Found problems: 85335
2017 Israel Oral Olympiad, 5
A mink is standing in the center of a field shaped like a regular polygon. The field is surrounded by a fence, and the mink can only exit through the vertices of the polygon. A dog is standing on one of the vertices, and can move along the fence. The mink wants to escape the field, while the dog tries to prevent it. Each of them moves with constant velocity. For what ratio of velocities could the mink escape if:
a. The field is a regular triangle?
b. The field is a square?
2001 SNSB Admission, 4
Let $ p,q $ be the two most distant points (in the Euclidean sense) of a closed surface $ M $ embedded in the Euclidean space.
[b]a)[/b] Show that the tangent planes of $ M $ at $ p $ and $ q $ are parallel.
[b]b)[/b] What happened if $ M $ would be a closed curve of $ \mathcal{C}^{\infty } \left(\mathbb{R}^3\right) $ class, instead?
2022-2023 OMMC, 19
Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$. Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$.
1969 IMO Longlists, 14
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and
$q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$
2013 Germany Team Selection Test, 1
$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.
2010 Indonesia TST, 2
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$.
[i]Hery Susanto, Malang[/i]
2014 Contests, 3
Prove that for every integer $S\ge100$ there exists an integer $P$ for which the following story could hold true:
The mathematician asks the shop owner: ``How much are the table, the cabinet and the bookshelf?'' The shop owner replies: ``Each item costs a positive integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and their product is $P$.''
The mathematician thinks and complains: ``This is not enough information to determine the three prices!''
(Proposed by Gerhard Woeginger, Austria)
1971 AMC 12/AHSME, 30
Given the linear fractional transformation of $x$ into $f_1(x)=\dfrac{2x-1}{x+1}$. Define $f_{n+1}(x)=f_1(f_n(x))$ for $n=1,2,3,\cdots$. Assuming that $f_{35}(x)=f_5(x)$, it follows that $f_{28}(x)$ is equal to
$\textbf{(A) }x\qquad\textbf{(B) }\frac{1}{x}\qquad\textbf{(C) }\frac{x-1}{x}\qquad\textbf{(D) }\frac{1}{1-x}\qquad \textbf{(E) }\text{None of these}$
2019 Peru EGMO TST, 6
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2013 AIME Problems, 10
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.
2023 Math Prize for Girls Olympiad, 3
Let $m$ be the product of the first 100 primes, and let $S$ denote the set of divisors of $m$ greater than 1 (hence $S$ has exactly $2^{100} - 1$ elements). We wish to color each element of $S$ with one of $k$ colors such that
$\ \bullet \ $ every color is used at least once; and
$\ \bullet \ $ any three elements of $S$ whose product is a perfect square have exactly two different colors used among them.
Find, with proof, all values of $k$ for which this coloring is possible.
2022 IFYM, Sozopol, 1
Let $ABC$ be a triangle for which the shortest side is $AC$. Its inscribed circle with center $I$ touches sides $AB$ and $BC$ in points $D$ and $E$ respectively. Point $M$ is the midpoint of $AC$. Points $F$ and $G$ lie on sides $BC$ and $AB$ respectively so that $FC=CA=AG$. The line through $I$ perpendicular to $MI$ intersects the line segments $AF$ and $CG$ in $P$ and $Q$ respectively. Prove that $AB=BC\Leftrightarrow PD=QE$.
2022 IMO Shortlist, C9
Let $\mathbb Z_{\ge 0}$ be the set of non-negative integers, and let $f:\mathbb Z_{\ge 0}\times \mathbb Z_{\ge 0} \to \mathbb Z_{\ge 0}$ be a bijection such that whenever $f(x_1,y_1) > f(x_2, y_2)$, we have $f(x_1+1, y_1) > f(x_2 + 1, y_2)$ and $f(x_1, y_1+1) > f(x_2, y_2+1)$.
Let $N$ be the number of pairs of integers $(x,y)$ with $0\le x,y<100$, such that $f(x,y)$ is odd. Find the smallest and largest possible values of $N$.
2011 Purple Comet Problems, 18
Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$
2007 Puerto Rico Team Selection Test, 6
The geometric mean of a set of $m$ non-negative numbers is the $m$-th root of the product of these numbers. For which positive values of $n$, is there a finite set $S_n$ of $n$ positive integers different such that the geometric mean of any subset of $S_n$ is an integer?
1983 IMO Shortlist, 2
Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$
Prove that there exists an infinity of [i]superabundant[/i] numbers.
2020 LMT Fall, 22
Find the area of a triangle with side lengths $\sqrt{13},\sqrt{29},$ and $\sqrt{34}.$ The area can be expressed as $\frac{m}{n}$ for $m,n$ relatively prime positive integers, then find $m+n.$
[i]Proposed by Kaylee Ji[/i]
2017 CCA Math Bonanza, I13
Toner Drum and Celery Hilton are both running for president. A total of $129$ million people cast their vote in a random order, with exactly $63$ million and $66$ million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page?
[i]2017 CCA Math Bonanza Individual Round #13[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 9.2
Let $A_1,$ $B_1$, $C_1$ be the touchpoints of the circle inscribed in the acute triangle $ABC$ ($A_1$ is the touchpoint with the side $BC$, etc.). Let $A_2$, $B_2$, $C_2$ be the intersection points of the altitudes of triangles $AB_1C_1$, $A_1BC_1$ and $A_1B_1C$ respectively. Prove that the lines $A_1A_2$ and $B_1B_2$ and $C_1C_2$ intersect at one point.
2011 IMO, 4
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.
[i]Proposed by Morteza Saghafian, Iran[/i]
2014 Postal Coaching, 3
The circles $\mathcal{K}_1,\mathcal{K}_2$ and $\mathcal{K}_3$ are pairwise externally tangent to each other; the point of tangency betwwen $\mathcal{K}_1$ and $\mathcal{K}_2$ is $T$. One of the external common tangents of $\mathcal{K}_1$ and $\mathcal{K}_2$ meets $\mathcal{K}_3$ at points $P$ and $Q$. Prove that the internal common tangent of $\mathcal{K}_1$ and $\mathcal{K}_2$ bisects the arc $PQ$ of $\mathcal{K}_3$ which is closer to $T$.
2009 Sharygin Geometry Olympiad, 7
Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.
(A.Glazyrin)
2014 Cuba MO, 7
Find all pairs of integers $(a, b)$ that satisfy the equation
$$(a + 1)(b- 1) = a^2b^2.$$
1993 Czech And Slovak Olympiad IIIA, 2
In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.