Found problems: 85335
1997 AMC 8, 11
Let $\boxed{N}$ mean the number of whole number divisors of $N$. For example, $\boxed{3}=2$ because 3 has two divisors, 1 and 3. Find the value of
\[\boxed{\boxed{11}\times\boxed{20}}.\]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 24$
2015 Purple Comet Problems, 18
You have many identical cube-shaped wooden blocks. You have four colors of paint to use, and you paint
each face of each block a solid color so that each block has at least one face painted with each of the four
colors. Find the number of distinguishable ways you could paint the blocks. (Two blocks are
distinguishable if you cannot rotate one block so that it looks identical to the other block.)
2008 Federal Competition For Advanced Students, P1, 3
Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$.
Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.
2016 ASDAN Math Tournament, 4
The radius $r$ of a circle is increasing at a rate of $2$ meters per minute. Find the rate of change, in $\text{meters}^2/\text{minute}$, of the area when $r$ is $6$ meters.
2020 Brazil National Olympiad, 6
Let $k$ be a positive integer. Arnaldo and Bernaldo play a game in a table $2020\times 2020$, initially all the cells are empty. In each round a player chooses a empty cell and put one red token or one blue token, Arnaldo wins if in some moment, there are $k$ consecutive cells in the same row or column with tokens of same color, if all the cells have a token and there aren't $k$ consecutive cells(row or column) with same color, then Bernaldo wins. If the players play alternately and Arnaldo goes first, determine for which values of $k$, Arnaldo has the winning strategy.
1990 India Regional Mathematical Olympiad, 4
Find the remainder when $2^{1990}$ is divided by $1990.$
Russian TST 2019, P3
Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.
1984 Putnam, B4
Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$, positive on $(0,\infty)$, such that for all $x>0$ the $y$-coordinate of the centroid of the region
$$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$is the same as the average value of $g$ on $[0,x]$.
2014 Contests, 3
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.
KoMaL A Problems 2018/2019, A. 750
Let $k_1,k_2,\ldots,k_5$ be five circles in the lane such that $k_1$ and $k_2$ are externally tangent to each other at point $T,$ $k_3$ and $k_4$ are exetrnally tangent to both $k_1$ and $k_2,$ $k_5$ is externally tangent to $k_3$ and $k_4$ at points $U$ and $V,$ respectively, and $k_5$ intersects $k_1$ at $P$ and $Q,$ like shown in the figure. Prove that \[\frac{PU}{QU}\cdot\frac{PV}{QV}=\frac{PT^2}{QT^2}.\]
1994 AMC 8, 2
$\dfrac{1}{10}+\dfrac{2}{10}+\dfrac{3}{10}+\dfrac{4}{10}+\dfrac{5}{10}+\dfrac{6}{10}+\dfrac{7}{10}+\dfrac{8}{10}+\dfrac{9}{10}+\dfrac{55}{10}=$
$\text{(A)}\ 4\dfrac{1}{2} \qquad \text{(B)}\ 6.4 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$
2021 Yasinsky Geometry Olympiad, 3
Prove that in triangle $ABC$, the foot of the altitude $AH$, the point of tangency of the inscribed circle with side $BC$ and projections of point $A$ on the bisectors $\angle B$ and $\angle C$ of the triangle lie on one circle.
(Dmitry Prokopenko)
2016 Chile TST IMO, 4
Let \( f \) and \( g \) be two nonzero polynomials with integer coefficients such that \( \deg(f) > \deg(g) \). Suppose that for infinitely many prime numbers \( p \), the polynomial \( pf + g \) has a rational root. Prove that \( f \) has a rational root.
Clarification: A rational root of a polynomial \( f \) is a number \( q \in \mathbb{Q} \) such that \( f(q) = 0 \).
1966 AMC 12/AHSME, 33
If $ab\ne0$ and $|a|\ne|b|$ the number of distinct values of $x$ satisfying the equation
\[\dfrac{x-a}{b}+\dfrac{x-b}{a}=\dfrac{b}{x-a}+\dfrac{a}{x-b}\]
is:
$\text{(A)}\ \text{zero}\qquad
\text{(B)}\ \text{one}\qquad
\text{(C)}\ \text{two}\qquad
\text{(D)}\ \text{three}\qquad
\text{(E)}\ \text{four}$
2021 JBMO Shortlist, G2
Let $P$ be an interior point of the isosceles triangle $ABC$ with $\hat{A} = 90^{\circ}$. If
$$\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},$$
prove that $AP \perp BC$.
Proposed by [i]Mehmet Akif Yıldız, Turkey[/i]
2011 ELMO Shortlist, 8
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
2000 JBMO ShortLists, 20
Let $ABC$ be a triangle and let $a,b,c$ be the lengths of the sides $BC, CA, AB$ respectively. Consider a triangle $DEF$ with the side lengths $EF=\sqrt{au}$, $FD=\sqrt{bu}$, $DE=\sqrt{cu}$. Prove that $\angle A >\angle B >\angle C$ implies $\angle A >\angle D >\angle E >\angle F >\angle C$.
2021 Junior Balkan Team Selection Tests - Romania, P5
Let $I$ be the incenter of triangle $ABC$. The circle of centre $A$ and radius $AI$ intersects the circumcircle of triangle $ABC$ in $M$ and $N$. Prove that the line $MN$ is tangent to the incircle of triangle $ABC$
2011 HMNT, 1
Five of James’ friends are sitting around a circular table to play a game of Fish. James chooses a place between two of his friends to pull up a chair and sit. Then, the six friends divide themselves into two disjoint teams, with each team consisting of three consecutive players at the table. If the order in which the three members of a team sit does not matter, how many possible (unordered) pairs of teams are possible?
2004 IMO Shortlist, 6
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[
f(x^2+y^2+2f(xy)) = (f(x+y))^2.
\] for all $x,y \in \mathbb{R}$.
2024 PErA, P3
Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$ where we denote $x_0=x_n$ and $x_{n+1}=x_1$.
1967 AMC 12/AHSME, 28
Given the two hypotheses: $\text{I}$ Some Mems are not Ens and $\text{II}$ No Ens are Veens. If "some" means "at least one," we can conclude that:
$\textbf{(A)}\ \text{Some Mems are not Veens}\qquad
\textbf{(B)}\ \text{Some Vees are not Mems}\\
\textbf{(C)}\ \text{No Mem is a Vee}\qquad
\textbf{(D)}\ \text{Some Mems are Vees}\\
\textbf{(E)}\ \text{Neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is deducible from the given statements}$
2005 AMC 10, 14
Equilateral $ \triangle ABC$ has side length $ 2$, $ M$ is the midpoint of $ \overline{AC}$, and $ C$ is the midpoint of $ \overline{BD}$. What is the area of $ \triangle CDM$?
[asy]size(200);defaultpen(linewidth(.8pt)+fontsize(8pt));
pair B = (0,0);
pair A = 2*dir(60);
pair C = (2,0);
pair D = (4,0);
pair M = midpoint(A--C);
label("$A$",A,NW);label("$B$",B,SW);label("$C$",C, SE);label("$M$",M,NE);label("$D$",D,SE);
draw(A--B--C--cycle);
draw(C--D--M--cycle);[/asy]$ \textbf{(A)}\ \frac {\sqrt {2}}{2}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {\sqrt {3}}{2}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ \sqrt {2}$
2015 HMMT Geometry, 6
In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees.
2018 Switzerland - Final Round, 7
Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$.
Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.