Found problems: 85335
2022 Yasinsky Geometry Olympiad, 5
Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$.
a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible.
b) Find the smallest possible value of $CK+KL$.
(Olexii Panasenko)
2018 Lusophon Mathematical Olympiad, 1
Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$.
2015 Junior Regional Olympiad - FBH, 2
One day students in school organised a exchange between them such that : $11$ strawberries change for $14$ raspberries, $22$ cherries change for $21$ raspberries, $10$ cherries change for $3$ bananas and $5$ pears for $2$ bananas. How many pears has Amila to give to get $7$ strawberries
1964 AMC 12/AHSME, 22
Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of triangle $DFE$ to the area of quadrilateral $ABEF$?
$ \textbf{(A)}\ 1:2 \qquad\textbf{(B)}\ 1:3 \qquad\textbf{(C)}\ 1:5 \qquad\textbf{(D)}\ 1:6 \qquad\textbf{(E)}\ 1:7 $
2022 EGMO, 2
Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(ab) = f(a)f(b)$, and
(2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.
2000 Belarus Team Selection Test, 3.3
Each edge of a graph with $15$ vertices is colored either red or blue in such a way that no three vertices are pairwise connected with edges of the same color. Determine the largest possible number of edges in the graph.
2009 AMC 10, 12
Distinct points $ A$, $ B$, $ C$, and $ D$ lie on a line, with $ AB\equal{}BC\equal{}CD\equal{}1$. Points $ E$ and $ F$ lie on a second line, parallel to the first, with $ EF\equal{}1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
2024 Romanian Master of Mathematics, 4
Fix integers $a$ and $b$ greater than $1$. For any positive integer $n$, let $r_n$ be the (non-negative) remainder that $b^n$ leaves upon division by $a^n$. Assume there exists a positive integer $N$ such that $r_n < \frac{2^n}{n}$ for all integers $n\geq N$. Prove that $a$ divides $b$.
[i]Pouria Mahmoudkhan Shirazi, Iran[/i]
1996 Austrian-Polish Competition, 2
A convex hexagon $ ABCDEF$ satisfies the following conditions:
1) $ AB\parallel DE$, $ BC\parallel EF$, and $ CD\parallel FA$.
2) The distances between these pairs of parallel lines are the same.
3) $ \angle FAB \equal{} \angle CDE \equal{} 90^\circ$
Prove that the diagonals $ BE$ and $ CF$ of the hexagon intersect with angle $ 45$ degrees.
$ \bullet$ Thank you dear [b]Babis Stergiou[/b] for your translation. :P
Novosibirsk Oral Geo Oly IX, 2021.2
The robot crawls the meter in a straight line, puts a flag on and turns by an angle $a <180^o$ clockwise. After that, everything is repeated. Prove that all flags are on the same circle.
2009 Baltic Way, 11
Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$, and let $K$ be a point on the ray $BA$ beyond $A$. The line $KM$ intersects the side $BC$ at the point $L$. $P$is the point on the segment $BM$ such that $PM$ is the bisector of the angle $LPK$. The line $\ell$ passes through $A$ and is parallel to $BM$. Prove that the projection of the point $M$ onto the line $\ell$ belongs to the line $PK$.
2020 Durer Math Competition Finals, 10
Soma has a tower of $63$ bricks , consisting of $6$ levels. On the $k$-th level from the top, there are $2k-1$ bricks (where $k = 1, 2, 3, 4, 5, 6$), and every brick which is not on the lowest level lies on precisely $2$ smaller bricks (which lie one level below) - see the figure. Soma takes away $7$ bricks from the tower, one by one. He can only remove a brick if there is no brick lying on it. In how many ways can he do this, if the order of removals is considered as well?
[img]https://cdn.artofproblemsolving.com/attachments/b/6/4b0ce36df21fba89708dd5897c43a077d86b5e.png[/img]
2000 Romania National Olympiad, 1
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence having $ x_1=3 $ and defined as $ x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor , $ for every natural number $ n. $ Find all values $ m $ for which the terms $ x_m,x_{m+1},x_{m+2} $ are in arithmetic progression, where $ \lfloor\rfloor $ denotes the integer part.
2001 National Olympiad First Round, 4
How many real solution does the equation $\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1985 Putnam, B4
Let $C$ be the unit circle $x^{2}+y^{2}=1 .$ A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$ and $y$-axes with diagonal $p q .$ What is the probability that no point of $R$ lies outside of $C ?$
2008 Cono Sur Olympiad, 5
Let $ABC$ be an isosceles triangle with base $AB$. A semicircle $\Gamma$ is constructed with its center on the segment AB and which is tangent to the two legs, $AC$ and $BC$. Consider a line tangent to $\Gamma$ which cuts the segments $AC$ and $BC$ at $D$ and $E$, respectively. The line perpendicular to $AC$ at $D$ and the line perpendicular to $BC$ at $E$ intersect each other at $P$. Let $Q$ be the foot of the perpendicular from $P$ to $AB$. Show that
$\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}$.
2007 Iran MO (2nd Round), 3
Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer $n$, it prints exactly one of the numbers $n,2n,3n$. We know that the machine prints $2$. Prove that it doesn't print $13824$.
2020 LMT Fall, 22
Find the area of a triangle with side lengths $\sqrt{13},\sqrt{29},$ and $\sqrt{34}.$ The area can be expressed as $\frac{m}{n}$ for $m,n$ relatively prime positive integers, then find $m+n.$
[i]Proposed by Kaylee Ji[/i]
2020 Balkan MO Shortlist, C1
Let $s \geq 2$ and $n \geq k \geq 2$ be integes, and let $A$ be a subset of $\{1, 2, . . . , n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $A$ share some entry. Prove that there are an integer $p \leq k$ and $s+2$ members $A_1, A_2, . . . , A_{s+2}$ of $A$ such that $A_i$ and $A_j$ share the $p$-th entry alone, whenever $i$ and $j$ are distinct.
[i]Miroslav Marinov, Bulgaria[/i]
1972 AMC 12/AHSME, 31
When the number $2^{1000}$ is divided by $13$, the remainder in the division is
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }7\qquad \textbf{(E) }11$
2005 Today's Calculation Of Integral, 11
Calculate the following indefinite integrals.
[1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$
[2] $\int \frac{e^x}{e^x+e^{a-x}}dx$
[3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$
[4] $\int x\ln (x^2-1)dx$
[5] $\int \frac{2(x+2)}{x^2+4x+1}dx$
2007 Princeton University Math Competition, 5
Round to the nearest tenth: $\log_6 (6^2-6+1) + 3\log_6 (5) - \frac{1}{2}\log_6 (9)$.
1975 All Soviet Union Mathematical Olympiad, 207
What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?
1957 AMC 12/AHSME, 18
Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\cdot AM$ is equal to:
[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);
pair O = origin;
pair A = (-1,0);
pair B = (1,0);
pair C = (0,1);
pair D = (0,-1);
pair M = dir(45);
pair P = intersectionpoint(O--C,A--M);
draw(Circle(O,1));
draw(A--B);
draw(C--D);
draw(A--M);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$D$",D,S);
label("$M$",M,NE);
label("$O$",O,NE);
label("$P$",P,NW);[/asy]$ \textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \textbf{(C)}\ CP\cdot CD \qquad \textbf{(D)}\ CP\cdot PD\qquad$
$ \textbf{(E)}\ CO\cdot OP$
1973 Canada National Olympiad, 3
Prove that if $p$ and $p+2$ are prime integers greater than 3, then 6 is a factor of $p+1$.