$
2022-2023 OMMC, 25
A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken?
$\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$
2007 Princeton University Math Competition, 7
How many ordered pairs of integers $(x, y)$ satisfy
\[8(x^3+x^2y+xy^2+y^3) = 15(x^2+y^2+xy+1)?\]
2019 AMC 12/AHSME, 8
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?
$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$
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?
\[ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 45 \qquad
\textbf{(E)}\ 50 \qquad
\]
2011 Saudi Arabia BMO TST, 4
Let $ABC$ be a triangle with circumcenter $O$. Points $P$ and $Q$ are interior to sides $CA$ and $AB$, respectively. Circle $\omega$ passes through the midpoints of segments $BP$, $CQ$, $PQ$. Prove that if line $PQ$ is tangent to circle $\omega$, then $OP = OQ$.
2023 ISI Entrance UGB, 5
There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t_n$ denote the number of ways this can be done. For example, clearly $t_1 = 2$ because we can have either a red or a blue tile. Also $t_2 = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
[list=a]
[*]Prove that $t_{2n+1} = t_n(t_{n-1} + t_{n+1})$ for all $n > 1$.
[*]Prove that $t_n = \sum_{d \ge 0} \binom{n-d}{d}2^{n-2d}$ for all $n >0$.
[/list]
Here,
\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]
for integers $m,r$.
Cono Sur Shortlist - geometry, 1993.1
Let $C_1$ and $C_2$ be two concentric circles and $C_3$ an outer circle to $C_1$ inner to $C_2$ and tangent to both. If the radius of $C_2$ is equal to $ 1$, how much must the radius of $C_1$ be worth, so that the area of is twice that of $C_3$?
2005 JHMT, Team Round
[b]p1.[/b] Consider the following function $f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}$.
Evaluate the infinite sum $f(1) + f(2) + f(3) + f(4) +...$
[b]p2.[/b] Find the area of the shape bounded by the following relations
$$y \le |x| -2$$
$$y \ge |x| - 4$$
$$y \le 0$$
where |x| denotes the absolute value of $x$.
[b]p3.[/b] An equilateral triangle with side length $6$ is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle?
[b]p4.[/b] Given $\sin x - \tan x = \sin x \tan x$, solve for $x$ in the interval $(0, 2\pi)$, exclusive.
[b]p5.[/b] How many rectangles are there in a $6$ by $6$ square grid?
[b]p6.[/b] Find the lateral surface area of a cone with radius $3$ and height $4$.
[b]p7.[/b] From $9$ positive integer scores on a $10$-point quiz, the mean is $ 8$, the median is $ 8$, and the mode is $7$. Determine the maximum number of perfect scores possible on this test.
[b]p8.[/b] If $i =\sqrt{-1}$, evaluate the following continued fraction:
$$2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}$$
[b]p9.[/b] The cubic polynomial $x^3-px^2+px-6$ has roots $p, q$, and $r$. What is $(1-p)(1-q)(1-r)$?
[b]p10.[/b] (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where $10\%$ of the merchants are thieves. The police utilize a lie detector that is $90\%$ accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 CCA Math Bonanza, I8
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$, what are all possible values of $a$?
[i]2019 CCA Math Bonanza Individual Round #8[/i]
2007 Bulgarian Autumn Math Competition, Problem 8.4
Let $ABCDEFG$ be a regular heptagon. We'll call the sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$ and $GA$ opposite to the vertices $E$, $F$, $G$, $A$, $B$, $C$ and $D$ respectively. If $M$ is a point inside the heptagon, we'll say that the line through $M$ and a vertex of the heptagon intersects a side of it (without the vertices) at a $\textit{perfect}$ point, if this side is opposite the vertex. Prove that for every choice of $M$, the number of $\textit{perfect}$ points is always odd.
2006 All-Russian Olympiad, 1
Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.
2004 Silk Road, 1
Find all $ f: \mathbb{R} \to \mathbb{R}$, such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$.
1954 AMC 12/AHSME, 26
The straight line $ \overline{AB}$ is divided at $ C$ so that $ AC\equal{}3CB$. Circles are described on $ \overline{AC}$ and $ \overline{CB}$ as diameters and a common tangent meets $ AB$ produced at $ D$. Then $ BD$ equals:
$ \textbf{(A)}\ \text{diameter of the smaller circle} \\
\textbf{(B)}\ \text{radius of the smaller circle} \\
\textbf{(C)}\ \text{radius of the larger circle} \\
\textbf{(D)}\ CB\sqrt{3}\\
\textbf{(E)}\ \text{the difference of the two radii}$
2017 Azerbaijan Senior National Olympiad, A1
Solve the system of equation for $(x,y) \in \mathbb{R}$
$$\left\{\begin{matrix}
\sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\
3x^2+4xy=24
\end{matrix}\right.$$ \\
Explain your answer
2004 Estonia National Olympiad, 4
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{1}{1+2ab}+\frac{1}{1+2bc}+\frac{1}{1+2ca}\ge 1$$
2019 Federal Competition For Advanced Students, P2, 4
Let $a, b, c$ be the positive real numbers such that $a+b+c+2=abc .$ Prove that $$(a+1)(b+1)(c+1)\geq 27.$$
2010 Romanian Master of Mathematics, 3
Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent.
[i]Pavel Kozhevnikov, Russia[/i]
2023 District Olympiad, P2
Let $A{}$ and $B$ be invertible $n\times n$ matrices with real entries. Suppose that the inverse of $A+B^{-1}$ is $A^{-1}+B$. Prove that $\det(AB)=1$. Does this property hold for $2\times 2$ matrices with complex entries?
2023 AMC 10, 3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
1982 Canada National Olympiad, 5
The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$, $B'$, $C'$, and $D'$, where $AA' = k/h_a$, $BB' = k/h_b$, $CC' = k/h_c$, and $DD' = k/h_d$. Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$, etc. Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$.
1972 Kurschak Competition, 2
A class has $n > 1$ boys and $n$ girls. For each arrangement $X$ of the class in a line let $f(X)$ be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with $f(X) = 0$ be $A$, and the number of arrangements with $f(X) = 1$ be $B$. Show that $B = 2A$.
1962 All-Soviet Union Olympiad, 5
An $n \times n$ array of numbers is given. $n$ is odd and each number in the array is $1$ or $-1$. Prove that the number of rows and columns containing an odd number of $-1$s cannot total $n$.
2017 Germany Team Selection Test, 3
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.