Found problems: 85335
2018 PUMaC Team Round, 9
There are numerous sets of $17$ distinct positive integers that sum to $2018$, such that each integer has the same sum of digits in base $10$. Let $M$ be the maximum possible integer that could exist in any such set. Find the sum of $M$ and the number of such sets that contain $M$.
2019 Romanian Masters In Mathematics, 6
Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds:
For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that
\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]
contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).
2001 Stanford Mathematics Tournament, 2
How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?
2020 Malaysia IMONST 2, 1
Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.
1954 AMC 12/AHSME, 29
If the ratio of the legs of a right triangle is $ 1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
$ \textbf{(A)}\ 1: 4 \qquad
\textbf{(B)}\ 1: \sqrt{2} \qquad
\textbf{(C)}\ 1: 2 \qquad
\textbf{(D)}\ 1: \sqrt{5} \qquad
\textbf{(E)}\ 1: 5$
2001 Romania National Olympiad, 1
a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots.
b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.
2010 AIME Problems, 4
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $ \frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $ \frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2018 Ramnicean Hope, 3
Consider two positive real numbers $ a,b $ and the function $ f:(0,\infty )\longrightarrow\left( \sqrt{ab} ,\frac{a+b}{2} \right) $ defined as $ f(x)=-x+\sqrt{x^2+(a+b)x+ab}. $ Prove that it's bijective.
[i]D.M. Bătineți-Giurgiu[/i] and [i]Neculai Stanciu[/i]
1999 National High School Mathematics League, 12
The bottom surface of triangular pyramid $S-ABC$ is a regular triangle. Projection of $A$ on plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. If $H-AB-C=30^{\circ},SA=2\sqrt3$, then the volume of $S-ABC$ is________.
1969 Spain Mathematical Olympiad, 6
Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities:
$$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$
Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time
$-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.
Kyiv City MO Juniors 2003+ geometry, 2011.9.41
The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.
2015 USA TSTST, 3
Let $P$ be the set of all primes, and let $M$ be a non-empty subset of $P$. Suppose that for any non-empty subset ${p_1,p_2,...,p_k}$ of $M$, all prime factors of $p_1p_2...p_k+1$ are also in $M$. Prove that $M=P$.
[i]Proposed by Alex Zhai[/i]
2021 Science ON Juniors, 3
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$.
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[i](Vlad Robu)[/i]
2001 District Olympiad, 4
Prove that:
a) the sequence $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1$ is monotonic.
b) there is a sequence $(a_n)_{n\ge 1}\in \{0,1\}$ such that:
\[\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}\]
[i]Radu Gologan[/i]
2004 Purple Comet Problems, 7
How many positive integers less that $200$ are relatively prime to either $15$ or $24$?
2017 CMIMC Algebra, 5
The set $S$ of positive real numbers $x$ such that
\[ \left\lfloor\frac{2x}{5}\right\rfloor + \left\lfloor\frac{3x}{5}\right\rfloor + 1 = \left\lfloor x\right\rfloor \]
can be written as $S = \bigcup_{j = 1}^{\infty} I_{j}$, where the $I_{i}$ are disjoint intervals of the form $[a_{i}, b_{i}) = \{x \, | \, a_i \leq x < b_i\}$ and $b_{i} \leq a_{i+1}$ for all $i \geq 1$. Find $\sum_{i=1}^{2017} (b_{i} - a_{i})$.
2010 Indonesia TST, 4
Given $3n$ cards, each of them will be written with a number from the following sequence:
$$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$
with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.
2006 MOP Homework, 6
In triangle $ABC, AB \ne AC$. Circle $\omega$ passes through $A$ and meets sides $AB$ and $AC$ at $M$ and $N$, respectively, and the side $BC$ at $P$ and $Q$ such that $Q$ lies in between $B$ and $P$. Suppose that $MP // AC, NQ // AB$, and $BP \cdot AC = CQ \cdot AB$. Find $\angle BAC$.
KoMaL A Problems 2019/2020, A. 766
Let $T$ be any triangle such that its side-lengths $a, b$ and $c$ and its circumradius $R$ are positive integers. Show that:
a) the inradius $r$ of $T$ is a positive integer;
b) the perimeter $P$ of $T$ is a multiple of $4$; and
c) all three of $a, b$ and $c$ are even.
2021 Science ON all problems, 3
Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$.
[i] (From "Radu Păun" contest, Radu Miculescu)[/i]
2016 HMNT, 6
The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
2013 NIMO Problems, 3
Let $a_1, a_2, \dots, a_{1000}$ be positive integers whose sum is $S$. If $a_n!$ divides $n$ for each $n = 1, 2, \dots, 1000$, compute the maximum possible value of $S$.
[i]Proposed by Michael Ren[/i]
2005 ISI B.Math Entrance Exam, 3
Let $ABCD$ be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides $ (AB+CD=AD+BC) $. Prove that the circles inscribed in triangles $ABC$ and $ACD$ are tangent to each other.
2007 Poland - Second Round, 3
$a$, $b$, $c$, $d$ are positive real numbers satisfying the following condition:
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=4\]
Prove that:
\[\sqrt[3]{\frac{a^{3}+b^{3}}{2}}+\sqrt[3]{\frac{b^{3}+c^{3}}{2}}+\sqrt[3]{\frac{c^{3}+d^{3}}{2}}+\sqrt[3]{\frac{d^{3}+a^{3}}{2}}\leq 2(a+b+c+d)-4\]
2023 VN Math Olympiad For High School Students, Problem 3
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$
Assume that there exists a prime number $p$ satisfying these conditions:
[i]i)[/i] $p|a_i$ for all $0\le i<n,$
[i]ii)[/i] $p\nmid a_n,$
[i]iii)[/i] $p^2\nmid a_0.$
Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$