Found problems: 85335
2004 Purple Comet Problems, 19
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$.
2023 Singapore Junior Math Olympiad, 4
Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.
2024 Bulgarian Winter Tournament, 12.3
Let $n$ be a positive integer and let $\mathcal{A}$ be a family of non-empty subsets of $\{1, 2, \ldots, n \}$ such that if $A \in \mathcal{A}$ and $A$ is subset of a set $B\subseteq \{1, 2, \ldots, n\}$, then $B$ is also in $\mathcal{A}$. Show that the function $$f(x):=\sum_{A \in \mathcal{A}} x^{|A|}(1-x)^{n-|A|}$$ is strictly increasing for $x \in (0,1)$.
2002 Baltic Way, 12
A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that
$|XY|=|XZ|+|XW|$
Prove that all four points lie on a line.
2000 Greece National Olympiad, 2
Find all prime numbers $p$ such that $1 +p+p^2 +p^3 +p^4$ is a perfect square.
Durer Math Competition CD 1st Round - geometry, 2022.C4
We inscribed in triangle $ABC$ the rectangle $DEFG$ such that $D$ and $E$ fall on side $AB$, $F$ on side $BC$, and $G$ on side $AC$. We know that $AF$ bisects angle $\angle BAC$, and that $\frac{AD}{DE} = \frac12$. What is the measure of angle $\angle CAB$?
1978 Kurschak Competition, 1
$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.
2010 Oral Moscow Geometry Olympiad, 3
Two circles $w_1$ and $w_2$ intersect at points $A$ and $B$. Tangents $\ell_1$ and $\ell_2$ respectively are drawn to them through point $A$. The perpendiculars dropped from point $B$ to $\ell_2$ and $\ell_1$ intersects the circles $w_1$ and $w_2$, respectively, at points $K$ and $N$. Prove that points $K, A$ and $N$ lie on one straight line.
2014 AMC 12/AHSME, 15
A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?
$\textbf{(A) }9\qquad
\textbf{(B) }18\qquad
\textbf{(C) }27\qquad
\textbf{(D) }36\qquad
\textbf{(E) }45\qquad$
II Soros Olympiad 1995 - 96 (Russia), 11.8
The following is known about the quadrilateral $ABCD$: triangles $ABC$ and $CDA$ are equal in area, the area of triangle $BCD$ is $k$ times greater than the area of triangle $DAB$, the bisectors of angles $ABC$ and $CDA$ intersect on the diagonal $AC$, straight lines $AC$ and $BD$ are not perpendicular. Find the ratio $AC/BD$.
2015 Purple Comet Problems, 2
The diagram below is made up of a rectangle AGHB, an equilateral triangle AFG, a rectangle ADEF, and a parallelogram ABCD. Find the degree measure of ∠ABC. For diagram go to http://www.purplecomet.org/welcome/practice, the 2015 middle school contest, and go to #2
2011 Balkan MO Shortlist, A1
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
1988 IMO Longlists, 79
Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.
2000 France Team Selection Test, 1
Points $P,Q,R,S$ lie on a circle and $\angle PSR$ is right. $H,K$ are the projections of $Q$ on lines $PR,PS$. Prove that $HK$ bisects segment $ QS$.
2008 China Northern MO, 2
The given triangular number table is as follows:
[img]https://cdn.artofproblemsolving.com/attachments/a/0/123b7511850047f3cc51494f107703f2757085.png[/img]
Among them, the numbers in the first row are $1, 2, 3, ..., 98, 99, 100$. Starting from the second row, each number is equal to the sum of the left and right numbers in the row above it. Find the value of $M$.
2020 Estonia Team Selection Test, 1
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
2008 Harvard-MIT Mathematics Tournament, 4
Positive real numbers $ x$, $ y$ satisfy the equations $ x^2 \plus{} y^2 \equal{} 1$ and $ x^4 \plus{} y^4 \equal{} \frac {17}{18}$. Find $ xy$.
2015 Tournament of Towns, 1
A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer.
[i]($3$ points)[/i]
2011 Mongolia Team Selection Test, 1
Let $t,k,m$ be positive integers and $t>\sqrt{km}$. Prove that
$\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}$
(proposed by B. Amarsanaa, folklore)
2019 Purple Comet Problems, 13
Squares $ABCD$ and $AEFG$ each with side length $12$ overlap so that $\vartriangle AED$ is an equilateral triangle as shown. The area of the region that is in the interior of both squares which is shaded in the diagram is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/c/2/a2f8d2a090a6342610c43b3fed8a87fa5d7f03.png[/img]
2019 Thailand TST, 2
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
2000 Harvard-MIT Mathematics Tournament, 7
A regular tetrahedron of volume $1$ is filled with water of total volume $\frac{7}{16}$. Is it possible that the center of the tetrahedron lies on the surface of the water? How about in a cube of volume $1$?
2006 Germany Team Selection Test, 1
Let $ ABC$ be an equilateral triangle, and $ P,Q,R$ three points in its interior satisfying
\[ \measuredangle PCA \equal{} \measuredangle CAR \equal{} 15^{\circ},\ \measuredangle RBC \equal{} \measuredangle BCQ \equal{} 20^{\circ},\ \measuredangle QAB \equal{} \measuredangle ABP \equal{} 25^{\circ}.\] Compute the angles of triangle $ PQR$.
2005 Iran MO (3rd Round), 2
Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:\[x^3\equiv1(\mbox{mod}\ m)\]
2014 Taiwan TST Round 1, 2
Determine whether there exist ten sets $A_1$, $A_2$, $\dots$, $A_{10}$ such that
(i) each set is of the form $\{a,b,c\}$, where $a \in \{1,2,3\}$, $b \in \{4,5,6\}$, $c \in \{7,8,9\}$,
(ii) no two sets are the same,
(iii) if the ten sets are arranged in a circle $(A_1, A_2, \dots, A_{10})$, then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note: $A_{10}$ is adjacent to $A_1$.)