Found problems: 85335
DMM Team Rounds, 2017
[b]p1.[/b] What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros?
[b]p2.[/b] Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$, for all $x, y \in R$. Find the sum of all possible $f(-2017)$.
[b]p3. [/b] What is the sum of prime factors of $1000027$?
[b]p4.[/b] Let $$\frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m},$$ where $n, m$ are relatively prime. Find $(m - n)$.
[b]p5.[/b] Determine the number of ordered pairs of real numbers $(x, y)$ such that $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$
[b]p6.[/b] Triangle $\vartriangle ABC$ has $\angle B = 120^o$, $AB = 1$. Find the largest real number $x$ such that $CA - CB > x$ for all possible triangles $\vartriangle ABC$.
[b]p7. [/b]Jung and Remy are playing a game with an unfair coin. The coin has a probability of $p$ where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round $ 1$. Whoever gets heads first wins the game. Given that Jung has the probability of $8/15$ , what is the value of $p$?
[b]p8.[/b] Consider a circle with $7$ equally spaced points marked on it. Each point is $ 1$ unit distance away from its neighbors and labelled $0,1,2,...,6$ in that order counterclockwise. Feng is to jump around the circle, starting at the point $0$ and making six jumps counterclockwise with distinct lengths $a_1, a_2, ..., a_6$ in a way such that he will land on all other six nonzero points afterwards. Let $s$ denote the maximum value of $a_i$. What is the minimum possible value of $s$?
[b]p9. [/b]Justin has a $4 \times 4 \times 4$ colorless cube that is made of $64$ unit-cubes. He then colors $m$ unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of $m$?
[b]p10.[/b] Yikai wants to know Liang’s secret code which is a $6$-digit integer $x$. Furthermore, let $d(n)$ denote the digital sum of a positive integer $n$. For instance, $d(14) = 5$ and $d(3) = 3$. It is given that $$x + d(x) + d(d(x)) + d(d(d(x))) = 999868.$$ Please find $x$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 Iran MO (3rd Round), 3
For each $k$ , find the least $n$ in terms of $k$ st the following holds:
There exists $n$ real numbers $a_1 , a_2 ,\cdot \cdot \cdot , a_n$ st for each $i$ :
$$0 < a_{i+1} - a_{i} < a_i - a_{i-1}$$
And , there exists $k$ pairs $(i,j)$ st $a_i - a_j = 1$.
1955 AMC 12/AHSME, 17
If $ \log x\minus{}5 \log 3\equal{}\minus{}2$, then $ x$ equals:
$ \textbf{(A)}\ 1.25 \qquad
\textbf{(B)}\ 0.81 \qquad
\textbf{(C)}\ 2.43 \qquad
\textbf{(D)}\ 0.8 \qquad
\textbf{(E)}\ \text{either 0.8 or 1.25}$
2009 Danube Mathematical Competition, 3
Let $n$ be a natural number. Determine the minimal number of equilateral triangles of side $1$ to cover the surface of an equilateral triangle of side $n +\frac{1}{2n}$.
2010 VJIMC, Problem 3
Let $A$ and $B$ be two $n\times n$ matrices with integer entries such that all of the matrices
$$A,\enspace A+B,\enspace A+2B,\enspace A+3B,\enspace\ldots,\enspace A+(2n)B$$are invertible and their inverses have integer entries, too. Show that $A+(2n+1)B$ is also invertible and that its inverse has integer entries.
1994 India Regional Mathematical Olympiad, 2
In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$.
2018 Purple Comet Problems, 8
On side $AE$ of regular pentagon $ABCDE$ there is an equilateral triangle $AEF$, and on side $AB$ of the pentagon there is a square $ABHG$ as shown. Find the degree measure of angle $AFG$.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/0d689d2665e67c9f9afdf193fb0a2db6dddb3d.png[/img]
2010 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Find the largest integer $N$ for which there exists a set of $n$ weights such that it is possible to determine the mass of all bodies with masses of $1, 2, ..., N$ using a balance scale .
(i.e. to determine whether a body with unknown mass has a mass $1, 2, ..., N$, and which namely).
2010 India IMO Training Camp, 11
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
2007 Olympic Revenge, 2
Let $a, b, c \in \mathbb{R}$ with $abc = 1$. Prove that
\[a^{2}+b^{2}+c^{2}+{1\over a^{2}}+{1\over b^{2}}+{1\over c^{2}}+2\left(a+b+c+{1\over a}+{1\over b}+{1\over c}\right) \geq 6+2\left({b\over a}+{c\over b}+{a\over c}+{c\over a}+{c\over b}+{b\over c}\right)\]
1949-56 Chisinau City MO, 48
Calculate $\sin^3 a + \cos^3 a$ if you know that $\sin a+ \cos a = m$.
2021 LMT Spring, B28
Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters.
[i]Proposed by Richard Chen[/i]
Novosibirsk Oral Geo Oly VII, 2019.3
Equal line segments are marked in triangle $ABC$. Find its angles.
[img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]
2009 China Western Mathematical Olympiad, 1
Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.
KoMaL A Problems 2021/2022, A. 813
Let $p$ be a prime number and $k$ be a positive integer. Let \[t=\sum_{i=0}^\infty\bigg\lfloor\frac{k}{p^i}\bigg\rfloor.\]a) Let $f(x)$ be a polynomial of degree $k$ with integer coefficients such that its leading coefficient is $1$ and its constant is divisible by $p.$ prove that there exists $n\in\mathbb{N}$ for which $p\mid f(n),$ but $p^{t+1}\nmid f(n).$
b) Prove that the statement above is sharp, i.e. there exists a polynomial $g(x)$ of degree $k,$ integer coefficients, leading coefficient $1$ and constant divisible by $p$ such that if $p\mid g(n)$ is true for a certain $n\in\mathbb{N},$ then $p^t\mid g(n)$ also holds.
[i]Proposed by Kristóf Szabó, Budapest[/i]
2021 AMC 10 Fall, 7
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$
[asy]
size(6cm);
pair A = (0,10);
label("$A$", A, N);
pair B = (0,0);
label("$B$", B, S);
pair C = (10,0);
label("$C$", C, S);
pair D = (10,10);
label("$D$", D, SW);
pair EE = (15,11.8);
label("$E$", EE, N);
pair F = (3,10);
label("$F$", F, N);
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
dot(A^^B^^C^^D^^EE^^F);
draw(A--B--C--D--cycle);
draw(D--EE--F--cycle);
label("$110^\circ$", (15,9), SW);
[/asy]
$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$
2010 ISI B.Math Entrance Exam, 3
Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .
Denmark (Mohr) - geometry, 2021.4
Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]
1965 German National Olympiad, 1
For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.
2023 BMT, 7
For an integer $n > 0$, let $p(n)$ be the product of the digits of $n$. Compute the sum of all integers $n$ such that $n - p(n) = 52$.
2018 LMT Fall, Individual
[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$.
[b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once.
[b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle.
[b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself.
[b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe?
[b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$?
[b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$.
[b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers.
[b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days.
[b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it.
[b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$.
[b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places.
[b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect.
[b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$.
[b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ .
[b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$.
[b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$?
[b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$.
[b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$.
[b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$.
[b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ .
[b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$.
[b]p23.[/b] Let $a,b,c$ be positive real numbers such that
(i) $c > a$
(ii) $10c = 7a +4b +2024$
(iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$.
Find $a +b +c$.
[b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$.
[b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2008 iTest Tournament of Champions, 4
Let \[f(n) = \sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\frac{1}{n-k}\binom{n-k}k,\] for each positive integer $n$. If $|f(2007) + f(2008)| = a/b$ for relatively prime positive integers $a$ and $b$, find the remainder when $a$ is divded by $1000$.
2021 AMC 12/AHSME Spring, 5
When a student multiplied the number $66$ by the repeating decimal,
$$1. \underline{a} \underline{b} \underline{a} \underline{b} … = 1.\overline{ab},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1. \underline{a} \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$- digit integer $\underline{a} \underline{b}$?
$\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$
2015 HMNT, 5
Kelvin the Frog is trying to hop across a river. The river has $10$ lilypads on it, and he must hop on them in a specific order (the order is unknown to Kelvin). If Kelvin hops to the wrong lilypad at any point, he will be thrown back to the wrong side of the river and will have to start over. Assuming Kelvin is infinitely intelligent, what is the minimum number of hops he will need to guarantee reaching the other side?
2011 Romania National Olympiad, 1
Find all real numbers $x, y,z,t \in [0, \infty)$ so that
$$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$