Found problems: 85335
Show how to divide space into
(a) congruent tetrahedra,
(b) congruent “equifaced” tetrahedra.
(A tetrahedron is called equifaced if all its faces are congruent triangles.)
(NB Vassiliev)
What is the largest prime factor of $729-64$?
[i]2016 CCA Math Bonanza Lightning #1.2[/i]
Let $V_0 = \varnothing$ be the empty set and recursively define $V_{n+1}$ to be the set of all $2^{|V_n|}$ subsets of $V_n$ for each $n=0,1,\dots$. For example \[
V_2 = \left\{ \varnothing, \left\{ \varnothing \right\} \right\}
\quad\text{and}\quad
V_3
=
\left\{
\varnothing,
\left\{ \varnothing \right\},
\left\{ \left\{ \varnothing \right\} \right\},
\left\{ \varnothing, \left\{ \varnothing \right\} \right\}
\right\}.
\] A set $x \in V_5$ is called [i]transitive[/i] if each element of $x$ is a subset of $x$. How many such transitive sets are there?
[i]Proposed by Evan Chen[/i]
Given a triangle $ABC$ and a plane $\pi$ having no common points with the triangle, find a point $M$ such that the triangle determined by the points of intersection of the lines $MA,MB,MC$ with $\pi$ is congruent to the triangle $ABC$.
We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$.
a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8;
b) Prove that any prism with 2000 vertices is [i]binary[/i].
Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR = 90^o$, and let $H,K$ be the projections of $Q$ on the lines $PR$ and $PS$, respectively. Prove that the line $HK$ passes through the midpoint of the segment $SQ$.
Let $d$ be a natural number. Given two natural numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if the $d$ numbers obtained substituting each one of the digits of $M$ by the digit of $N$ which is on the same position are all multiples of $7$. Find all the values of $d$ for which the following condition is valid:
For any two numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if $N$ is a friend of $M$.
In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .
For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] \equal{} 6, [\pi] \equal{} 3, [\minus{}1.5] \equal{} \minus{}2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 \plus{} [y]^2 \equal{} 4$.
Jones covered a distance of $ 50$ miles on his first trip. On a later trip he traveled $ 300$ miles while going three times as fast. His new time compared with the old time was:
$ \textbf{(A)}\ \text{three times as much} \qquad\textbf{(B)}\ \text{twice as much} \qquad\textbf{(C)}\ \text{the same}$
$ \textbf{(D)}\ \text{half as much} \qquad\textbf{(E)}\ \text{a third as much}$
Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$
A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken?
$\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
[asy]/* AMC8 2003 #22 Problem */
size(3inch, 2inch);
unitsize(1cm);
pen outline = black+linewidth(1);
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline);
filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline);
filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1));
filldraw(Circle((1,1), 1), white, outline);
filldraw(Circle((3.5,.5), .5), white, outline);
filldraw(Circle((4.5,.5), .5), white, outline);
filldraw(Circle((3.5,1.5), .5), white, outline);
filldraw(Circle((4.5,1.5), .5), white, outline);
filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline);
label("A", (1, 2), N);
label("B", (4, 2), N);
label("C", (7, 2), N);
draw((0,-.5)--(.5,-.5), BeginArrow);
draw((1.5, -.5)--(2, -.5), EndArrow);
label("2 cm", (1, -.5));
draw((3,-.5)--(3.5,-.5), BeginArrow);
draw((4.5, -.5)--(5, -.5), EndArrow);
label("2 cm", (4, -.5));
draw((6,-.5)--(6.5,-.5), BeginArrow);
draw((7.5, -.5)--(8, -.5), EndArrow);
label("2 cm", (7, -.5));
draw((6,1)--(6,-.5), linetype("4 4"));
draw((8,1)--(8,-.5), linetype("4 4"));[/asy]
$ \textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}$
Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.
Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following :
[list]
(a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$
(b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list]
How many polynomials are in $\mathbf{W}?$
Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$, for all $n\ge 1$, where $e$ is the natural logarithm. Prove that for any natural $n$, $x_n\ge 0$.
Suppose that $X$ is a set with $n$ elements and $\mathcal F\subseteq X^{(k)}$ and $X_1,X_2,...,X_s$ is a partition of $X$. We know that for every $A,B\in \mathcal F$ and every $1\le j\le s$, $E=B\cap (\bigcup_{i=1}^{j}X_i)\neq A\cap (\bigcup_{i=1}^{j} X_i)=F$ shows that none of $E,F$ contains the other one. Prove that
\[|\mathcal F|\le \max_{\sum\limits_{i=1}^{S}w_i=k}\prod_{j=1}^{s}\binom{|X_j|}{w_j}\]
(15 points)
Exam time was 5 hours and 20 minutes.
Does there exist a set of $2020$ distinct positive whole numbers with the property that the product of any $101$ of them is divisible by the sum of those $101$ numbers?
Let there be a square $ABCD$. Points $E$ and $F$ are on sides $(BC)$ and $(AB)$ such that $BF=CE$. LInes $AE$ and $CF$ intersect in point $G$. Prove that $EF$ and $DG$ are perpendicular.
Of the following statements, the only one that is incorrect is:
$ \textbf{(A)}$ An inequality will remain true after each side is increased, decreased, multiplied or divided (zero excluded) by the same positive quantity.
$ \textbf{(B)}$ The arithmetic mean of two unequal positive quantities is greater than their geometric mean.
$ \textbf{(C)}$ If the sum of two positive quantities is given, ther product is largest when they are equal.
$ \textbf{(D)}$ If $ a$ and $ b$ are positive and unequal, $ \frac {1}{2}(a^2 \plus{} b^2)$ is greater than $ [\frac {1}{2}(a \plus{} b)]^2$.
$ \textbf{(E)}$ If the product of two positive quantities is given, their sum is greatest when they are equal.
Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.