Found problems: 85335
2015 Iran Team Selection Test, 1
Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.
2012 Bosnia and Herzegovina Junior BMO TST, 4
If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that:
$a^2+b^2+c^2+4abc<\frac{1}{2}$
2011 Baltic Way, 16
Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$
\[x_n=2x_{n-1}-4x_{n-2}+3.\]
Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.
2023 Euler Olympiad, Round 1, 2
A student took a rectangular piece of paper with length equal to one meter and width equal to five centimeters. The student brought the ends together, turning one end 180 degrees and gluing the surfaces to create a figure called a Möbius strip. On one side of this strip, the student placed a flea and an ant. It is known that if the flea and the ant move in different directions on the Möbius strip, they will meet each other in 2 minutes. However, if they move in the same direction, they will meet in 7 minutes. Given that the flea is faster than the ant and both move at constant speeds, determine the speed of the flea.
[i]Proposed by Lia Chitishvili, Georgia[/i]
2021 AMC 12/AHSME Spring, 18
Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0?$
$\textbf{(A) } \frac{17}{32} \qquad \textbf{(B) } \frac{11}{16} \qquad \textbf{(C) } \frac{7}{9} \qquad \textbf{(D) } \frac{7}{6} \qquad \textbf{(E) } \frac{25}{11}$
1968 Spain Mathematical Olympiad, 2
Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases .
a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$.
b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$.
c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.
2005 iTest, 23
$\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+ \sqrt[3]{x ...}}}}= 8$. Find $x$.
2005 ISI B.Math Entrance Exam, 6
Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that
$\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$
Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$
1968 AMC 12/AHSME, 2
The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:
$\textbf{(A)}\ -\dfrac{2}{3} \qquad
\textbf{(B)}\ -\dfrac{1}{3} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \dfrac{1}{4} \qquad
\textbf{(E)}\ \dfrac{3}{8} $
LMT Team Rounds 2021+, 13
Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$.
Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.
1992 Poland - First Round, 11
Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$.
1972 Vietnam National Olympiad, 1
Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$).
i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$.
Thus we can write $y$ as a function of $x, y = T_n(x)$.
Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$.
From this it follows that $T_n(x)$ is a polynomial of degree $n$.
ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.
2025 AIME, 15
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]
There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
Durer Math Competition CD Finals - geometry, 2020.D2
Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.
1965 All Russian Mathematical Olympiad, 066
The tourist has come to the Moscow by train. All-day-long he wandered randomly through the streets. Than he had a supper in the cafe on the square and decided to return to the station only through the streets that he has passed an odd number of times. Prove that he is always able to do that.
2011 All-Russian Olympiad Regional Round, 11.6
$\omega$ is the circumcirle of an acute triangle $ABC$. The tangent line passing through $A$ intersects the tangent lines passing through points $B$ and $C$ at points $K$ and $L$, respectively. The line parallel to $AB$ through $K$ and the line parallel to $AC$ through $L$ intersect at point $P$. Prove that $BP=CP$.
(Author: P. Kozhevnikov)
MOAA Team Rounds, 2021.9
Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Raina Yang[/i]
2019 New Zealand MO, 6
Let $V$ be the set of vertices of a regular $21$-gon. Given a non-empty subset $U$ of $V$ , let $m(U)$ be the number of distinct lengths that occur between two distinct vertices in $U$. What is the maximum value of $\frac{m(U)}{|U|}$ as $U$ varies over all non-empty subsets of $V$ ?
2023 Harvard-MIT Mathematics Tournament, 27
Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_1, x_2, \ldots, x_n$ for which
[list]
[*]$x_1^k+x_2^k+\ldots+x_n^k=1$ for $k=1, 2, \ldots, n-1;$
[*]$x_1^n+x_2^n+\ldots+x_n^n=2;$ and
[*]$x_1^m+x_2^m+\ldots+x_n^m= 4.$
[/list]
Compute the smallest possible value of $m+n.$
2024 Assara - South Russian Girl's MO, 6
The points $A, B, C, D$ are marked on the straight line in this order. Circle $\omega_1$ passes through points $A$ and $C$, and the circle $\omega_2$ passes through points $B$ and $D$. On the circle $\omega_2$, the point $E$ is marked so that $AB = BE$, and on the circle $\omega_1$, the point $F$ is marked so that $CD = CF$. The line $AE$ intersects the circle $\omega_2$ a second time at point $X$, and the line $DF$ intersects the circle $\omega_1$ at point $Y$. Prove that the $XY$ lines and $AD$ is perpendicular.
[i]A.D.Tereshin[/i]
2012 Purple Comet Problems, 16
The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is
\[
\frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots.
\]
Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. Find $m+n$.
1990 IMO Longlists, 67
Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$
2020 CMIMC Algebra & Number Theory, 1
Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.
2007 Tournament Of Towns, 3
A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal.
[i](2 points)[/i]
2010 AMC 12/AHSME, 14
Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter?
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 33 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 37$