Found problems: 85335
2024 HMNT, 21
Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino.
[center]
[img]
https://cdn.artofproblemsolving.com/attachments/b/1/17565ba86dbc2358f546fa57145a7726d1b0a9.png
[/img]
[/center]
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.
2022 HMNT, 7
Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation
$$gcd(a, b) \cdot a + b^2 = 10000.$$
2021 Iran Team Selection Test, 5
Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$.
Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$.
Proposed by [i]Alireza Dadgarnia[/i]
2016 Azerbaijan Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2010 Dutch BxMO TST, 4
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
Russian TST 2017, P1
A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city
2017 Dutch IMO TST, 4
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$
for all $x, y \in \mathbb{R}$.
1982 Vietnam National Olympiad, 1
Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.
2022 IMAR Test, 1
Find all pairs of primes $p, q<2023$ such that $p \mid q^2+8$ and $q \mid p^2+8$.
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
2013 Vietnam National Olympiad, 3
Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.
[b]a)[/b] Prove that $D,I,J$ collinear.
[b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.
Kvant 2020, M1000
A polyline $AMB$ is inscribed in the arc $AB{}$, consisting of two segments, and $AM>MB$. Let $K$ be the midpoint of the arc $AB{}$. Prove that the foot $H{}$ of the perpendicular from $K$ onto $AM$ divides the polyline in two equal segments: \[AH=HM+MB.\][i]Discovered by Archimedes[/i]
LMT Guts Rounds, 33
Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$
2018 Kyiv Mathematical Festival, 4
Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?
2016 Korea Winter Program Practice Test, 3
$p, q, r$ are natural numbers greater than 1.
There are $pq$ balls placed on a circle, and one number among $0, 1, 2, \cdots , pr-1$ is written on each ball, satisfying following conditions.
(1) If $i$ and $j$ is written on two adjacent balls, $|i-j|=1$ or $|i-j|=pr-1$.
(2) $i$ is written on a ball $A$. If we skip $q-1$ balls clockwise from $A$ and see $q^{th}$ ball, $i+r$ or $i-(p-1)r$ is written on it. (This condition is satisfied for every ball.)
If $p$ is even, prove that the number of pairs of two adjacent balls with $1$ and $2$ written on it is odd.
2009 AMC 8, 24
The letters $ A$, $ B$, $ C$ and $ D$ represent digits. If $ \begin{tabular}{ccc} &A&B \\ \plus{}&C&A \\ \hline &D&A \end{tabular}$ and $ \begin{tabular}{ccc} &A&B \\ \minus{}&C&A \\ \hline &&A \end{tabular}$, what digit does $ D$ represent?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
2019 Novosibirsk Oral Olympiad in Geometry, 3
Equal line segments are marked in triangle $ABC$. Find its angles.
[img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]
2024 Azerbaijan BMO TST, 5
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2022 Brazil National Olympiad, 2
The nonzero real numbers $a, b, c$ satisfy the following system: $$\begin{cases} a+ab=c\\ b+bc=a\\ c+ca=b \end{cases}$$ Find all possible values of the $abc$.
2024 Australian Mathematical Olympiad, P8
Let $r=0.d_0d_1d_2\ldots$ be a real number. Let $e_n$ denote the number formed by the digits $d_n, d_{n-1}, \ldots, d_0$ written from left to right (leading zeroes are permitted). Given that $d_0=6$ and for each $n \geq 0$, $e_n$ is equal to the number formed by the $n+1$ rightmost digits of $e_n^2$. Show that $r$ is irrational.
2011 Today's Calculation Of Integral, 675
In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$.
[i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]
1993 Czech And Slovak Olympiad IIIA, 1
Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$
2022 Stanford Mathematics Tournament, 1
If $f(x)=x^4+4x^3+7x^2+6x+2022$, compute $f'(3)$.
2005 Vietnam National Olympiad, 2
Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:
a) \[ CD\leq R \]
b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).