Found problems: 85335
2019 AMC 10, 3
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$
1993 AIME Problems, 5
Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?
1999 Romania Team Selection Test, 7
Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that
\[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \]
Give an example of two such progressions having at least five terms.
[i]Mihai Baluna[/i]
2012 Sharygin Geometry Olympiad, 6
Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$.
(V.Yassinsky)
2022 Rioplatense Mathematical Olympiad, 5
The quadrilateral $ABCD$ has the following equality $\angle ABC=\angle BCD=150^{\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\triangle APB,\triangle BQC,\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following equality is true $P(APQRD)=P(ABCD)+32$. Determine the length of the side $CD$.
PEN H Problems, 83
Find all pairs $(a, b)$ of positive integers such that \[(\sqrt[3]{a}+\sqrt[3]{b}-1 )^{2}= 49+20 \sqrt[3]{6}.\]
2007 Peru IMO TST, 3
Let $T$ a set with 2007 points on the plane, without any 3 collinear points.
Let $P$ any point which belongs to $T$.
Prove that the number of triangles that contains the point $P$ inside and
its vertices are from $T$, is even.
1994 Korea National Olympiad, Problem 1
Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that
$ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and
$ g(1)\equal{}h(1)\equal{}1$.
2001 China Team Selection Test, 3
Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$, $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ($c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$).
Prove: $\left| S - 10^k AB \right| \leq 9k.$
2005 Postal Coaching, 13
Let $a_1 < a_2 < .... < a_n < 2n$ ne $n$ positive integers such that $a_j$ does not divide $a_k$ or $j \not= k$. Prove that $a_1 \geq 2^{k}$ where $k$ is defined by the condition $3^{k} < 2n < 3^{k+1}$ and show that it is the best estimate for $a_1$
2021 Iranian Geometry Olympiad, 4
In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.
[i]Proposed by Iman Maghsoudi - Iran[/i]
2013 CHMMC (Fall), 4
The numbers $25$ and $76$ have the property that when squared in base 10, their squares also end in the same two digits. A positive integer that has at most $3$ digits when expressed in base 21 and also has the property that its base $21$ square ends in the same $3$ digits is called amazing. Find the sum of all amazing numbers. Express your answer in base $21$.
2008 Postal Coaching, 4
Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.
1979 AMC 12/AHSME, 4
For all real numbers $x$, $x[x\{x(2-x)-4\}+10]+1=$
$\textbf{(A) }-x^4+2x^3+4x^2+10x+1$
$\textbf{(B) }-x^4-2x^3+4x^2+10x+1$
$\textbf{(C) }-x^4-2x^3-4x^2+10x+1$
$\textbf{(D) }-x^4-2x^3-4x^2-10x+1$
$\textbf{(E) }-x^4+2x^3-4x^2+10x+1$
2012 Cuba MO, 1
If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.
2012 Estonia Team Selection Test, 2
For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold:
(1) $a_i = a_{i+n}$ for any $i$,
(2) $a_i$ is not divisible by $n$ for any $i$,
(3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$.
For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?
Durer Math Competition CD Finals - geometry, 2023.D2
Let $ABCD$ be a isosceles trapezoid. Base $AD$ is $11$ cm long while the other three sides are each $5$ cm long. We draw the line that is perpendicular to $BD$ and contains $C$ and the line that is perpendicular to $AC$ and contains$ B$. We mark the intersection of these two lines with $E$. What is the distance between point $E$ and line $AD$?
1982 IMO Longlists, 24
Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence $a_0, a_1, \ldots$ of descendants (i.e., $a = a_0$ and for all $n \geq 1, a_{n+1}$ is always a child of $a_n$). It is assumed that no-one can have infinitely many children.
[i]Variant 1[/i]. Prove that if $a$ has infinitely many ancestors, then $a$ has an infinite descending sequence of ancestors (i.e., $a_0, a_1, \ldots$ where $a = a_0$ and $a_n$ is always a child of $a_{n+1}$).
[i]Variant 2.[/i] Prove that if someone has infinitely many ancestors, then all people cannot descend from $A(dam)$ and $E(ve)$.
2023 MMATHS, 10
Consider the recurrence relation $x_{n+2}=2x_{n+1}+x_n,$ with $x_0=0, x_1=1.$ What is the greatest common divisor of $x_{2023}$ and $x_{721}$?
PEN P Problems, 4
Determine all positive integers that are expressible in the form \[a^{2}+b^{2}+c^{2}+c,\] where $a$, $b$, $c$ are integers.
2023 BMT, 8
Define a family of functions $S_k(n)$ for positive integers $n$ and $k$ by the following two rules:
$$S_0(n) = 1,$$
$$S_k(n) = \sum_{d | n} dS_{k-1}(d).$$
Compute the remainder when $S_{30}(30)$ is divided by $1001$.
2006 Oral Moscow Geometry Olympiad, 3
On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$.
(M. Volchkevich)
2003 AMC 10, 17
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?
$ \textbf{(A)}\ 2: 1 \qquad
\textbf{(B)}\ 3: 1 \qquad
\textbf{(C)}\ 4: 1 \qquad
\textbf{(D)}\ 16: 3 \qquad
\textbf{(E)}\ 6: 1$
2002 Abels Math Contest (Norwegian MO), 3b
Six line segments of lengths $17, 18, 19, 20, 21$ and $23$ form the side edges of a triangular pyramid (also called a tetrahedron). Can there exist a sphere tangent to all six lines?
2013 NIMO Problems, 4
On side $\overline{AB}$ of square $ABCD$, point $E$ is selected. Points $F$ and $G$ are located on sides $\overline{AB}$ and $\overline{AD}$, respectively, such that $\overline{FG} \perp \overline{CE}$. Let $P$ be the intersection point of segments $\overline{FG}$ and $\overline{CE}$. Given that $[EPF] = 1$, $[EPGA] = 8$, and $[CPFB] = 15$, compute $[PGDC]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)
[i]Proposed by Aaron Lin[/i]