Found problems: 85335
1970 AMC 12/AHSME, 11
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is
$\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$
2023 Ukraine National Mathematical Olympiad, 8.8
You are given a set of $m$ integers, all of which give distinct remainders modulo some integer $n$. Show that for any integer $k \le m$ you can split this set into $k$ nonempty groups so that the sums of elements in these groups are distinct modulo $n$.
[i]Proposed by Anton Trygub[/i]
2022 Pan-African, 3
Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$. For $k = 1, 2, \dots, 2n$, set $a_{2n + k} = a_k$, and define
$$
A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.
$$
Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$.
1956 AMC 12/AHSME, 32
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time?
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac {1}{2} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7\frac {1}{2} \qquad\textbf{(E)}\ 9$
2021 Alibaba Global Math Competition, 6
Let $M(t)$ be measurable and locally bounded function, that is,
\[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\]
with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that
\[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\]
Show that
\[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]
2020 Dutch BxMO TST, 5
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
2016 China Team Selection Test, 3
Let $n \geq 2$ be a natural. Define
$$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$.
For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define
$$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$
$$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$
Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.
2018 Peru Iberoamerican Team Selection Test, P2
Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.
2019 Kosovo National Mathematical Olympiad, 1
Does there exist a triangle with length $a,b,c$ such that:
[b]a)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$
[b]b)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$
1996 All-Russian Olympiad, 3
Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.
[i]N. Agakhanov, N. Tereshin[/i]
2013 Princeton University Math Competition, 4
Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).
Note: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.
2023 AIME, 2
If $\sqrt{\log_bn}=\log_b\sqrt n$ and $b\log_bn=\log_bbn,$ then the value of $n$ is equal to $\frac jk,$ where $j$ and $k$ are relatively prime. What is $j+k$?
2002 Federal Competition For Advanced Students, Part 1, 3
Let $f(x)=\frac{9^x}{9^x+3}$. Compute $\sum_{k} f \biggl( \frac{k}{2002} \biggr)$, where $k$ goes over all integers $k$ between $0$ and $2002$ which are coprime to $2002$.
2012 Brazil Team Selection Test, 2
To a sheet of paper, we glue $2011$ “handles” that do not intersect, that is, strips of paper glued to the sheet in your ends. No handle can be twisted. Prove that the surface boundary thus formed has at least two cycles (closed curves). That is, an ant that only walks along the edge of the paper never runs through the entire surface boundary.
For example, the configuration represented in the figure has three cycles: one in dashed lines, one in lines dotted lines and another in a continuous line (this cycle passes under a tab twice).
[img]https://cdn.artofproblemsolving.com/attachments/f/e/121146a240215f241278b3aabde13a67544e7a.png[/img]
2023 JBMO Shortlist, G4
Let $ABCD$ be a cyclic quadrilateral, for which $B$ and $C$ are acute angles. $M$ and $N$ are the projections of the vertex $B$ on the lines $AC$ and $AD$, respectively, $P$ and $T$ are the projections of the vertex $D$ on the lines $AB$ and $AC$ respectively, $Q$ and $S$ are the intersections of the pairs of lines $MN$ and $CD$, and $PT$ and $BC$, respectively. Prove the following statements:
a) $NS \parallel PQ \parallel AC$;
b) $NP=SQ$;
c) $NPQS$ is a rectangle if, and only if, $AC$ is a diamteter of the circumscribed circle of quadrilateral $ABCD$.
2004 Iran Team Selection Test, 1
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that:
\[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]
2023 Yasinsky Geometry Olympiad, 4
The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$.
(Oleksii Karliuchenko)
2010 Princeton University Math Competition, 3
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?
2017 Sharygin Geometry Olympiad, 1
Let $ABCD$ be a cyclic quadrilateral with $AB=BC$ and $AD = CD$. A point $M$ lies on the minor arc $CD$ of its circumcircle. The lines $BM$ and $CD$ meet at point $P$, the lines $AM$ and $BD$ meet at point $Q$. Prove that $PQ \parallel AC$.
2017 IFYM, Sozopol, 8
Let $\Delta ABC$ be a scalene triangle with center $I$ of its inscribed circle. Points $A_1$,$B_1$, and $C_1$ are the points of tangency of the same circle with $BC$,$CA$, and $AB$ respectively. Prove that the circumscribed circles of $\Delta AIA_1$,$\Delta BIB_1$, and $\Delta CIC_1$ intersect in a common point, different from $I$.
2006 IMC, 1
Let $f: \mathbb{R}\to \mathbb{R}$ be a real function. Prove or disprove each of the following statements.
(a) If f is continuous and range(f)=$\mathbb{R}$ then f is monotonic
(b) If f is monotonic and range(f)=$\mathbb{R}$ then f is continuous
(c) If f is monotonic and f is continuous then range(f)=$\mathbb{R}$
1997 Cono Sur Olympiad, 3
Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that:
$2a^2 + 3b^2 - 5c^2 = 1997$
2017 Germany, Landesrunde - Grade 11/12, 1
Solve the equation \[ x^5+x^4+x^3+x^2=x+1 \] in $\mathbb{R}$.
2021 MOAA, 18
Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$.
[i]Proposed by Andy Xu[/i]
1964 AMC 12/AHSME, 24
Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum?
$ \textbf{(A)}\ \frac{a+b}{2} \qquad\textbf{(B)}\ a+b \qquad\textbf{(C)}\ \sqrt{ab} \qquad\textbf{(D)}\ \sqrt{\frac{a^2+b^2}{2}}\qquad\textbf{(E)}\ \frac{a+b}{2ab} $