Found problems: 85335
2018 Saudi Arabia IMO TST, 3
Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.
1959 IMO, 3
Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.
2009 HMNT, 2
You start with a number. Every second, you can add or subtract any number of the form $n!$ to your current number to get a new number. In how many ways can you get from $0$ to $100$ in $4$ seconds?
($n!$ is de ned as $n\times (n -1)\times(n - 2) ... 2\times1$, so $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, etc.)
2012 Saint Petersburg Mathematical Olympiad, 6
On the coordinate plane in the first quarter there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.
2023 Chile Classification NMO Seniors, 3
In the convex quadrilateral $ABCD$, $M$ is the midpoint of side $AD$, $AD = BD$, lines $CM$ and $AB$ are parallel, and $3\angle LBAC = \angle LACD$. Find the measure of angle $\angle ACB$.
2007 Moldova Team Selection Test, 1
Let $a_{1}, a_{2}, \ldots, a_{n}\in [0;1]$. If $S=a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}$ then prove that \[\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+\ldots+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}\]
2004 AMC 12/AHSME, 19
Circles $ A$, $ B$ and $ C$ are externally tangent to each other and internally tangent to circle $ D$. Circles $ B$ and $ C$ are congruent. Circle $ A$ has radius $ 1$ and passes through the center of $ D$. What is the radius of circle $ B$?
[asy]
unitsize(15mm);
pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
draw(Circle(D,2));
draw(Circle(A,1));
draw(Circle(B,8/9));
draw(Circle(C,8/9));
label("\(A\)", A);
label("\(B\)", B);
label("\(C\)", C);
label("D", (-1.2,1.8));[/asy]
$ \textbf{(A)}\ \frac23 \qquad \textbf{(B)}\ \frac {\sqrt3}{2} \qquad \textbf{(C)}\ \frac78 \qquad \textbf{(D)}\ \frac89 \qquad \textbf{(E)}\ \frac {1 \plus{} \sqrt3}{3}$
2019 Switzerland Team Selection Test, 10
Let $n \geq 5$ be an integer. A shop sells balls in $n$ different colors. Each of $n + 1 $ children bought three balls with different colors, but no two children bought exactly the same color combination. Show that there are at least two children who bought exactly one ball of the same color.
2006 Iran Team Selection Test, 3
Let $l,m$ be two parallel lines in the plane.
Let $P$ be a fixed point between them.
Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$.
(By angle $EPF$ we mean the directed angle)
Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.
2019 CCA Math Bonanza, I5
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
[i]2019 CCA Math Bonanza Individual Round #5[/i]
2003 Tournament Of Towns, 4
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
2015 JBMO TST - Turkey, 5
A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number.
2012 Singapore Junior Math Olympiad, 5
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
2013 Brazil Team Selection Test, 2
Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.
2016 Israel Team Selection Test, 1
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
2014 Purple Comet Problems, 18
Find the number of subsets of $\{1,3,5,7,9,11,13,15,17,19\}$ where the elements in the subset add to $49$.
2003 VJIMC, Problem 1
Let $d(k)$ denote the number of natural divisors of a natural number $k$. Prove that for any natural number $n_0$ the sequence $\left\{d(n^2+1)\right\}^\infty_{n=n_0}$ is not strictly monotone.
2006 Australia National Olympiad, 2
Let $f$ be a function defined on the positive integers, taking positive integral values, such that
$f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$,
$f(a) < f(b)$ if $a < b$,
$f(3) \geq 7$.
Find the smallest possible value of $f(3)$.
1963 AMC 12/AHSME, 21
The expression $x^2-y^2-z^2+2yz+x+y-z$ has:
$\textbf{(A)}\ \text{no linear factor with integer coeficients and integer exponents} \qquad$
$
\textbf{(B)}\ \text{the factor }-x+y+z \qquad$
$
\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad$
$
\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad$
$
\textbf{(E)}\ \text{the factor }x-y+z+1$
2015 Princeton University Math Competition, A5
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that the sum
\[a+a^2+a^3+\cdots+a^{(p-2)!} \]is not divisible by $p$?
2007 Indonesia TST, 2
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
\[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\]
for all real numbers $x$ and $y$.
1959 AMC 12/AHSME, 3
If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:
$ \textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these} $
IV Soros Olympiad 1997 - 98 (Russia), 11.5
Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.
2021 Federal Competition For Advanced Students, P2, 3
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied.
(Walther Janous)
1991 Bulgaria National Olympiad, Problem 3
Prove that for every prime number $p\ge5$,
(a) $p^3$ divides $\binom{2p}p-2$;
(b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.