Found problems: 85335
Let $ABC$ be a triangle, with $AB < AC$, $D$ the foot of the altitude from $A, M$ the midpoint of $BC$, and $B'$ the symmetric of $B$ with respect to $D$. The perpendicular line to $BC$ at $B'$ intersects $AC$ at point $P$ . Prove that if $BP$ and $AM$ are perpendicular then triangle $ABC$ is right-angled.
Liana Topan
Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$
Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.
$ A$ and $ B$ together can do a job in $ 2$ days; $ B$ and $ C$ can do it in four days; and $ A$ and $ C$ in $ 2\frac{2}{5}$ days. The number of days required for $ A$ to do the job alone is:
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 2.8$
With an unmarked ruler only, reconstruct the trapezoid $ABCD$ ($AD \parallel BC$) given the vertices $A$ and $B$, the intersection point $O$ of the diagonals of the trapezoid and the midpoint $M$ of the base $AD$.
(Yukhim Rabinovych)
Given natural numbers $p < k < n$. On an endless checkered plane some cells are marked so that in any rectangle $(k + 1) \times n$ ($n$ cells horizontally, $k + 1$ vertically) marked exactly $p$ cells. Prove that there is a $k \times (n + 1)$ rectangle ($n + 1$ cell horizontally, $k$ - vertically), in which no less than $p + 1$ cells.
Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]
$33$ counters are shown in the left grid below. Choose a counter to start at and remove it from the grid. At each subsequent step, choose a direction (up, down, left, or right), move along the grid line from your current position to the nearest counter in that direction, and remove that counter. You cannot choose a direction that reverses your previous one (e.g., left then right is not allowed). Your goal is to pick up all $33$ counters in a single sequence of steps. When you find the right sequence, write the numbers $1$ to $33$ on the counters so that $N$ is written on the $N$th counter you removed.
A smaller example of a solved grid is shown to the right below. (Note that the final move from $8$ to $9$ is possible because counters $3, 4,$ and $5$ have been removed in earlier steps.) There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
unitsize(0.5cm);
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{"0","0","0","0","0","0","0","0","0","0","-"}};
void drawcircle(int x, int y) {
filldraw(circle((x,y),0.45),white);
}
for(int i = 0; i < 11; ++i) {
draw((i,-0.7)--(i,13.7),dashed);
}
for(int i = 0; i < 14; ++i) {
draw((-0.7,i)--(10.7,i),dashed);
}
drawcircle(10,0);
drawcircle(10,1);
drawcircle(10,2);
drawcircle(10,3);
drawcircle(10,4);
drawcircle(10,5);
drawcircle(9,4);
drawcircle(8,4);
drawcircle(7,4);
drawcircle(6,4);
drawcircle(6,5);
drawcircle(7,6);
drawcircle(8,6);
drawcircle(9,7);
drawcircle(8,8);
drawcircle(7,8);
drawcircle(6,8);
drawcircle(5,8);
drawcircle(4,7);
drawcircle(5,9);
drawcircle(5,10);
drawcircle(4,11);
drawcircle(3,10);
drawcircle(3,9);
drawcircle(3,8);
drawcircle(2,9);
drawcircle(1,9);
drawcircle(0,9);
drawcircle(0,10);
drawcircle(1,11);
drawcircle(2,12);
drawcircle(1,13);
drawcircle(0,13);
for(int k = 0; k<14; ++k){
for(int l = 0; l<11; ++l){
if(x[k][l]!="0"){
label((x[k][l]),(l,-k+13),fontsize(10pt));
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}
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[/asy]
A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$.
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
Let $ABC$ be a triangle, let $I$ be its incenter, let $\Omega$ be its circumcircle, and let $\omega$ be the $A$- mixtilinear incircle. Let $D,E$ and $T$ be the intersections of $\omega$ and $AB,AC$ and $\Omega$, respectively, let the line $IT$ cross $\omega$ again at $P$, and let lines $PD$ and $PE$ cross the line $BC$ at $M$ and $N$ respectively. Prove that points $D,E,M,N$ are concyclic. What is the center of this circle?
A prime number $p$ and a positive integer $n$ are given. Prove that one can colour every one of the numbers $1,2,\ldots,p-1$ using one of the $2n$ colours so that for any $i=2,3,\ldots,n$ the sum of any $i$ numbers of the same colour is not divisible by $p$.
Find all non-negative integers $k,n$ which satisfy $2^{2k+1} + 9\cdot 2^k+5=n^2$.
The lines $a$ and $b$ are parallel and the point $A$ lies on $a$. One chooses one circle $\gamma$ through A tangent to $b$ at $B$. $a$ intersects $\gamma$ for the second time at $T$. The tangent line at $T$ of $\gamma$ is called $t$.
Prove that independently of the choice of $\gamma$, there is a fixed point $P$ such that $BT$ passes through $P$.
Prove that independently of the choice of $\gamma$, there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$.
For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then
$\text{(A)}$ there is no such $p$
$\text{(B)}$ there in only one such $p$
$\text{(C)}$ there is more than one such $p$, but finitely many
$\text{(D)}$ there are infinitely many such $p$
For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit
\[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} \]
exists and determine its value.
Clau writes all four-digit natural numbers where $3$ and $7$ are always together. How many digits does she write in total?
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $
[i]Ion Tecu[/i]
If, for $k=1,2,\dots ,n$, $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$ and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$
Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$
Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore.
Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that
\[
n \geq 2^{n-k-1} - 1.
\]
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.