Found problems: 85335
2024 UMD Math Competition Part I, #23
For how many pairs of integers $(m, n)$ with $0 < m \le n \le 50$ do there exist precisely four triples of integers $(x, y, z)$ satisfying the following system?
\[\begin{cases} x^2 + y+ z = m \\ x + y^2 + z = n\end{cases}\]
\[\rm a. ~180\qquad \mathrm b. ~182\qquad \mathrm c. ~186 \qquad\mathrm d. ~188\qquad\mathrm e. ~190\]
2003 Bosnia and Herzegovina Team Selection Test, 5
It is given regular polygon with $2n$ sides and center $S$. Consider every quadrilateral with vertices as vertices of polygon. Let $u$ be number of such quadrilaterals which contain point $S$ inside and $v$ number of remaining quadrilaterals. Find $u-v$
2019 239 Open Mathematical Olympiad, 1
The following fractions are written on the board $\frac{1}{n}, \frac{2}{n-1}, \frac{3}{n-2}, \ldots , \frac{n}{1}$ where $n$ is a natural number. Vasya calculated the differences of the neighboring fractions in this row and found among them $10000$ fractions of type $\frac{1}{k}$ (with natural $k$). Prove that he can find even $5000$ more of such these differences.
2012 Iran MO (3rd Round), 3
Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that
\[\frac{TP}{TQ}\ge \frac{b}{a}.\]
[i]Proposed by Morteza Saghafian[/i]
2020 Purple Comet Problems, 7
The diagram below shows $\vartriangle ABC$ with area $64$, where $D, E$, and $F$ are the midpoints of $BC, CA$, and $AB$, respectively. Point $G$ is the intersection of $DF$ and $BE$. Find the area of quadrilateral $AFGE$.
[img]https://cdn.artofproblemsolving.com/attachments/d/0/056f9c856973b4efc96e77e54afb16ed8cc216.png[/img]
2020 CMIMC Geometry, 4
Triangle $ABC$ has a right angle at $B$. The perpendicular bisector of $\overline{AC}$ meets segment $\overline{BC}$ at $D$, while the perpendicular bisector of segment $\overline{AD}$ meets $\overline{AB}$ at $E$. Suppose $CE$ bisects acute $\angle ACB$. What is the measure of angle $ACB$?
2010 Victor Vâlcovici, 2
$ \sum_{cyc}\frac{1}{\left(\text{tg} y+\text{tg} z\right) \text{cos}^2 x} \ge 3, $ for any $ x,y,z\in (0,\pi/2) $
[i]Carmen[/i] and [i]Viorel Botea[/i]
2007 ITest, 34
Let $a/b$ be the probability that a randomly selected divisor of $2007$ is a multiple of $3$. If $a$ and $b$ are relatively prime positive integers, find $a+b$.
2012 NIMO Problems, 4
The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$.
[i]Proposed by Lewis Chen[/i]
1969 Putnam, A1
Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$
2005 Today's Calculation Of Integral, 7
Calculate the following indefinite integrals.
[1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$
[2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$
[3] $\int (\sin ^2 x+\cos x)\sin x dx$
[4] $\int x\sqrt{2-x} dx$
[5] $\int x\ln x dx$
2017 All-Russian Olympiad, 4
Are there infinite increasing sequence of natural numbers, such that sum of every 2 different numbers are relatively prime with sum of every 3 different numbers?
1984 Kurschak Competition, 2
$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$,$B_1A_2$, ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?
2010 ITAMO, 2
Every non-negative integer is coloured white or red, so that:
• there are at least a white number and a red number;
• the sum of a white number and a red number is white;
• the product of a white number and a red number is red.
Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.
2016 Kyrgyzstan National Olympiad, 2
The number $N$ consists only $2's$ and $1's$ in its [b]decimal representation[/b].We know that,after deleting digits from N,we can get any number consisting $9999$- $1's$ and $one$ - $2's$ in its [b]decimal representation[/b].[b][u]Find the least number of digits in the decimal representation of N[/u][/b]
2021 Belarusian National Olympiad, 8.1
Prove that there exists a $2021$-digit positive integer $\overline{a_1a_2\ldots a_{2021}}$, with all its digits being non-zero, such that for every $1 \leq n \leq 2020$ the following equality holds
$$\overline{a_1a_2\ldots a_n} \cdot \overline{a_{n+1}a_{n+2}\ldots a_{2021}}=\overline{a_1a_2\ldots a_{2021-n}} \cdot \overline{a_{2022-n}a_{2023-n}\ldots a_{2021}}$$
and all four numbers in the equality are pairwise different.
1955 Miklós Schweitzer, 5
[b]5.[/b] Show that a ring $R$ is commutative if for every $x \in R$ the element $x^{2}-x$ belongs to the centre of $R$. [b](A. 18)[/b]
2011 BMO TST, 4
Find all prime numbers p such that $2^p+p^2 $ is also a prime number.
2017 IMO Shortlist, A7
Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of [i]positive[/i] integers such that $a_0=0,a_1=1$, and
\[
a_{n+1} =
\begin{cases}
a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\
a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$}
\end{cases}\qquad\text{for }n=1,2,\ldots.
\]
for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.
1995 Bundeswettbewerb Mathematik, 3
Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.
2005 National High School Mathematics League, 3
$\triangle ABC$ is inscribed to unit circle. Bisector of $\angle A,\angle B,\angle C$ intersect the circle at $A_1,B_1,C_1$ respectively. The value of $\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}$ is
$\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8$
1998 Iran MO (3rd Round), 1
Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$
\[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\]
where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.
Kyiv City MO Juniors 2003+ geometry, 2011.8.41
The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$.
(Veklich Bogdan)
2006 Chile National Olympiad, 6
Let $ \vartriangle ABC $ be an acute triangle and scalene, with $ BC $ its smallest side. Let $ P, Q $ points on $ AB, AC $ respectively, such that $ BQ = CP = BC $. Let $ O_1, O_2 $ be the centers of the circles circumscribed to $ \vartriangle AQB, \vartriangle APC $, respectively. Sean $ H, O $ the orthocenter and circumcenter of $ \vartriangle ABC $
a) Show that $ O_1O_2 = BC $.
b) Show that $ BO_2, CO_1 $ and $ HO $ are concurrent
2012 NIMO Summer Contest, 11
Let $a$ and $b$ be two positive integers satisfying the equation
\[
20\sqrt{12} = a\sqrt{b}.
\]
Compute the sum of all possible distinct products $ab$.
[i]Proposed by Lewis Chen[/i]