Found problems: 85335
1991 All Soviet Union Mathematical Olympiad, 544
Does there exist a triangle in which two sides are integer multiples of the median to that side?
Does there exist a triangle in which every side is an integer multiple of the median to that side?
2014 JHMMC 7 Contest, 3
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.
2016 China Second Round Olympiad, 4
Let $p>3$ and $p+2$ are prime numbers,and define sequence
$$a_{1}=2,a_{n}=a_{n-1}+\lfloor \dfrac{pa_{n-1}}{n}\rfloor$$
show that:for any $n=3,4,\cdots,p-1$ have $$n|pa_{n-1}+1$$
2020 Taiwan TST Round 2, 6
Let $I, O, \omega, \Omega$ be the incenter, circumcenter, the incircle, and the circumcircle, respectively, of a scalene triangle $ABC$. The incircle $\omega$ is tangent to side $BC$ at point $D$. Let $S$ be the point on the circumcircle $\Omega$ such that $AS, OI, BC$ are concurrent. Let $H$ be the orthocenter of triangle $BIC$. Point $T$ lies on $\Omega$ such that $\angle ATI$ is a right angle. Prove that the points $D, T, H, S$ are concyclic.
[i]Proposed by ltf0501[/i]
2020 Putnam, B3
Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$.
2020 HK IMO Preliminary Selection Contest, 3
A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end?
2014 BMT Spring, P1
Let $ABC$ be a triangle. Let $ r$ denote the inradius of $\vartriangle ABC$. Let $r_a$ denote the $A$-exradius of $\vartriangle ABC$. Note that the $A$-excircle of $\vartriangle ABC$ is the circle that is tangent to segment $BC$, the extension of ray $AB$ beyond $ B$ and the extension of $AC$ beyond $C$. The $A$-exradius is the radius of the $A$-excircle. Define $ r_b$ and $ r_c$ analogously. Prove that $$\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$$
2007 District Olympiad, 3
Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation:
$$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$
2000 Croatia National Olympiad, Problem 4
We are given coins of $1,2,5,10,20,50$ lipas and of $1$ kuna (Croatian currency: $1$ kuna = $100$ lipas). Prove that if a bill of $M$ lipas can be paid by $N$ coins, then a bill of $N$ kunas can be paid by M coins.
2017 Princeton University Math Competition, 11
For a sequence of $10$ coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails-Heads”, and “Tails-Tails” sequences is recorded. These four numbers are then multiplied to get the [i]Tiger number[/i] of the sequence of flips. How many such sequences have a [i]Tiger number [/i] of $24$?
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
2023 Harvard-MIT Mathematics Tournament, 26
Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$
2020 OMMock - Mexico National Olympiad Mock Exam, 2
We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$.
[i]Proposed by Ariel García[/i]
2010 Grand Duchy of Lithuania, 4
In the triangle $ABC$ angle $C$ is a right angle. On the side $AC$ point $D$ has been found, and on the segment $BD$ point K has been found such that $\angle ABC = \angle KAD = \angle AKD$. Prove that $BK = 2DC$.
2017 Saudi Arabia IMO TST, 2
Let $ABCD$ be a quadrilateral inscribed a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E, AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN,Q$ be the intersection of $MN$ and $GH$.
a) Prove that $IK$ and $OE$ are parallel.
b) Prove that $PK$ is perpendicular to $IQ$.
1960 AMC 12/AHSME, 29
Five times $A$'s money added to $B$'s money is more than $\$51.00$. Three times $A$'s money minus $B$'s money is $\$21.00$. If $a$ represents $A$'s money in dollars and $b$represents $B$'s money in dollars, then:
$ \textbf{(A)}\ a>9, b>6 \qquad\textbf{(B)}\ a>9, b<6 \qquad\textbf{(C)}\ a>9, b=6\qquad$
$\textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad\textbf{(E)}\ 2a=3b $
2019 Thailand TST, 3
Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]
where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.
[i]Proposed by Evan Chen, Taiwan[/i]
2001 All-Russian Olympiad Regional Round, 8.3
All sides of a convex pentagon are equal, and all angles are different. Prove that the maximum and minimum angles are adjacent to the same side of the pentagon.
2024 Iranian Geometry Olympiad, 4
An inscribed $n$-gon ($n > 3$), is divided into $n-2$ triangles by diagonals which meet only in vertices. What is the maximum possible number of congruent triangles obtained?
(An inscribed $n$-gon is an $n$-gon where all its vertices lie on a circle)
[i]Proposed by Boris Frenkin - Russia[/i]
2010 AMC 10, 18
Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
$ \textbf{(A)}\ \frac {47}{72}\qquad
\textbf{(B)}\ \frac {37}{56}\qquad
\textbf{(C)}\ \frac {2}{3}\qquad
\textbf{(D)}\ \frac {49}{72}\qquad
\textbf{(E)}\ \frac {39}{56}$
2023 Auckland Mathematical Olympiad, 10
Find the maximum of the expression
$$||...||x_1 - x_2|- x_3| -... | - x_{2023}|,$$
where $x_1,x_2,..., x_{2023}$ are distinct natural numbers between $1$ and $2023$.
2006 India IMO Training Camp, 2
Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.
1994 ITAMO, 2
solve this diophantine equation y^2 = x^3 - 16
2005 Colombia Team Selection Test, 3
Let $A_1A_2A_3\ldots A_n$ be a regular $n$-gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$. Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$. Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$, and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$.
Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$.
[i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]
2021 Yasinsky Geometry Olympiad, 3
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm, and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepiped $m$ parallel to line $A_1C_1$ is drawn on the plane $\gamma$.
a) Draw a section of a parallelepiped with plane $\gamma$.
b) Justify what geometric figure is this section, and find its area.
(Alexander Shkolny)