Found problems: 85335
2011 Oral Moscow Geometry Olympiad, 6
Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.
2016 AMC 12/AHSME, 4
The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$
2010 F = Ma, 6
A projectile is launched across flat ground at an angle $\theta$ to the horizontal and travels in the absence of air resistance. It rises to a maximum height $H$ and lands a horizontal distance $R$ away. What is the ratio $H/R$?
(A) $\tan \theta$
(B) $2 \tan \theta$
(C) $\frac{2}{\tan \theta}$
(D) $\frac{1}{2}\tan \theta$
(E) $\frac{1}{4}\tan \theta$
2021 Romania National Olympiad, 4
Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$.
[i]Andrei Bâra[/i]
DMM Team Rounds, 2019
[b]p1.[/b] Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together?
[b]p2. [/b] Suppose we wrote the integers between $0001$ and $2019$ on a blackboard as such: $$000100020003 · · · 20182019.$$ How many $0$’s did we write?
[b]p3.[/b] Duke’s basketball team has made $x$ three-pointers, $y$ two-pointers, and $z$ one-point free throws, where $x, y, z$ are whole numbers. Given that $3|x$, $5|y$, and $7|z$, find the greatest number of points that Duke’s basketball team could not have scored.
[b]p4.[/b] Find the minimum value of $x^2 + 2xy + 3y^2 + 4x + 8y + 12$, given that $x$ and $y$ are real numbers.
Note: calculus is not required to solve this problem.
[b]p5.[/b] Circles $C_1, C_2$ have radii $1, 2$ and are centered at $O_1, O_2$, respectively. They intersect at points $ A$ and $ B$, and convex quadrilateral $O_1AO_2B$ is cyclic. Find the length of $AB$. Express your answer as $x/\sqrt{y}$ , where $x, y$ are integers and $y$ is square-free.
[b]p6.[/b] An infinite geometric sequence $\{a_n\}$ has sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n} $.
[b]p7.[/b] Let there be a sequence of numbers $x_1, x_2, x_3,...$ such that for all $i$, $$x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.$$ Find the largest value of $n$ such that $$\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.$$
[b]p8.[/b] Let $X$ be a $9$-digit integer that includes all the digits $1$ through $9$ exactly once, such that any $2$-digit number formed from adjacent digits of $X$ is divisible by $7$ or $13$. Find all possible values of $X$.
[b]p9.[/b] Two $2025$-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$ , form the legs of a right triangle. Find the sum of the digits in the hypotenuse.
[b]p10.[/b] Suppose that the side lengths of $\vartriangle ABC$ are positive integers and the perimeter of the triangle is $35$. Let $G$ the centroid and $I$ be the incenter of the triangle. Given that $\angle GIC = 90^o$ , what is the length of $AB$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2013 Macedonian Team Selection Test, Problem 2
a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational.
b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.
2014 CentroAmerican, 3
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
2024 HMIC, 4
Given a positive integer $n$, let $[n] = \{1,2,\dots,n\}$. Let
[list]
[*] $a_n$ denote the number of functions $f: [n] \to [n]$ such that $f(f(i))\ge i$ for all $i$; and
[*] $b_n$ denote the number of ordered set partitions of $[n]$, i.e., the number of ways to pick an integer $k$ and an ordered $k$-tuple of pairwise disjoint nonempty sets $(A_1,\dots,A_k)$ whose union is $[n]$.
[/list]
Prove that $a_n=b_n$.
[i]Derek Liu[/i]
1987 Balkan MO, 1
Let $a$ be a real number and let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(0)=\frac{1}{2}$ and
\[f(x+y)=f(x)f(a-y)+f(y)f(a-x), \quad \forall x,y \in \mathbb{R}.\]
Prove that $f$ is constant.
2015 AIME Problems, 11
The circumcircle of acute $\triangle ABC$ has center $O$. The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$, respectively. Also $AB=5$, $BC=4$, $BQ=4.5$, and $BP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2017 Pakistan TST, Problem 3
Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$
$f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$
2023 Tuymaada Olympiad, 5
A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.
1997 Yugoslav Team Selection Test, Problem 1
Given a natural number $k$, find the smallest natural number $C$ such that
$$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$.
2018 PUMaC Live Round, Calculus 2
Three friends are trying to meet for lunch at a cafe. Each friend will arrive independently at random between $1\!:\!00$ pm and $2\!:\!00$ pm. Each friend will only wait for $5$ minutes by themselves before leaving. However, if another friend arrives within those $5$ minutes, the pair will wait $15$ minutes from the time the second friend arrives. If the probability that the three friends meet for lunch can be expressed in simplest form as $\tfrac{m}{n}$, what is $m+n$?
2014 Harvard-MIT Mathematics Tournament, 12
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
2019 Novosibirsk Oral Olympiad in Geometry, 7
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
2003 Tuymaada Olympiad, 1
Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$
\[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\]
\[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\]
[i]Proposed by A. Khrabrov[/i]
2019 Belarusian National Olympiad, 11.3
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.
Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$.
[i](I. Gorodnin)[/i]
2002 SNSB Admission, 6
Find a Galois extension of the field $ \mathbb{Q} $ whose Galois group is isomorphic with $ \mathbb{Z}/3\mathbb{Z} . $
2012 ELMO Shortlist, 5
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
2014 BMT Spring, 10
Suppose that $x^3-x+10^{-6}=0$. Suppose that $x_1<x_2<x_3$ are the solutions for $x$. Find the integers $(a,b,c)$ closest to $10^8x_1$, $10^8x_2$, and $10^8x_3$ respectively.
2018 Germany Team Selection Test, 2
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
2004 Tournament Of Towns, 7
Let AOB and COD be angles which can be identified by a rotation of the plane (such that rays OA and OC are identified). A circle is inscribed in each of these angles; these circles intersect at points E and F. Show that angles AOE and DOF are equal.
2017 Miklós Schweitzer, 6
Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$. Show that if $f:I\to J$ is a continuous solution of the functional inequality
$$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),$$then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.
2004 Harvard-MIT Mathematics Tournament, 3
A swimming pool is in the shape of a circle with diameter $60$ ft. The depth varies linearly along the east-west direction from $3$ ft at the shallow end in the east to $15$ ft at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at all along the north-south direction. What is the volume of the pool, in ft$^3$?