Found problems: 85335
1990 Iran MO (2nd round), 3
We want to cover a rectangular $5 \times 137$ with the following figures, prove that this is impossible.
\[\text{Squars are the same and all are } \Huge{1 \times 1}\]
[asy]
import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);
draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt));
label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle);
[/asy]
2018 Iran MO (2nd Round), 6
Two circles $\omega_1,\omega_2$ intersect at $P,Q $. An arbitrary line passing through $P $ intersects $\omega_1 , \omega_2$ at $A,B $ respectively. Another line parallel to $AB $ intersects $\omega_1$ at $D,F $ and $\omega_2$ at $E,C $ such that $E,F $ lie between $C,D $.Let $X\equiv AD\cap BE $ and $Y\equiv BC\cap AF $. Let $R $ be the reflection of $P $ about $CD$. Prove that:
a. $R $ lies on $XY $.
b. PR is the bisector of $\hat {XPY}$.
2009 Spain Mathematical Olympiad, 6
Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.
2004 All-Russian Olympiad, 1
Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.
2004 Croatia National Olympiad, Problem 4
A frog jumps on the coordinate lattice, starting from the point $(1,1)$, according to the following rules:
(i) From point $(a,b)$ the frog can jump to either $(2a,b)$ or $(a,2b)$;
(ii) If $a>b$, the frog can also jump from $(a,b)$ to $(a-b,b)$, while for $a<b$ it can jump from $(a,b)$ to $(a,b-a)$.
Can the frog get to the point: (a) $(24,40)$; (b) $(40,60)$; (c) $(24,60)$; (d) $(200,4)$?
2020 CMIMC Team, Estimation
Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.
2018 Purple Comet Problems, 24
Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1986 AMC 12/AHSME, 25
If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor = \]
$ \textbf{(A)}\ 8192\qquad\textbf{(B)}\ 8204\qquad\textbf{(C)}\ 9218\qquad\textbf{(D)}\ \lfloor \log_{2}(1024!)\rfloor\qquad\textbf{(E)}\ \text{none of these} $
2017 Iran MO (3rd round), 3
Let $a,b$ and $c$ be positive real numbers. Prove that
$$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$
1996 Estonia Team Selection Test, 1
Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.
2013 JBMO TST - Turkey, 2
[b]a)[/b] Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares.
[b]b)[/b] Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?
2024 India IMOTC, 17
Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that
1. $f$ is a monic polynomial with $\deg f \ge 1$.
2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$.
Find all such possible triples.
[i]Proposed by Mainak Ghosh and Rijul Saini[/i]
2022 AMC 12/AHSME, 15
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2000 Moldova Team Selection Test, 3
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
2005 Today's Calculation Of Integral, 47
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$.
\[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]
2012 AMC 10, 17
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
$ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $
2020 BMT Fall, 20
Non-degenerate quadrilateral $ABCD$ with $AB = AD$ and $BC = CD$ has integer side lengths, and $\angle ABC = \angle BCD = \angle CDA$. If $AB = 3$ and $B \ne D$, how many possible lengths are there for $BC$?
2004 Spain Mathematical Olympiad, Problem 4
Does there exist such a power of ${2}$, that when written in the decimal system its digits are all different than zero and it is possible to reorder the other digits to form another power of ${2}$? Justify your answer.
2017 Miklós Schweitzer, 5
For every non-constant polynomial $p$, let $H_p=\big\{z\in \mathbb{C} \, \big| \, |p(z)|=1\big\}$. Prove that if $H_p=H_q$ for some polynomials $p,q$, then there exists a polynomial $r$ such that $p=r^m$ and $q=\xi\cdot r^n$ for some positive integers $m,n$ and constant $|\xi|=1$.
2017 APMO, 2
Let $ABC$ be a triangle with $AB < AC$. Let $D$ be the intersection point of the internal bisector of angle $BAC$ and the circumcircle of $ABC$. Let $Z$ be the intersection point of the perpendicular bisector of $AC$ with the external bisector of angle $\angle{BAC}$. Prove that the midpoint of the segment $AB$ lies on the circumcircle of triangle $ADZ$.
[i]Olimpiada de Matemáticas, Nicaragua[/i]
1991 AIME Problems, 9
Suppose that $\sec x+\tan x=\frac{22}7$ and that $\csc x+\cot x=\frac mn,$ where $\frac mn$ is in lowest terms. Find $m+n.$
2019 Purple Comet Problems, 20
Harold has $3$ red checkers and $3$ black checkers. Find the number of distinct ways that Harold can place these checkers in stacks. Two ways of stacking checkers are the same if each stack of the rst way matches a corresponding stack in the second way in both size and color arrangement. So, for example, the $3$ stack arrangement $RBR, BR, B$ is distinct from $RBR, RB, B$, but the $4$ stack arrangement $RB, BR, B, R$ is the same as $B, BR, R, RB$.
2017 Vietnamese Southern Summer School contest, Problem 1
Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by:
$$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$
for all positive integers $n$.
1. For $a=0$, prove that $(x_n)$ converges.
2. Determine the largest possible value of $a$ such that $(x_n)$ converges.
2024 CCA Math Bonanza, I12
Given that \[a^2+b^2+c^2+d^2+\frac{5}{4}=e+\sqrt{a+b+c+d-e},\] the value of $a+b+c+d+e$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Individual #12[/i]
2017 CHMMC (Fall), 5
Felix picks four points uniformly at random inside a unit circle $\mathcal{C}$. He then draws the four possible triangles which can be formed using these points as vertices. Finally, he randomly chooses of the six possible pairs of the triangles he just drew.
What is the probability that the center of the circle $\mathcal{C}$ is contained in the union of the interiors of the two triangles that Felix chose?