Found problems: 85335
2023 VN Math Olympiad For High School Students, Problem 7
Given a polynomial with integer coefficents$$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0,n\ge 1$$satisfying these conditions:
i) $|a_0|$ is not a perfect square.
ii) $P(x)$ is irreducible in $\mathbb{Q}[x].$
Prove that: $P(x^2)$ is irreducible in $\mathbb{Q}[x].$
2000 District Olympiad (Hunedoara), 4
Consider the pyramid $ VABCD, $ where $ V $ is the top and $ ABCD $ is a rectangular base. If $ \angle BVD = \angle AVC, $ then prove that the triangles $ VAC $ and $ VBD $ share the same perimeter and area.
2021 Latvia TST, 2.1
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
2021 MOAA, 13
Determine the greatest power of $2$ that is a factor of $3^{15}+3^{11}+3^{6}+1$.
[i]Proposed by Nathan Xiong[/i]
1989 All Soviet Union Mathematical Olympiad, 492
$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.
1986 AMC 8, 22
Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?
\[ \textbf{(A)} \text{Alan, Beth} \qquad
\textbf{(B)} \text{Beth, Carlos} \qquad
\textbf{(C)} \text{Carlos, Diana} \qquad
\textbf{(D)} \text{Alan, Diana} \qquad
\textbf{(E)} \text{Beth, Diana}
\]
2020 CMIMC Geometry, Estimation
Gunmay picks $6$ points uniformly at random in the unit square. If $p$ is the probability that their convex hull is a hexagon, estimate $p$ in the form $0.abcdef$ where $a,b,c,d,e,f$ are decimal digits. (A [i]convex combination[/i] of points $x_1, x_2, \dots, x_n$ is a point of the form $\alpha_1x_1 + \alpha_2x_2 + \dots + \alpha_nx_n$ with $0 \leq \alpha_i \leq 1$ for all $i$ and $\alpha_1 + \alpha_2 + \dots + \alpha_n = 1$. [i]The convex hull[/i] of a set of points $X$ is the set of all possible convex combinations of all subsets of $X$.)
2010 Saudi Arabia BMO TST, 4
Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$
2012 AMC 10, 8
The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $
2017 Iran MO (3rd round), 1
Find all polynomials $P(x)$ and $Q(x)$ with real coefficients such that
$$P(Q(x))=P(x)^{2017}$$
for all real numbers $x$.
1995 AMC 12/AHSME, 15
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point
[asy]
size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle);
for(int i = 0; i < 5; ++i) { pair P = dir(90+i*72); dot(P); label("$"+string(i+1)+"$",P,1.4*P); }[/asy]
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2007 Postal Coaching, 3
Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.
2007 AMC 10, 16
A teacher gave a test to a class in which $ 10\%$ of the students are juniors and $ 90\%$ are seniors. The average score on the test was $ 84$. The juniors all received the same score, and the average score of the seniors was $ 83$. What score did each of the juniors receive on the test?
$ \textbf{(A)}\ 85 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 93 \qquad \textbf{(D)}\ 94 \qquad \textbf{(E)}\ 98$
2020 CCA Math Bonanza, T2
The base $4$ repeating decimal $0.\overline{12}_4$ can be expressed in the form $\frac{a}{b}$ in base 10, where $a$ and $b$ are relatively prime positive integers. Compute the sum of $a$ and $b$.
[i]2020 CCA Math Bonanza Team Round #2[/i]
2024 CMIMC Geometry, 3
Circles $C_1$, $C_2$, and $C_3$ are inside a rectangle $WXYZ$ such that $C_1$ is tangent to $\overline{WX}$, $\overline{ZW}$, and $\overline{YZ}$; $C_2$ is tangent to $\overline{WX}$ and $\overline{XY}$; and $C_3$ is tangent to $\overline{YZ}$, $C_1$, and $C_2$. If the radii of $C_1$, $C_2$, and $C_3$ are $1$, $\tfrac 12$, and $\tfrac 23$ respectively, compute the area of the triangle formed by the centers of $C_1$, $C_2$, and $C_3$.
[i]Proposed by Connor Gordon[/i]
1994 North Macedonia National Olympiad, 4
$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$
2014 Junior Regional Olympiad - FBH, 5
Let $ABCDEF$ be a hexagon. Sides and diagonals of hexagon are colored in two colors: blue and yellow. Prove that there exist a triangle with vertices from set $\{A,B,C,D,E,F\}$ which sides are all same colour
2009 Jozsef Wildt International Math Competition, W. 8
If $n,p,q \in \mathbb{N}, p<q $ then $${{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}} $$
2018 India IMO Training Camp, 3
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
2021 Purple Comet Problems, 1
The diagram shows two intersecting line segments that form some of the sides of two squares with side lengths $3$ and $6$. Two line segments join vertices of these squares. Find the area of the region enclosed by the squares and segments.
2015 Junior Regional Olympiad - FBH, 2
Seller reduced price of one shirt for $20\%$,and they raised it by $10\%$. Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by $10\%$ from the original price
2005 Tuymaada Olympiad, 8
Let $a,b,c$ be positive reals s.t. $a^2+b^2+c^2=1$. Prove the following inequality \[ \sum \frac{a}{a^3+bc} >3 . \]
[i]Proposed by A. Khrabrov[/i]
2010 Contests, 3
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.
For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.
Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).
Sorry for my bad English.
2005 JBMO Shortlist, 5
Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.
2016 PUMaC Team, 9
Let $\vartriangle ABC$ be a right triangle with $AB = 4, BC = 5$, and hypotenuse $AC$. Let I be the incenter of $\vartriangle ABC$ and $E$ be the excenter of $\vartriangle ABC$ opposite $A$ (the center of the circle tangent to $BC$ and the extensions of segments $AB$ and $AC$). Suppose the circle with diameter $IE$ intersects line $AB$ beyond $B$ at $D$. If $BD =\sqrt{a}- b$, where a and b are positive integers. Find $a + b$.