Found problems: 85335
2009 Korea National Olympiad, 1
Let $I, O$ be the incenter and the circumcenter of triangle $ABC$, and $D,E,F$ be the circumcenters of triangle $ BIC, CIA, AIB$. Let $ P, Q, R$ be the midpoints of segments $ DI, EI, FI $. Prove that the circumcenter of triangle $PQR $, $M$, is the midpoint of segment $IO$.
2005 Korea Junior Math Olympiad, 5
In $\triangle ABC$, let the bisector of $\angle BAC$ hit the circumcircle at $M$. Let $P$ be the intersection of $CM$ and $AB$. Denote by $(V,WX,YZ)$ the intersection of the line passing $V$ perpendicular to $WX$ with the line $YZ$. Prove that the points $(P,AM,AC), (P,AC,AM), (P,BC,MB)$ are collinear.
[hide=Restatement]In isosceles triangle $APX$ with $AP=AX$, select a point $M$ on the altitude. $PM$ intersects $AX$ at $C$. The circumcircle of $ACM$ intersects $AP$ at $B$. A line passing through $P$ perpendicular to $BC$ intersects $MB$ at $Z$. Show that $XZ$ is perpendicular to $AP$.[/hide]
2010 CHMMC Fall, 5
The three positive integers $a, b, c$ satisfy the equalities $gcd(ab, c^2) = 20$, $gcd(ac, b^2) = 18$, and $gcd(bc, a^2) = 75$. Compute the minimum possible value of $a + b + c$.
2022 Indonesia MO, 1
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have
\[ f(f(f(x)) + f(y)) = f(y) - f(x) \]
ABMC Speed Rounds, 2023
[i]25 problems for 30 minutes[/i]
[b]p1.[/b] Compute $2^2 + 0 \cdot 0 + 2^2 + 3^3$.
[b]p2.[/b] How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey?
[b]p3.[/b] What is the remainder when $20232023$ is divided by $50$?
[b]p4.[/b] Let $ABCD$ be a square. The fraction of the area of $ABCD$ that is the area of the intersection of triangles $ABD$ and $ABC$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$.
[b]p5.[/b] Raymond is playing basketball. He makes a total of $15$ shots, all of which are either worth $2$ or $3$ points. Given he scored a total of $40$ points, how many $2$-point shots did he make?
[b]p6.[/b] If a fair coin is flipped $4$ times, the probability that it lands on heads more often than tails is $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$.
[b]p7.[/b] What is the sum of the perfect square divisors of $640$?
[b]p8.[/b] A regular hexagon and an equilateral triangle have the same perimeter. The ratio of the area between the hexagon and equilateral triangle can be expressed in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
[b]p9.[/b] If a cylinder has volume $1024\pi$, radius of $r$ and height $h$, how many ordered pairs of integers $(r, h)$ are possible?
[b]p10.[/b] Pump $A$ can fill up a balloon in $3$ hours, while pump $B$ can fill up a balloon in $5$ hours. Pump $A$ starts filling up a balloon at $12:00$ PM, and pump $B$ is added alongside pump $A$ at a later time. If the balloon is completely filled at $2:00$ PM, how many minutes after $12:00$ PM was Pump $B$ added?
[b]p11.[/b] For some positive integer $k$, the product $81 \cdot k$ has $20$ factors. Find the smallest possible value of $k$.
[b]p12.[/b] Two people wish to sit in a row of fifty chairs. How many ways can they sit in the chairs if they do not want to sit directly next to each other and they do not want to sit with exactly one empty chair between them?
[b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $2$ and $M$ be the midpoint of $BC$. Let $P$ be a point in the same plane such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}-b$, where $a$ and $b$ are positive integers such that $a$ is not divisible by any perfect square aside from $1$. Find $a + b$.
[b]p14.[/b] What are the $2022$nd to $2024$th digits after the decimal point in the decimal expansion of $\frac{1}{27}$ , expressed as a $3$ digit number in that order (i.e the $2022$nd digit is the hundreds digit, $2023$rd digit is the tens digit, and $2024$th digit is the ones digit)?
[b]p15.[/b] After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$?
[b]p16.[/b] Let $ABCD$ be a trapezoid with $AB \parallel CD$ where $AB > CD$, $\angle B = 90^o$, and $BC = 12$. A line $k$ is dropped from $A$, perpendicular to line $CD$, and another line $\ell$ is dropped from $C$, perpendicular to line $AD$. $k$ and $\ell$ intersect at $X$. If $\vartriangle AXC$ is an equilateral triangle, the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers such that $n$ is not divisible by any perfect square aside from $1$. Find $m + n$.
[b]p17.[/b] If real numbers $x$ and $y$ satisfy $2x^2 + y^2 = 8x$, maximize the expression $x^2 + y^2 + 4x$.
[b]p18.[/b] Let $f(x)$ be a monic quadratic polynomial with nonzero real coefficients. Given that the minimum value of $f(x)$ is one of the roots of $f(x)$, and that $f(2022) = 1$, there are two possible values of $f(2023)$. Find the larger of these two values.
[b]p19.[/b] I am thinking of a positive integer. After realizing that it is four more than a multiple of $3$, four less than a multiple of $4$, four more than a multiple of 5, and four less than a multiple of $7$, I forgot my number. What is the smallest possible value of my number?
[b]p20.[/b] How many ways can Aston, Bryan, Cindy, Daniel, and Evan occupy a row of $14$ chairs such that none of them are sitting next to each other?
[b]p21.[/b] Let $x$ be a positive real number. The minimum value of $\frac{1}{x^2} +\sqrt{x}$ can be expressed in the form \frac{a}{b^{(c/d)}} , where $a$, $b$, $c$, $d$ are all positive integers, $a$ and $b$ are relatively prime, $c$ and $d$ are relatively prime, and $b$ is not divisible by any perfect square. Find $a + b + c + d$.
[b]p22.[/b] For all $x > 0$, the function $f(x)$ is defined as $\lfloor x \rfloor \cdot (x + \{x\})$. There are $24$ possible $x$ such that $f(x)$ is an integer between $2000$ and $2023$, inclusive. If the sum of these $24$ numbers equals $N$, then find $\lfloor N \rfloor$.
Note: Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, called the floor function. Also, $\{x\}$ is defined as $x - \lfloor x \rfloor$, called the fractional part function.
[b]p23.[/b] Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$.
Note: $[ABCD]$ denotes the area of the polygon $ABCD$.
[b]p24.[/b] Alice writes all of the three-digit numbers on a blackboard (it’s a pretty big blackboard). Let $X_a$ be the set of three-digit numbers containing a somewhere in its representation, where a is a string of digits. (For example, $X_{12}$ would include $12$, $121$, $312$, etc.) If Bob then picks a value of $a$ at random so $0 \le a \le 999$, the expected number of elements in $X_a$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find$ m + n$.
[b]p25.[/b] Let $f(x) = x^5 + 2x^4 - 2x^3 + 4x^2 + 5x + 6$ and $g(x) = x^4 - x^3 + x^2 - x + 1$. If $a$, $b$, $c$, $d$ are the roots of $g(x)$, then find $f(a) + f(b) + f(c) + f(d)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2005 Harvard-MIT Mathematics Tournament, 3
Let $ABCD$ be a rectangle with area $1$, and let $E$ lie on side $CD$. What is the area of the triangle formed by the centroids of triangles $ABE$, $BCE$, and $ADE$?
2013 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle which it not right-angled. De fine a sequence of triangles $A_iB_iC_i$, with $i \ge 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$, $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$,$C_iA_i$,$A_iB_i$, respectively.
Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$. Prove that $\angle A = 60^{\circ}$.
2019 PUMaC Algebra A, 7
A doubly-indexed sequence $a_{m,n}$, for $m$ and $n$ nonnegative integers, is defined as follows:
[list]
[*]$a_{m,0}=0$ for all $m>0$ and $a_{0,0}=1$.
[*]$a_{m,1}=0$ for all $m>1$, $a_{1,1}=1$, and $a_{0,1}=0$.
[*]$a_{0,n}=a_{0,n-1}+a_{0,n-2}$ for all $n\geq 2$.
[*]$a_{m,n}=a_{m,n-1}+a_{m,n-2}+a_{m-1,n-1}-a_{m-1,n-2}$ for all $m>0$, $n\geq 2$.
[/list]
Then there exists a unique value of $x$ so $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{a_{m,n}x^m}{3^{n-m}}=1$. Find $\lfloor 1000x^2 \rfloor$.
2009 Kazakhstan National Olympiad, 2
In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$.
Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.
2008 Princeton University Math Competition, B1
Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange?
2016 IMO Shortlist, A2
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
2013 Stanford Mathematics Tournament, 7
Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\tfrac12$ the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?
1953 Putnam, A1
Prove that for every positive integer $n$
$$ \frac{2}{3} n \sqrt{n} < \sqrt{1} + \sqrt{2} +\ldots +\sqrt{n} < \frac{4n+3}{6} \sqrt{n}.$$
1993 Brazil National Olympiad, 5
Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.
2013 Tournament of Towns, 6
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.
2023 Turkey Olympic Revenge, 6
In triangle $ABC$, $D$ is a variable point on line $BC$. Points $E,F$ are on segments $AC, AB$ respectively such that $BF=BD$ and $CD=CE$. Circles $(AEF)$ and $(ABC)$ meet again at $S$. Lines $EF$ and $BC$ meet at $P$ and circles $(PDS)$ and $(AEF)$ meet again at $Q$. Prove that, as $D$ varies, isogonal conjugate of $Q$ with respect to triangle $ ABC$ lies on a fixed circle.
[i]Proposed by Serdar Bozdag[/i]
2018 Centroamerican and Caribbean Math Olympiad, 5
Let $n$ be a positive integer, $1<n<2018$. For each $i=1, 2, \ldots ,n$ we define the polynomial $S_i(x)=x^2-2018x+l_i$, where $l_1, l_2, \ldots, l_n$ are distinct positive integers. If the polynomial $S_1(x)+S_2(x)+\cdots+S_n(x)$ has at least an integer root, prove that at least one of the $l_i$ is greater or equal than $2018$.
2007 Croatia Team Selection Test, 5
Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.
1987 AIME Problems, 7
Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$
2011 Akdeniz University MO, 2
Let $O$ is a point in a plane $P$ and let $[OX,[OY,[OZ$ is distinct ray in $P$. Prove that, if $A \in [OX$ , $B \in [OY$ and $C \in [OZ$ points such that $\triangle OAB$ , $\triangle OBC$ and $\triangle OCA$ 's perimeter is 2, there is only one $(A,B,C)$ triple
2000 Harvard-MIT Mathematics Tournament, 6
Prove that every multiple of $3$ can be written as a sum of four cubes (positive or negatives).
2023 Iran Team Selection Test, 2
Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$.
[i]Proposed by Navid Safaei [/i]
ICMC 8, 5
A positive integer is a non-trivial perfect power if it can be expressed as $n^k$ where $n$ and $k$ are positive integers and $k>1$. Show that there exist arbitrarily large consecutive square numbers with no other non-trivial perfect powers between them.
2011 Kyiv Mathematical Festival, 5
Pete claims that he can draw $3$ segments of length $1$ and a circle of radius less than $\sqrt3 / 3$ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $3 $ segments. Is Pete right?
2013 HMNT, 6
Points $A,B,C$ lie on a circle $\omega$ such that $BC$ is a diameter. $AB$ is extended past $B$ to point $B'$ and $AC$ is extended past $C$ to point $C'$ such that line $B'C'$ is parallel to $BC$ and tangent to $\omega$ at point $D$. If $B'D = 4$ and $C'D = 6$, compute $BC$.