Found problems: 85335
Kvant 2022, M2717
In an acute triangle $ABC$ the heights $AD, BE$ and $CF$ intersecting at $H{}$. Let $O{}$ be the circumcenter of the triangle $ABC$. The tangents to the circle $(ABC)$ drawn at $B{}$ and $C{}$ intersect at $T{}$. Let $K{}$ and $L{}$ be symmetric to $O{}$ with respect to $AB$ and $AC$ respectively. The circles $(DFK)$ and $(DEL)$ intersect at a point $P{}$ different from $D{}$. Prove that $P, D$ and $T{}$ lie on the same line.
[i]Proposed by Don Luu (Vietnam)[/i]
1951 AMC 12/AHSME, 19
A six place number is formed by repeating a three place number; for example, $ 256256$ or $ 678678$, etc. Any number of this form is always exactly divisible by:
$ \textbf{(A)}\ 7 \text{ only} \qquad\textbf{(B)}\ 11 \text{ only} \qquad\textbf{(C)}\ 13 \text{ only} \qquad\textbf{(D)}\ 101 \qquad\textbf{(E)}\ 1001$
2007 AMC 8, 23
What is the area of the shaded pinwheel shown in the $5\times 5$ grid?
[asy]
filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black);
int i;
for(i=0; i<6; i=i+1) {
draw((i,0)--(i,5));
draw((0,i)--(5,i));
}[/asy]
$\textbf{(A)}\: 4\qquad \textbf{(B)}\: 6\qquad \textbf{(C)}\: 8\qquad \textbf{(D)}\: 10\qquad \textbf{(E)}\: 12$
2008 District Olympiad, 2
Consider the positive reals $ x$, $ y$ and $ z$. Prove that:
a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$.
b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.
1986 All Soviet Union Mathematical Olympiad, 421
Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for
a) $n=6$
b) $n=1986$ ?
2008 iTest Tournament of Champions, 3
Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by
\begin{align*}
f(0) &= 1\\
f(2x) &= \lfloor\phi f(x)\rfloor\\
f(2x+1) &= f(2x) + f(x).
\end{align*}
Find the remainder when $f(2007)$ is divided by $2008$.
2017 CCA Math Bonanza, L5.3
How many ways are there to fill a $3\times3\times6$ rectangular prism with $1\times1\times2$ blocks? Rotations are not distinct. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\max\left(2\left(1-\left|\frac{C-A}{C}\right|\right),0\right)$.
[i]2017 CCA Math Bonanza Lightning Round #5.3[/i]
2019 Caucasus Mathematical Olympiad, 1
In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?
2009 Postal Coaching, 5
Let $ABCD$ be a quadrilateral that has an incircle with centre $O$ and radius $r$. Let $P = AB \cap CD$, $Q = AD \cap BC$, $E = AC \cap BD$. Show that $OE \cdot d = r^2$, where $d$ is the distance of $O$ from $PQ$.
2005 Mediterranean Mathematics Olympiad, 4
Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$.
Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations
1963 German National Olympiad, 3
It has to be proven: If at least two of the real numbers $a, b, c$ are different from zero, then the inequality holds
$$\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32$$
Under what conditions does equality occur?
2022 MMATHS, 12
Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$, $BC = 21$, and $CA = 17$. Points $D$ and E lie on side $BC$ such that $BD = 4$, $DE = 6$, and $EC = 11$. The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$, and line $IP$ meets the altitude from $A$ to $BC$ at $X$. Find $(DX \cdot EX)^2$.
1988 Mexico National Olympiad, 8
Compute the volume of a regular octahedron circumscribed about a sphere of radius $1$.
2012 USA Team Selection Test, 3
Determine all positive integers $n$, $n\ge2$, such that the following statement is true:
If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.
KoMaL A Problems 2020/2021, A. 794
A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$.
How many caterpillars of area $n=10^{6}+1$ are there?
Proposed by Nikolai Beluhov, Bulgaria
2020 Taiwan TST Round 2, 5
A finite set $K$ consists of at least 3 distinct positive integers. Suppose that $K$ can be partitioned into two nonempty subsets $A,B\in K$ such that $ab+1$ is always a perfect square whenever $a\in A$ and $b\in B$. Prove that
\[\max_{k\in K}k\geq \left\lfloor (2+\sqrt{3})^{\min\{|A|,|B|\}-1}\right\rfloor+1,\]where $|X|$ stands for the cartinality of the set $X$, and for $x\in \mathbb{R}$, $\lfloor x\rfloor$ is the greatest integer that does not exceed $x$.
IV Soros Olympiad 1997 - 98 (Russia), 9.1
Through vertices $A$ and $B$ of the unit square $ABCD$ , passes a circle intersecting lines $AD$ and $AC$ at points $K$ and $M$, other than $A$. Find the length of the projection $KM$ onto $AC$.
2014 Contests, 2
Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that
(a) $M$ is the midpoint of $AB$;
(b) $N$ is the midpoint of $AC$;
(c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$.
Prove that $\angle APM = \angle PBA$.
2025 District Olympiad, P3
Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$
[list=a]
[*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$.
[*] Give an example of function which satisfies the hypothesis.
2005 Purple Comet Problems, 4
A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?
2010 Belarus Team Selection Test, 3.1
Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that
a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$
b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$
(D. Pirshtuk)
1995 Chile National Olympiad, 2
In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area.
[img]https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg[/img]
2002 HKIMO Preliminary Selection Contest, 20
A rectangular piece of paper has integer side lengths. The paper is folded so that a pair of diagonally opposite vertices coincide, and it is found that the crease is of length 65. Find a possible value of the perimeter of the paper.
2013 Stanford Mathematics Tournament, 20
Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)
2021 AIME Problems, 15
Let $f(n)$ and $g(n)$ be functions satisfying
$$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$and
$$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$.