Found problems: 85335
2016 Mexico National Olmypiad, 1
Let $C_1$ and $C_2$ be two circumferences externally tangents at $S$ such that the radius of $C_2$ is the triple of the radius of $C_1$. Let a line be tangent to $C_1$ at $P \neq S$ and to $C_2$ at $Q \neq S$. Let $T$ be a point on $C_2$ such that $QT$ is diameter of $C_2$. Let the angle bisector of $\angle SQT$ meet $ST$ at $R$. Prove that $QR=RT$
2001 Pan African, 2
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
2020 DMO Stage 1, 1.
[b]Q[/b] Let $p,q,r$ be non negative reals such that $pqr=1$. Find the maximum value for the expression
$$\sum_{cyc} p[r^{4}+q^{4}-p^{4}-p]$$
[i]Proposed by Aritra12[/i]
2021 Romania Team Selection Test, 2
For any positive integer $n>1$, let $p(n)$ be the greatest prime factor of $n$. Find all the triplets of distinct positive integers $(x,y,z)$ which satisfy the following properties: $x,y$ and $z$ form an arithmetic progression, and $p(xyz)\leq 3.$
2012 EGMO, 8
A [i]word[/i] is a finite sequence of letters from some alphabet. A word is [i]repetitive[/i] if it is a concatenation of at least two identical subwords (for example, $ababab$ and $abcabc$ are repetitive, but $ababa$ and $aabb$ are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)
[i]Romania (Dan Schwarz)[/i]
2013 Purple Comet Problems, 1
The diagram shows two congruent isosceles triangles in a $20\times20$ square which has been partitioned into four $10\times10$ squares. Find the area of the shaded region.
[asy]
import graph;
size(4.4cm);
real labelscalefactor = 0.5;
pen dotstyle = black;
fill((-2,5)--(0,1)--(1,3)--(1,5)--cycle,gray);
draw((-3,5)--(1,5), linewidth(2.2));
draw((1,5)--(1,1), linewidth(2.2));
draw((1,1)--(-3,1), linewidth(2.2));
draw((-3,1)--(-3,5), linewidth(2.2));
draw((-1,5)--(-1,1), linewidth(2.2));
draw((-3,3)--(1,3), linewidth(2.2));
draw((-2,5)--(-3,3), linewidth(1.4));
draw((-2,5)--(0,1), linewidth(1.4));
draw((0,1)--(1,3), linewidth(1.4));
draw((-2,5)--(0,1));
draw((0,1)--(1,3));
draw((1,3)--(1,5));
draw((1,5)--(-2,5));[/asy]
2018 Argentina National Olympiad Level 2, 2
There are $n^2$ empty boxes, each with a square base. The height and width of each box are integers between $1$ and $n$ inclusive, and no two boxes are identical. One box [i]fits inside[/i] another if its height and width are both smaller, and additionally, one of its dimensions is at least $2$ units smaller. In this way, we can form sequences of boxes (the first inside the second, the second inside the third, and so on). We place each of these sequences on a different shelf. How many shelves are needed to store all the boxes, with certainty?
MOAA Team Rounds, 2022.8
Raina the frog is playing a game in a circular pond with six lilypads around its perimeter numbered clockwise from $1$ to $6$ (so that pad $1$ is adjacent to pad $6$). She starts at pad $1$, and when she is on pad i, she may jump to one of its two adjacent pads, or any pad labeled with $j$ for which $j - i$ is even. How many jump sequences enable Raina to hop to each pad exactly once?
2016 Irish Math Olympiad, 4
Let $ABC$ be a triangle with $|AC| \ne |BC|$. Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$, so that $P$ is between $A$ and $B$. Prove that if $M$ is any point on the circle with diameter $PQ$, then $\angle AMP = \angle BMP$.
1992 Poland - Second Round, 5
Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.
1999 National High School Mathematics League, 6
Points $A(1,2)$, a line that passes $(5,-2)$ intersects the parabola $y^2=4x$ at two points $B,C$. Then, $\triangle ABC$ is
$\text{(A)}$ an acute triangle
$\text{(B)}$ an obtuse triangle
$\text{(C)}$ a right triangle
$\text{(D)}$ not sure
2012 Indonesia TST, 1
Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$.
(A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)
2021 JBMO TST - Turkey, 8
$w_1$ and $w_2$ circles have different diameters and externally tangent to each other at $X$. Points $A$ and $B$ are on $w_1$, points $C$ and $D$ are on $w_2$ such that $AC$ and $BD$ are common tangent lines of these two circles. $CX$ intersects $AB$ at $E$ and $w_1$ at $F$ second time. $(EFB)$ intersects $AF$ at $G$ second time. If $AX \cap CD =H$, show that points $E, G, H$ are collinear.
2020 BMT Fall, 24
For positive integers $N$ and $m$, where $m \le N$, define $$a_{m,N} =\frac{1}{{N+1 \choose m}} \sum_{i=m-1}^{N-1} \frac{ {i \choose m-1}}{N - i}$$ Compute the smallest positive integer $N$ such that $$\sum^N_{m=1}a_{m,N} >\frac{2020N}{N +1}$$
2013 MTRP Senior, 5
A function f : $R$ $\rightarrow$ $R$ satisfies the property $f(x^2) - f^2(x) \geq 1/4$ for all x.
Verify if the function is one-one.
2018 Costa Rica - Final Round, N4
Let $p$ be a prime number such that $p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1$. Show that $d$ is a prime.
2018 China Team Selection Test, 2
There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$.
2014 China Team Selection Test, 6
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
2010 IMO, 5
Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;
Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.
[i]Proposed by Hans Zantema, Netherlands[/i]
1994 All-Russian Olympiad, 3
Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at A and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that the common chord of the circumcircles of triangles $ABC$ and $BDE$ passes through point $F$.
(A. Kalinin)
Maryland University HSMC part II, 2015
[b]p1.[/b] Nine coins are placed in a row, alternating between heads and tails as follows: $H T H T H T H T H$. A legal move consists of turning over any two adjacent coins.
(a) Give a sequence of legal moves that changes the configuration into $H H H H H H H H H$.
(b) Prove that there is no sequence of legal moves that changes the original configuration into $T T T T T T T T T$.
[b]p2.[/b] Find (with proof) all integers $k $that satisfy the equation
$$\frac{k - 15}{2000}+\frac{k - 12}{2003}+\frac{k - 9}{2006}+\frac{k - 6}{2009}+\frac{k - 3}{2012}
= \frac{k - 2000}{15}+\frac{k - 2003}{12}+\frac{k - 2006}{9}+\frac{k - 2009}{6}+\frac{k - 2012}{3}.$$
[b]p3.[/b] Some (not necessarily distinct) natural numbers from $1$ to $2015$ are written on $2015$ lottery tickets, with exactly one number written on each ticket. It is known that the sum of the numbers on any nonempty subset of tickets (including the set of all tickets) is not divisible by $2016$. Prove that the same number is written on all of the tickets.
[b]p4.[/b] A set of points $A$ is called distance-distinct if every pair of points in $A$ has a different distance.
(a) Show that for all infinite sets of points $B$ on the real line, there exists an infinite distance-distinct set A contained in $B$.
(b) Show that for all infinite sets of points $B$ on the real plane, there exists an infinite distance-distinct set A contained in $B$.
[b]p5.[/b] Let $ABCD$ be a (not necessarily regular) tetrahedron and consider six points $E, F, G, H, I, J$ on its edges $AB$, $BC$, $AC$, $AD$, $BD$, $CD$, respectively, such that $$|AE| \cdot |EB| = |BF| \cdot |FC| = |AG| \cdot |GC| = |AH| \cdot |HD| = |BI| \cdot |ID| = |CJ| \cdot |JD|.$$ Prove that the points $E, F, G, H, I$, and $J$ lie on the surface of a sphere.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2021 Durer Math Competition Finals, 16
Consider a table consisting of $2\times 7$ squares. Each little square is surrounded by walls (each internal wall belongs to two squares). We would like to remove some internal walls to make it possible to get from any square to any other one without crossing walls. How many ways can we do this while removing the minimal possible number of internal walls?
[i]The figure shows a possible configuration, the remaining walls are marked in red, the removed ones are marked in light pink. Two configurations are considered the same if the same walls are removed.[/i]
[img]https://cdn.artofproblemsolving.com/attachments/d/c/1a3d9ab0d0971929e6d484a970d4b1f36f0031.png[/img]
1990 Baltic Way, 1
Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?
1999 Junior Balkan Team Selection Tests - Romania, 1
Let be a natural number $ n. $ Prove that there is a polynomial $ P\in\mathbb{Z} [X,Y] $ such that $ a+b+c=0 $ implies
$$ a^{2n+1}+b^{2n+1}+c^{2n+1}=abc\left( P(a,b)+P(b,c)+P(c,a)\right) $$
[i]Dan Brânzei[/i]
PEN E Problems, 7
Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.