Found problems: 85335
Denmark (Mohr) - geometry, 2001.3
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$.
[img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]
2021 Science ON all problems, 2
In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$.
[i] (Andra Elena Mircea)[/i]
2007 China National Olympiad, 1
Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that
\[\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}\]
2014 IMO, 3
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
2019 Iran Team Selection Test, 4
Consider triangle $ABC$ with orthocenter $H$. Let points $M$ and $N$ be the midpoints of segments $BC$ and $AH$. Point $D$ lies on line $MH$ so that $AD\parallel BC$ and point $K$ lies on line $AH$ so that $DNMK$ is cyclic. Points $E$ and $F$ lie on lines $AC$ and $AB$ such that $\angle EHM=\angle C$ and $\angle FHM=\angle B$. Prove that points $D,E,F$ and $K$ lie on a circle.
[i]Proposed by Alireza Dadgarnia[/i]
2012 Switzerland - Final Round, 5
Let n be a natural number. Let $A_1, A_2, . . . , A_k$ be distinct $3$-element subsets of $\{1, 2, . . . , n\}$ such that $|A_i \cap A_j | \ne 1$ for all $1 \le i, j \le k$. Determine all $n$ for which there are $n$ such that these subsets exist.
[hide=original wording of last sentence]Bestimme alle n, fur die es n solche Teilmengen gibt.[/hide]
2002 ITAMO, 1
Find all $3$-digit positive integers that are $34$ times the sum of their digits.
1979 Vietnam National Olympiad, 3
$ABC$ is a triangle. Find a point $X$ on $BC$ such that :
area $ABX$ / area $ACX$ = perimeter $ABX$ / perimeter $ACX$.
1968 AMC 12/AHSME, 35
In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ become arbitrarily close to:
[asy]
size((270));
draw((0,0)--(10,0)..(5,5)..(0,0));
draw((5,0)--(5,5));
draw((9,3)--(1,3)--(1,1)--(9,1)--cycle);
draw((9.9,1)--(.1,1));
label("O", (5,0), S);
label("a", (7.5,0), S);
label("G", (5,1), SE);
label("J", (5,5), N);
label("H", (5,3), NE);
label("E", (1,3), NW);
label("L", (1,1), S);
label("C", (.1,1), W);
label("F", (9,3), NE);
label("M", (9,1), S);
label("D", (9.9,1), E);
[/asy]
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{1}{\sqrt{2}}+\frac{1}{2} \qquad\textbf{(E)}\ \frac{1}{\sqrt{2}}+1$
2009 China Team Selection Test, 1
Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$
1982 Putnam, B3
Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $c$ and $d$ are selected independently at random from the set $\{1,2,\ldots,n\}$. Show that $\lim_{n\to\infty}p_n\sqrt n$ exists and express this limit in the form $r(\sqrt s-t)$, where $s$ and $t$ are integers and $r$ is a rational number.
2018 Harvard-MIT Mathematics Tournament, 8
For how many pairs of sequences of nonnegative integers $(b_1,b_2,\ldots, b_{2018})$ and $(c_1,c_2,\ldots, c_{2018})$ does there exist a sequence of nonnegative integers $(a_0,\ldots, a_{2018})$ with the following properties:
[list]
[*] For $0\leq i\leq 2018,$ $a_i<2^{2018}.$
[*] For $1\leq i \leq 2018, b_i=a_{i-1}+a_i$ and $c_i=a_{i-1}|a_i$;
[/list]
where $|$ denotes the bitwise or operation?
2002 Dutch Mathematical Olympiad, 5
In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.
2001 China Team Selection Test, 2
Let \( \varphi \) be the Euler's totient function.
1. For any given integer \( a > 1 \), does there exist \( l \in \mathbb{N}_+ \) such that for any \( k \in \mathbb{N}_+ \), \( l \mid k \) and \( a^2 \nmid l \), \( \frac{\varphi(k)}{\varphi(l)} \) is a non-negative power of \( a \)?
2. For integer \( x > a \), are there integers \( k_1 \) and \( k_2 \) satisfying:
\[
\varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2).
\]
And these two different \( k_i \) correspond to the same \( l_1 \) and \( l_2 \) as described in (1), yet \( \varphi(l_1) = \varphi(l_2) \).
3. Define \( \#E \) as the number of elements in set \( E \). For integer \( x > a \), let \( V(x) = \#\{v \in \mathbb{N}_+ \mid v = \varphi(k) \leq x\} \) and \( W(x) = \#\{w \in \mathbb{N}_+ \mid w = \varphi(l) \leq x, a^2 \mid l\} \). Compare \( V\left( \frac{x}{a} \right) \) with \( W(x) \).
2016 PUMaC Number Theory A, 3
For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.
2013 ELMO Shortlist, 6
Let $a, b, c$ be positive reals such that $a+b+c=3$. Prove that \[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \][i]Proposed by David Stoner[/i]
2005 China Western Mathematical Olympiad, 4
Given is the positive integer $n > 2$. Real numbers $\mid x_i \mid \leq 1$ ($i = 1, 2, ..., n$) satisfying $\mid \sum_{i=1}^{n}x_i \mid > 1$. Prove that there exists positive integer $k$ such that $\mid \sum_{i=1}^{k}x_i - \sum_{i=k+1}^{n}x_i \mid \leq 1$.
1954 Poland - Second Round, 6
Prove that if $ x_1, x_2, \ldots, x_n $ are angles between $ 0^\circ $ and $ 180^\circ $, and $ n $ is any natural number greater than $ 1 $, then $$
\sin (x_1 + x_2 + \ldots + x_n) < \sin x_1 + \sin x_2 + \ldots + \sin x_n.$$
2021 Balkan MO Shortlist, A6
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xy) = f(x)f(y) + f(f(x + y))$$
holds for all $x, y \in \mathbb{R}$.
2013 May Olympiad, 1
Sofia summed all the page numbers from a book starting at $1$ and getting $2013$. Pablo saw how she did this and realized Sofia skipped a page. How many pages does the book have, and what page did Sofia skip?
DMM Individual Rounds, 2009
[b]p1.[/b] Let $p > 5$ be a prime. It is known that the average of all of the prime numbers that are at least $5$ and at most $p$ is $12$. Find $p$.
[b]p2.[/b] The numbers $1, 2,..., n$ are written down in random order. What is the probability that $n-1$ and $n$ are written next to each other? (Give your answer in term of $n$.)
[b]p3.[/b] The Duke Blue Devils are playing a basketball game at home against the UNC Tar Heels. The Tar Heels score $N$ points and the Blue Devils score $M$ points, where $1 < M,N < 100$. The first digit of $N$ is $a$ and the second digit of $N$ is $b$. It is known that $N = a+b^2$. The first digit of $M$ is $b$ and the second digit of $M$ is $a$. By how many points do the Blue Devils win?
[b]p4.[/b] Let $P(x)$ be a polynomial with integer coefficients. It is known that $P(x)$ gives a remainder of $1$ upon polynomial division by $x + 1$ and a remainder of $2$ upon polynomial division by $x + 2$. Find the remainder when $P(x)$ is divided by $(x + 1)(x + 2)$.
[b]p5.[/b] Dracula starts at the point $(0,9)$ in the plane. Dracula has to pick up buckets of blood from three rivers, in the following order: the Red River, which is the line $y = 10$; the Maroon River, which is the line $y = 0$; and the Slightly Crimson River, which is the line $x = 10$. After visiting all three rivers, Dracula must then bring the buckets of blood to a castle located at $(8,5)$. What is the shortest distance that Dracula can walk to accomplish this goal?
[b]p6.[/b] Thirteen hungry zombies are sitting at a circular table at a restaurant. They have five identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, only those two zombies may reach it. In how many ways can we arrange the plates of food around the circle so that each zombie can reach exactly one plate of food? (All zombies are distinct.)
[b]p7.[/b] Let $R_I$ , $R_{II}$ ,$R_{III}$ ,$R_{IV}$ be areas of the elliptical region $$\frac{(x - 10)^2}{10}+ \frac{(y-31)^2}{31} \le 2009$$ that lie in the first, second, third, and fourth quadrants, respectively. Find $R_I -R_{II} +R_{III} -R_{IV}$ .
[b]p8.[/b] Let $r_1, r_2, r_3$ be the three (not necessarily distinct) solutions to the equation $x^3+4x^2-ax+1 = 0$. If $a$ can be any real number, find the minimum possible value of
$$\left(r_1 +\frac{1}{r_1} \right)^2+ \left(r_2 +\frac{1}{r_2} \right)^2+ \left(r_3 +\frac{1}{r_3} \right)^2$$
[b]p9.[/b] Let $n$ be a positive integer. There exist positive integers $1 = a_1 < a_2 <... < a_n = 2009$ such that the average of any $n - 1$ of elements of $\{a_1, a_2,..., a_n\}$ is a positive integer. Find the maximum possible value of $n$.
[b]p10.[/b] Let $A(0) = (2, 7, 8)$ be an ordered triple. For each $n$, construct $A(n)$ from $A(n - 1)$ by replacing the $k$th position in $A(n - 1)$ by the average (arithmetic mean) of all entries in $A(n - 1)$, where $k \equiv n$ (mod $3$) and $1 \le k \le 3$. For example, $A(1) = \left( \frac{17}{3} , 7, 8 \right)$ and $A(2) = \left( \frac{17}{3} , \frac{62}{9}, 8\right)$. It is known that all entries converge to the same number $N$. Find the value of $N$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1998 Croatia National Olympiad, Problem 4
Among any $79$ consecutive natural numbers there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which the above statement fails.
1974 AMC 12/AHSME, 4
What is the remainder when $x^{51}+51$ is divided by $x+1$?
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 49 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 51 $
2022 Irish Math Olympiad, 6
6. Suppose [i]a[/i], [i]b[/i], [i]c[/i] are real numbers such that [i]a[/i] + [i]b[/i] + [i]c[/i] = 1. Prove that
\[a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.\]
2024 AMC 10, 5
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$