Found problems: 85335
2019 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
1981 Swedish Mathematical Competition, 3
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
2012 Czech-Polish-Slovak Junior Match, 4
A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.
2000 AMC 10, 24
Let $f$ be a function for which $f\left(\frac x3\right)=x^2+x+1$. Find the sum of all values of $z$ for which $f(3z)=7$.
$\text{(A)}\ -\frac13\qquad\text{(B)}\ -\frac19 \qquad\text{(C)}\ 0 \qquad\text{(D)}\ \frac59 \qquad\text{(E)}\ \frac53$
2003 Iran MO (3rd Round), 5
Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$.
If not, then what is $S$ congruent to $\mod p$ ?
2018 Canadian Open Math Challenge, C1
Source: 2018 Canadian Open Math Challenge Part C Problem 1
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At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center]
$\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$,
[color=transparent](A.)[/color]layer of this pyramid?
$\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers.
[color=transparent](B.)[/color]How many cans are on the bottom layer of the prism?
$\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle.
[color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...)
[color=transparent](C.)[/color]For example, a prism could be composed of the following layers:
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center]
Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.
2019 Brazil Team Selection Test, 4
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.
2023 SG Originals, Q3
Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.
2016 CCA Math Bonanza, I1
Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$
[i]2016 CCA Math Bonanza Individual Round #1[/i]
2006 IberoAmerican Olympiad For University Students, 7
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.
Kyiv City MO Juniors 2003+ geometry, 2017.9.5
Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle.
(Danilo Hilko)
1982 Tournament Of Towns, (031) 5
The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.
2024 Harvard-MIT Mathematics Tournament, 10
Suppose point $P$ is inside quadrilateral $ABCD$ such that
$\angle{PAB} = \angle{PDA}, \angle{PAD} = \angle{PDC}, \angle{PBA} = \angle{PCB}, \angle{PBC} = \angle{PCD}.$
If $PA = 4, PB = 5,$ and $PC = 10$, compute the perimeter of $ABCD$.
2018 Ukraine Team Selection Test, 8
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
2022 MIG, 21
Let $T(p)$ denote the number of right triangles with integer side lengths and one of its side lengths being $p$. Which of the following values of $p$ produces the greatest possible value of $T(p)$ among all five answer choices?
$\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }54$
1999 Moldova Team Selection Test, 16
Define functions $f,g: \mathbb{R}\to \mathbb{R}$, $g$ is injective, satisfy:
\[f(g(x)+y)=g(f(y)+x)\]
1976 Bulgaria National Olympiad, Problem 1
In a circle with a radius of $1$ is an inscribed hexagon (convex). Prove that if the multiple of all diagonals that connects vertices of neighboring sides is equal to $27$ then all angles of hexagon are equals.
[i]V. Petkov, I. Tonov[/i]
1986 AMC 12/AHSME, 6
Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table?
[asy]
size(300);
defaultpen(linewidth(0.8)+fontsize(13pt));
path table = origin--(1,0)--(1,6)--(6,6)--(6,0)--(7,0)--(7,7)--(0,7)--cycle;
path block = origin--(3,0)--(3,1.5)--(0,1.5)--cycle;
path rotblock = origin--(1.5,0)--(1.5,3)--(0,3)--cycle;
draw(table^^shift((14,0))*table);
filldraw(shift((7,0))*block^^shift((5.5,7))*rotblock^^shift((21,0))*rotblock^^shift((18,7))*block,gray);
draw((7.25,1.75)--(8.5,3.5)--(8.5,8)--(7.25,9.75),Arrows(size=5));
draw((21.25,3.25)--(22,3.5)--(22,8)--(21.25,8.25),Arrows(size=5));
unfill((8,5)--(8,6.5)--(9,6.5)--(9,5)--cycle);
unfill((21.5,5)--(21.5,6.5)--(23,6.5)--(23,5)--cycle);
label("$r$",(8.5,5.75));
label("$s$",(22,5.75));
[/asy]
$\textbf{(A) }28\text{ inches}\qquad\textbf{(B) }29\text{ inches}\qquad\textbf{(C) }30\text{ inches}\qquad\textbf{(D) }31\text{ inches}\qquad\textbf{(E) }32\text{ inches}$
2024 Harvard-MIT Mathematics Tournament, 1
Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^2 + rx + s$ has $s$ and $t$ as roots, and the polynomial $x^2 + tx + r$ has $5$ as a root. Compute $s$.
1940 Putnam, A8
A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$.
Show that its area, disregarding sign, is
$$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$
where $\Delta$ is the discriminant of the matrix
$$M=\begin{pmatrix}
a_1 & b_1 &c_1\\
a_2 & b_2 &c_2\\
a_3 & b_3 &c_3
\end{pmatrix}.$$
1987 National High School Mathematics League, 7
$k(k>1)$ is an integer, and $a$ is a solution to the equation $x^2-kx+1=0$. For any integer $n(n>10)$, the last digit number of $a^{2^n}+a^{-2^n}$ is always $7$, then the last digit number of $k$ is________.
2019 Online Math Open Problems, 19
Arianna and Brianna play a game in which they alternate turns writing numbers on a paper. Before the game begins, a referee randomly selects an integer $N$ with $1 \leq N \leq 2019$, such that $i$ has probability $\frac{i}{1 + 2 + \dots + 2019}$ of being chosen. First, Arianna writes $1$ on the paper. On any move thereafter, the player whose turn it is writes $a+1$ or $2a$, where $a$ is any number on the paper, under the conditions that no number is ever written twice and any number written does not exceed $N$. No number is ever erased. The winner is the person who first writes the number $N$. Assuming both Arianna and Brianna play optimally, the probability that Brianna wins can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n.$
[i]Proposed by Edward Wan[/i]
2024 Mongolian Mathematical Olympiad, 3
Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\]
[i]Proposed by Otgonbayar Uuye. [/i]
1995 Putnam, 1
For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi^{\prime}$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi^{\prime}(x) = \pi^{\prime}(y)$.
2022 Nigerian Senior MO Round 2, Problem 2
Let $G$ be the centroid of $\triangle ABC $ and let $D, E $ and $F$ be the midpoints of the line segments $BC, CA $ and $AB$ respectively. Suppose the circumcircle of $\triangle ABC $ meets $AD $ again at $X$, the circumcircle of $\triangle DEF $ meets $BE$ again at $Y$ and the circumcircle of $\triangle DEF $ meets $CF$ again at $Z$. Show that $G, X, Y $ and $Z$ are concyclic.