Found problems: 85335
2002 Kazakhstan National Olympiad, 5
On the plane is given the acute triangle $ ABC $. Let $ A_1 $ and $ B_1 $ be the feet of the altitudes of $ A $ and $ B $ drawn from those vertices, respectively. Tangents at points $ A_1 $ and $ B_1 $ drawn to the circumscribed circle of the triangle $ CA_1B_1 $ intersect at $ M $. Prove that the circles circumscribed around the triangles $ AMB_1 $, $ BMA_1 $ and $ CA_1B_1 $ have a common point.
2001 Romania National Olympiad, 3
We consider the points $A,B,C,D$, not in the same plane, such that $AB\perp CD$ and $AB^2+CD^2=AD^2+BC^2$.
a) Prove that $AC\perp BD$.
b) Prove that if $CD<BC<BD$, then the angle between the planes $(ABC)$ and $(ADC)$ is greater than $60^{\circ}$.
2019 Auckland Mathematical Olympiad, 2
There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.
2017 CMIMC Team, 5
We have four registers, $R_1,R_2,R_3,R_4$, such that $R_i$ initially contains the number $i$ for $1\le i\le4$. We are allowed two operations:
[list]
[*] Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.
[*] Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. (For example if we do this once then $(R_1,R_2,R_3,R_4)=(1,4,2,3)$.)
[/list]
Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
2015 Rioplatense Mathematical Olympiad, Level 3, 2
Let $a , b , c$ positive integers, coprime. For each whole number $n \ge 1$, we denote by $s ( n )$ the number of elements in the set $\{ a , b , c \}$ that divide $n$. We consider $k_1< k_2< k_3<...$ .the sequence of all positive integers that are divisible by some element of $\{ a , b , c \}$. Finally we define the characteristic sequence of $( a , b , c )$ like the succession $ s ( k_1) , s ( k_2) , s ( k_3) , .... $ .
Prove that if the characteristic sequences of $( a , b , c )$ and $( a', b', c')$ are equal, then $a = a', b = b'$ and $c=c'$
2007 AMC 10, 10
Two points $ B$ and $ C$ are in a plane. Let $ S$ be the set of all points $ A$ in the plane for which $ \triangle ABC$ has area $ 1$. Which of the following describes $ S$?
$ \textbf{(A)}\ \text{two parallel lines}\qquad
\textbf{(B)}\ \text{a parabola}\qquad
\textbf{(C)}\ \text{a circle}\qquad
\textbf{(D)}\ \text{a line segment}\qquad
\textbf{(E)}\ \text{two points}$
Gheorghe Țițeica 2025, P3
Two regular pentagons $ABCDE$ and $AEKPL$ are given in space, such that $\angle DAK = 60^{\circ}$. Let $M$, $N$ and $S$ be the midpoints of $AE$, $CD$ and $EK$. Prove that:
[list=a]
[*] $\triangle NMS$ is a right triangle;
[*] planes $(ACK)$ and $(BAL)$ are perpendicular.
[/list]
[i]Ukraine Olympiad[/i]
1984 IMO Shortlist, 13
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
2021 IMO Shortlist, G5
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$.
Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.
2014 BMT Spring, 1
For the team, power, and tournament rounds, BMT divided up the teams into $14$ rooms. You sign up to proctor all $3$ rounds, but you cannot proctor in the same room more than once.
How many ways can you be assigned for rooms for the $3$ rounds?
2022-IMOC, A5
Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$
1988 AIME Problems, 5
Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.
PEN E Problems, 21
Prove that if $p$ is a prime, then $p^{p}-1$ has a prime factor that is congruent to $1$ modulo $p$.
2016 Korea National Olympiad, 1
$n$ is a positive integer. The number of solutions of $x^2+2016y^2=2017^n$ is $k$. Write $k$ with $n$.
2007 Moldova Team Selection Test, 1
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.
2024 AMC 10, 6
A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle?
$
\textbf{(A) }160 \qquad
\textbf{(B) }180 \qquad
\textbf{(C) }222 \qquad
\textbf{(D) }228 \qquad
\textbf{(E) }390 \qquad
$
1956 AMC 12/AHSME, 31
In our number system the base is ten. If the base were changed to four you would count as follows: $ 1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be:
$ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 44 \qquad\textbf{(D)}\ 104 \qquad\textbf{(E)}\ 110$
2017 AMC 10, 21
In $\triangle ABC,$ $AB=6, AC=8, BC=10,$ and $D$ is the midpoint of $\overline{BC}.$ What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC?$
$\textbf{(A)} \sqrt{5} \qquad \textbf{(B)} \frac{11}{4}\qquad \textbf{(C)} 2\sqrt{2} \qquad \textbf{(D)} \frac{17}{6} \qquad \textbf{(E)} 3$
2013 Swedish Mathematical Competition, 4
A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of $60^o$. A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than $10$ meters from its starting position, given that it is facing north when it starts. How long must the fence be?
1992 IMO Longlists, 11
Let $\phi(n,m), m \neq 1$, be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$, where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality:
\[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\]
for every positive integer $n.$
2020 IMO Shortlist, A2
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
2014-2015 SDML (Middle School), 12
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
1997 Miklós Schweitzer, 1
Define a class of graphs $G_k$ for each positive integer k as follows. A graph G = ( V , E ) is an element of $G_k$ if and only if there exists an edge coloring $\psi: E\to [ k ] = \{1,2, ..., k\}$ such that for all vertex coloring $\phi: V\to [ k ]$ there exist an edge e = { x , y } such that $\phi ( x ) = \phi( y ) = \psi( e )$. Prove that there exist $c_1< c_2$ positive constants with the following two properties:
(i) each graph in $G_k$ has at least $c_1 k^2$ vertices;
(ii) there is a graph in $G_k$ which has at most $c_2 k^2$ vertices.
1960 AMC 12/AHSME, 3
Applied to a bill for $\$10,000$ the difference between a discount of $40\%$ and two successive discounts of $36\%$ and $4\%$, expressed in dollars, is:
$ \textbf{(A) }0\qquad\textbf{(B) }144\qquad\textbf{(C) }256\qquad\textbf{(D) }400\qquad\textbf{(E) }416 $
1954 Czech and Slovak Olympiad III A, 1
Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.