Found problems: 85335
1976 IMO Longlists, 9
Find all (real) solutions of the system
\[3x_1-x_2-x_3-x_5 = 0,\]\[-x_1+3x_2-x_4-x_6= 0,\]\[-x_1 + 3x_3 - x_4 - x_7 = 0,\]\[-x_2 - x_3 + 3x_4 - x_8 = 0,\]\[-x_1 + 3x_5 - x_6 - x_7 = 0,\]\[-x_2 - x_5 + 3x_6 - x_8 = 0,\]\[-x_3 - x_5 + 3x_7 - x_8 = 0,\]\[-x_4 - x_6 - x_7 + 3x_8 = 0.\]
2012 Sharygin Geometry Olympiad, 1
Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$. Let $N$ be the reflection of $M$ in $BC$. The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$. Determine the value of $\angle AKC$.
(A.Blinkov)
1993 IMO Shortlist, 4
Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$
$n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$
2024 Iran MO (3rd Round), 2
For all positive integers $n$ Prove that one can find pairwise coprime integers $a,b,c>n$ such that the set of prime divisors of the numbers $a+b+c$ and $ab+bc+ac$ coincides.
Proposed by [i]Mohsen Jamali[/i] and [i]Hesam Rajabzadeh[/i]
1953 AMC 12/AHSME, 43
If the price of an article is increased by percent $ p$, then the decrease in percent of sales must not exceed $ d$ in order to yield the same income. The value of $ d$ is:
$ \textbf{(A)}\ \frac{1}{1\plus{}p} \qquad\textbf{(B)}\ \frac{1}{1\minus{}p} \qquad\textbf{(C)}\ \frac{p}{1\plus{}p} \qquad\textbf{(D)}\ \frac{p}{p\minus{}1} \qquad\textbf{(E)}\ \frac{1\minus{}p}{1\plus{}p}$
LMT Guts Rounds, 2
If you increase a number $X$ by $20\%,$ you get $Y.$ By what percent must you decrease $Y$ to get $X?$
2020 Princeton University Math Competition, A1
Let $a_1, . . . , a_{2020}$ be a sequence of real numbers such that $a_1 = 2^{-2019}$, and $a^2_{n-1}a_n = a_n-a_{n-1}$.
Prove that $a_{2020} <\frac{1}{2^{2019} -1}$
2010 Mathcenter Contest, 5
The set $X$ of integers is called [i]good[/i] If for each pair $a,b\in X$ , only one of $a+b,\mid a-b\mid$ is a member of $X$ ($a,b$ may be equal). Find the total number of sets with $2008$ as member.
[i](tatari/nightmare)[/i]
2008 IMC, 4
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?
2006 Indonesia MO, 2
Let $ a,b,c$ be positive integers. If $ 30|a\plus{}b\plus{}c$, prove that $ 30|a^5\plus{}b^5\plus{}c^5$.
2009 Putnam, A3
Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\
\cos4 & \cos5 & \cos 6 \\
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.)
Evaluate $ \lim_{n\to\infty}d_n.$
2020 LMT Fall, 17
In a regular square room of side length $2\sqrt{2}$ ft, two cats that can see $2$ feet ahead of them are randomly placed into the four corners such that they do not share the same corner. If the probability that they don't see the mouse, also placed randomly into the room can be expressed as $\frac{a-b\pi}{c},$ where $a,b,c$ are positive integers with a greatest common factor of $1,$ then find $a+b+c.$
[i]Proposed by Ada Tsui[/i]
2002 Olympic Revenge, 1
Show that there is no function \(f:\mathbb{N}^* \rightarrow \mathbb{N}^*\) such that \(f^n(n)=n+1\) for all \(n\) (when \(f^n\) is the \(n\)th iteration of \(f\))
2005 Germany Team Selection Test, 1
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
2022 BMT, 10
In triangle $\vartriangle ABC$, $E$ and $F$ are the feet of the altitudes from $B$ to $\overline{AC}$ and $C$ to $\overline{AB}$, respectively. Line $\overleftrightarrow{BC}$ and the line through $A$ tangent to the circumcircle of $ABC$ intersect at $X$. Let $Y$ be the intersection of line $\overleftrightarrow{EF}$ and the line through $A$ parallel to $\overline{BC}$. If $XB = 4$, $BC = 8$, and $EF = 4\sqrt3$, compute $XY$.
2017 BMT Spring, 7
Determine the maximal area triangle such that all of its vertices satisfy $\frac{x^2}{9} + \frac{y^2}{16} = 1$.
2023 IMO, 6
Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and
$$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$
Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$
Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$
and $CC_1C_2$ all pass through two common points.
(Note: a scalene triangle is one where no two sides have equal length.)
[i]Proposed by Ankan Bhattacharya, USA[/i]
2019 Peru Cono Sur TST, P5
Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.
[b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal.
[b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.
1978 IMO Longlists, 28
Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy
\[c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0\]
Prove that:
$(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$;
$(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$).
Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.
2013 Tournament of Towns, 2
Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.
1971 AMC 12/AHSME, 16
After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The ratio of the second average to the true average was
$\textbf{(A) }1:1\qquad\textbf{(B) }35:36\qquad\textbf{(C) }36:35\qquad\textbf{(D) }2:1\qquad \textbf{(E) }\text{None of these}$
1992 AMC 12/AHSME, 9
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is
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draw((1,0)--(2,sqrt(3))--(3,0)--(4,sqrt(3))--(5,0));
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$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 10\sqrt{3}\qquad\textbf{(E)}\ 12\sqrt{3} $
1983 AIME Problems, 8
What is the largest 2-digit prime factor of the integer $n = \binom{200}{100}$?
EMCC Team Rounds, 2017
[b]p1.[/b] Compute $2017 + 7201 + 1720 + 172$.
[b]p2. [/b]A number is called [i]downhill [/i]if its digits are distinct and in descending order. (For example, $653$ and $8762$ are downhill numbers, but $97721$ is not.) What is the smallest downhill number greater than 86432?
[b]p3.[/b] Each vertex of a unit cube is sliced off by a planar cut passing through the midpoints of the three edges containing that vertex. What is the ratio of the number of edges to the number of faces of the resulting solid?
[b]p4.[/b] In a square with side length $5$, the four points that divide each side into five equal segments are marked. Including the vertices, there are $20$ marked points in total on the boundary of the square. A pair of distinct points $A$ and $B$ are chosen randomly among the $20$ points. Compute the probability that $AB = 5$.
[b]p5.[/b] A positive two-digit integer is one less than five times the sum of its digits. Find the sum of all possible such integers.
[b]p6.[/b] Let $$f(x) = 5^{4^{3^{2^{x}}}}.$$ Determine the greatest possible value of $L$ such that $f(x) > L$ for all real numbers $x$.
[b]p7.[/b] If $\overline{AAAA}+\overline{BB} = \overline{ABCD}$ for some distinct base-$10$ digits $A, B, C, D$ that are consecutive in some order, determine the value of $ABCD$. (The notation $\overline{ABCD}$ refers to the four-digit integer with thousands digit $A$, hundreds digit $B$, tens digit $C$, and units digit $D$.)
[b]p8.[/b] A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube?
[b]p9.[/b] Define $\lfloor x \rfloor$ as the greatest integer less than or equal to x, and ${x} = x - \lfloor x \rfloor$ as the fractional part of $x$. If $\lfloor x^2 \rfloor =2 \lfloor x \rfloor$ and $\{x^2\} =\frac12 \{x\}$, determine all possible values of $x$.
[b]p10.[/b] Find the largest integer $N > 1$ such that it is impossible to divide an equilateral triangle of side length $ 1$ into $N$ smaller equilateral triangles (of possibly different sizes).
[b]p11.[/b] Let $f$ and $g$ be two quadratic polynomials. Suppose that $f$ has zeroes $2$ and $7$, $g$ has zeroes $1$ and $ 8$, and $f - g$ has zeroes $4$ and $5$. What is the product of the zeroes of the polynomial $f + g$?
[b]p12.[/b] In square $PQRS$, points $A, B, C, D, E$, and $F$ are chosen on segments $PQ$, $QR$, $PR$, $RS$, $SP$, and $PR$, respectively, such that $ABCDEF$ is a regular hexagon. Find the ratio of the area of $ABCDEF$ to the area of $PQRS$.
[b]p13.[/b] For positive integers $m$ and $n$, define $f(m, n)$ to be the number of ways to distribute $m$ identical candies to $n$ distinct children so that the number of candies that any two children receive differ by at most $1$. Find the number of positive integers n satisfying the equation $f(2017, n) = f(7102, n)$.
[b]p14.[/b] Suppose that real numbers $x$ and $y$ satisfy the equation $$x^4 + 2x^2y^2 + y^4 - 2x^2 + 32xy - 2y^2 + 49 = 0.$$ Find the maximum possible value of $\frac{y}{x}$.
[b]p15.[/b] A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC, AC, AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1972 USAMO, 4
Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]