Found problems: 85335
1973 Dutch Mathematical Olympiad, 5
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
2013 Purple Comet Problems, 28
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
[asy]
import graph; size(12.57cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
real xmin = -3.79, xmax = 8.79, ymin = 0.32, ymax = 4.18; /* image dimensions */
pen ffqqtt = rgb(1,0,0.2); pen ffzzzz = rgb(1,0.6,0.6); pen zzzzff = rgb(0.6,0.6,1);
draw((6,3.5)--(8,1.5), zzzzff);
draw((7,3)--(5,1), blue);
draw((6,3.5)--(7,3), blue);
draw((6,3.5)--(5,1), blue);
draw((5,1)--(8,1.5), blue);
draw((7,3)--(8,1.5), blue);
draw((4,3.5)--(2,1.5), ffzzzz);
draw((1,3)--(2,1.5), ffqqtt);
draw((2,1.5)--(3,1), ffqqtt);
draw((1,3)--(3,1), ffqqtt);
draw((4,3.5)--(1,3), ffqqtt);
draw((4,3.5)--(3,1), ffqqtt);
draw((-3,3)--(-3,1), linewidth(1.6));
draw((-3,3)--(-1,3), linewidth(1.6));
draw((-1,3)--(-1,1), linewidth(1.6));
draw((-3,1)--(-1,1), linewidth(1.6));
draw((-3,3)--(-2,3.5), linewidth(1.6));
draw((-2,3.5)--(0,3.5), linewidth(1.6));
draw((0,3.5)--(-1,3), linewidth(1.6));
draw((0,3.5)--(0,1.5), linewidth(1.6));
draw((0,1.5)--(-1,1), linewidth(1.6));
draw((-3,1)--(-2,1.5));
draw((-2,1.5)--(0,1.5));
draw((-2,3.5)--(-2,1.5));
draw((1,3)--(1,1), linewidth(1.6));
draw((1,3)--(3,3), linewidth(1.6));
draw((3,3)--(3,1), linewidth(1.6));
draw((1,1)--(3,1), linewidth(1.6));
draw((1,3)--(2,3.5), linewidth(1.6));
draw((2,3.5)--(4,3.5), linewidth(1.6));
draw((4,3.5)--(3,3), linewidth(1.6));
draw((4,3.5)--(4,1.5), linewidth(1.6));
draw((4,1.5)--(3,1), linewidth(1.6));
draw((1,1)--(2,1.5));
draw((2,3.5)--(2,1.5));
draw((2,1.5)--(4,1.5));
draw((5,3)--(5,1), linewidth(1.6));
draw((5,3)--(6,3.5), linewidth(1.6));
draw((5,3)--(7,3), linewidth(1.6));
draw((7,3)--(7,1), linewidth(1.6));
draw((5,1)--(7,1), linewidth(1.6));
draw((6,3.5)--(8,3.5), linewidth(1.6));
draw((7,3)--(8,3.5), linewidth(1.6));
draw((7,1)--(8,1.5));
draw((5,1)--(6,1.5));
draw((6,3.5)--(6,1.5));
draw((6,1.5)--(8,1.5));
draw((8,3.5)--(8,1.5), linewidth(1.6));
label("$ A $",(-3.4,3.41),SE*labelscalefactor);
label("$ D $",(-2.16,4.05),SE*labelscalefactor);
label("$ H $",(-2.39,1.9),SE*labelscalefactor);
label("$ E $",(-3.4,1.13),SE*labelscalefactor);
label("$ F $",(-1.08,0.93),SE*labelscalefactor);
label("$ G $",(0.12,1.76),SE*labelscalefactor);
label("$ B $",(-0.88,3.05),SE*labelscalefactor);
label("$ C $",(0.17,3.85),SE*labelscalefactor);
label("$ A $",(0.73,3.5),SE*labelscalefactor);
label("$ B $",(3.07,3.08),SE*labelscalefactor);
label("$ C $",(4.12,3.93),SE*labelscalefactor);
label("$ D $",(1.69,4.07),SE*labelscalefactor);
label("$ E $",(0.60,1.15),SE*labelscalefactor);
label("$ F $",(2.96,0.95),SE*labelscalefactor);
label("$ G $",(4.12,1.67),SE*labelscalefactor);
label("$ H $",(1.55,1.82),SE*labelscalefactor);
label("$ A $",(4.71,3.47),SE*labelscalefactor);
label("$ B $",(7.14,3.10),SE*labelscalefactor);
label("$ C $",(8.14,3.82),SE*labelscalefactor);
label("$ D $",(5.78,4.08),SE*labelscalefactor);
label("$ E $",(4.6,1.13),SE*labelscalefactor);
label("$ F $",(6.93,0.96),SE*labelscalefactor);
label("$ G $",(8.07,1.64),SE*labelscalefactor);
label("$ H $",(5.65,1.90),SE*labelscalefactor);
dot((-3,3),dotstyle);
dot((-3,1),dotstyle);
dot((-1,3),dotstyle);
dot((-1,1),dotstyle);
dot((-2,3.5),dotstyle);
dot((0,3.5),dotstyle);
dot((0,1.5),dotstyle);
dot((-2,1.5),dotstyle);
dot((1,3),dotstyle);
dot((1,1),dotstyle);
dot((3,3),dotstyle);
dot((3,1),dotstyle);
dot((2,3.5),dotstyle);
dot((4,3.5),dotstyle);
dot((4,1.5),dotstyle);
dot((2,1.5),dotstyle);
dot((5,3),dotstyle);
dot((5,1),dotstyle);
dot((6,3.5),dotstyle);
dot((7,3),dotstyle);
dot((7,1),dotstyle);
dot((8,3.5),dotstyle);
dot((8,1.5),dotstyle);
dot((6,1.5),dotstyle); [/asy]
2018 MOAA, 10
Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.
II Soros Olympiad 1995 - 96 (Russia), 10.7
Three straight lines $\ell_1$, $\ell_2$ and $\ell_3$, forming a triangle, divide the plane into $7$ parts. Each of the points $M_1$, $M_2$ and $M_3$ lies in one of the angles, vertical to some angle of the triangle. The distance from $M_1$ to straight lines $\ell_1$, $\ell_2$ and $\ell_3$ are equal to $7,3$ and $1$ respectively The distance from $M_2$ to the same lines are $4$, $1$ and $3$ respectively. For $M_3$ these distances are $3$, $5$ and $2$. What is the radius of the circle inscribed in the triangle?
[hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.[/hide]
2002 Moldova National Olympiad, 4
Let $ ABCD$ be a convex quadrilateral and let $ N$ on side $ AD$ and $ M$ on side $ BC$ be points such that $ \dfrac{AN}{ND}\equal{}\dfrac{BM}{MC}$. The lines $ AM$ and $ BN$ intersect at $ P$, while the lines $ CN$ and $ DM$ intersect at $ Q$. Prove that if $ S_{ABP}\plus{}S_{CDQ}\equal{}S_{MNPQ}$, then either $ AD\parallel BC$ or $ N$ is the midpoint of $ DA$.
1990 AIME Problems, 5
Let $n$ be the smallest positive integer that is a multiple of $75$ and has exactly $75$ positive integral divisors, including $1$ and itself. Find $n/75$.
1957 AMC 12/AHSME, 17
A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:
$ \textbf{(A)}\ 24\text{ in.}\qquad
\textbf{(B)}\ 12\text{ in.}\qquad
\textbf{(C)}\ 30\text{ in.}\qquad
\textbf{(D)}\ 18\text{ in.}\qquad
\textbf{(E)}\ 36\text{ in.}$
2019 Romania Team Selection Test, 2
Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $
[i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]
2017 Brazil National Olympiad, 4.
[b]4.[/b] We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules:
[i]i.[/i] The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3).
[i]ii.[/i] Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet.
[i]iii.[/i] If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used.
[i]iv.[/i] Every password has at least four digits.
Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password.
Determine the smallest $n (n \geq 4)$ such that, given any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order.
2007 IMC, 1
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.
2020-2021 Winter SDPC, #5
Suppose that the positive divisors of a positive integer $n$ are $1=d_1<d_2<\ldots<d_k=n$, where $k \geq 5$. Given that $k \leq 1000$ and $n={d_2}^{d_3}{d_4}^{d_5}$, compute, with proof, all possible values of $k$.
2018 Kyiv Mathematical Festival, 1
A square of size $2\times2$ with one of its cells occupied by a tower is called a castle. What maximal number of castles one can place on a board of size $7\times7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?
2001 Poland - Second Round, 2
In a triangle $ABC$, $I$ is the incentre and $D$ the intersection point of $AI$ and $BC$. Show that $AI+CD=AC$ if and only if $\angle B=60^{\circ}+\frac{_1}{^3}\angle C$.
2002 Moldova National Olympiad, 1
Integers $ a_1,a_2,\ldots a_9$ satisfy the relations $ a_{k\plus{}1}\equal{}a_k^3\plus{}a_k^2\plus{}a_k\plus{}2$ for $ k\equal{}1,2,...,8$. Prove that among these numbers there exist three with a common divisor greater than $ 1$.
1975 Chisinau City MO, 103
Prove the inequality: $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}$$
2018 Taiwan TST Round 1, 2
In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.
1999 China Team Selection Test, 3
For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau \equal{} (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|$. Let $ x_{11} \equal{} x_1$. Find
[b]I.[/b] The maximum and minimum values of $ S(\tau)$.
[b]II.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its maximum.
[b]III.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.
2018 China Team Selection Test, 4
Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.
2006 JBMO ShortLists, 7
Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2$.
2015 Postal Coaching, Problem 4
Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of
$$\frac{KM + LN}{AC + BD}$$
.
1989 AMC 12/AHSME, 11
Hi guys,
I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this:
1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though.
2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary.
3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions:
A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh?
B. Do NOT go back to the previous problem(s). This causes too much confusion.
C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for.
4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving!
Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D
2001 India IMO Training Camp, 3
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:
\[AL+BM+CN \leq 3(AD+BE+CF)\]
When does equality occur?
1896 Eotvos Mathematical Competition, 2
Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$ imply the equation $xy-12x+15y=0$.
2004 Pre-Preparation Course Examination, 1
A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that:
1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$.
2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\]
a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have:
\[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\]
b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.
2019 Romania National Olympiad, 4
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$