Olympiad problems

2006 Vietnam National Olympiad, 4

Tags: function , calculus , derivative , limit , algebra unsolved , algebra

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

2021 Korea - Final Round, P5

Tags: geometry , tangent circles , incenter , circumcircle , FKMO

The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$. Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.

2016 Costa Rica - Final Round, N1

Tags: number theory , Diophantine equation , diophantine

Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

MathLinks Contest 6th, 7.1

Tags: algebra , polynomial , 6th edition

Write the following polynomial as a product of irreducible polynomials in $\mathbb{Z}[X]$ \[ f(X) = X^{2005} - 2005 X + 2004 . \]Justify your answer.

2010 Princeton University Math Competition, 7

Tags: trigonometry , algebra , polynomial , symmetry , geometric series

The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.

1967 IMO Shortlist, 6

Tags: geometry , construction , bisector , midpoint , Center , IMO Shortlist , IMO Longlist

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: [b]a)[/b] The bisector of a given angle. [b]b)[/b] The midpoint of a given rectilinear line segment. [b]c)[/b] The center of a circle through three given non-collinear points. [b]d)[/b] A line through a given point parallel to a given line.

2020 AMC 10, 3

Tags: AMC , AMC 12 , 2020 AMC 12B , 2020 AMC , ratio , 2020 AMC 10B , AMC 10

The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$? $\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $

2016 Ukraine Team Selection Test, 10

Tags: algebra , polynomial , calculus , TST

Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$ Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.

2017 India IMO Training Camp, 3

Tags: geometry , IMO Shortlist

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

2013 Harvard-MIT Mathematics Tournament, 10

Tags: geometry

Triangle $ABC$ is inscribed in a circle $\omega$. Let the bisector of angle $A$ meet $\omega$ at $D$ and $BC$ at $E$. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$, respectively. Suppose that $\angle A = 60^o$, $AB = 3$, and $AE = 4$. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of APD' meets line $BC$ at $F$ (other than $P$), compute $FC'$.

2017 Flanders Math Olympiad, 3

Tags: combinatorics

In a closed rectangular neighborhood there are: $S$ streets (these are straight roads of maximum length), $V$ four-arm intersections ( [img]https://cdn.artofproblemsolving.com/attachments/e/4/6a5974a30dc182b59a519a8ef4eb4f1412e05e.png[/img]), $H$ city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and $T$ represents the number of $T$-intersections ([img]https://cdn.artofproblemsolving.com/attachments/0/a/b390a30a0b27d83db681f70f633bdeed697163.png[/img] ). For example, in the neighborhood below, there are $15$ streets, $8$ four-arm intersections, $20$ city blocks and $22$ $T$-intersections. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c1a5e463d0fb5671ac0702c91cfc2272d4e2c3.png[/img] Prove that in each district $S + V = H + 3$.

2022 Yasinsky Geometry Olympiad, 1

Tags: geometry , construction , areas

From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$. (Gryhoriy Filippovskyi)

2012 Today's Calculation Of Integral, 792

Tags: calculus , integration , limit , logarithms , trigonometry , calculus computations

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2002 Turkey Team Selection Test, 3

Tags: inequalities proposed , inequalities

A positive integer $n$ and real numbers $a_1,\dots, a_n$ are given. Show that there exists integers $m$ and $k$ such that \[|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.\]

1986 AIME Problems, 9

Tags: analytic geometry , geometry , AMC , AIME , ratio , parallelogram , symmetry

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

2009 Princeton University Math Competition, 2

Tags:

Find the number of subsets of $\{1,2,\ldots,7\}$ that do not contain two consecutive integers.

2017 Iran MO (3rd round), 1

Tags: geometry , circumcircle , incenter

Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.

CVM 2020, Problem 3

Tags: geometry

In $\triangle ABC$ we consider the points $A',B',C'$ in sides $BC,AC,AB$ such that $$3BA'=CA',~3CB'=AB',~3AC'=BA'$$$\triangle DEF$ is defined by the intersections of $AA',BB',CC'$. If the are of $\triangle ABC$ is $2020$ find the area of $\triangle DEF$. [i]Proposed by Alejandro Madrid, Valle[/i]

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4

Tags:

We have a square grid of 4 times 4 points. How many triangles are there with vertices on the points? (The three vertices may not lie on a line.) A. 252 B. 256 C. 360 D. 516 E. 560

1986 AMC 12/AHSME, 12

Tags: AMC

John scores $93$ on this year's AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. How many questions does he leave unanswered? (In the new scoring system one receives 5 points for correct answers, 0 points for wrong answers, and 2 points for unanswered questions. In the old system, one started with 30 points, received 4 more for each correct answer, lost one point for each wrong answer, and neither gained nor lost points for unanswered questions. There are 30 questions in the 1986 AHSME.) $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ \text{Not uniquely determined} $

2004 Putnam, B4

Tags: Putnam , rotation , analytic geometry , geometry , geometric transformation , college contests , Putnam complex

Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.

2024 Korea Summer Program Practice Test, 1

Tags: algebra , polynomial

Find all polynomials $P$ with integer coefficients such that $P(P(x))-x$ is irreducible over $\mathbb{Z}[x]$.

2008 JBMO Shortlist, 10

Tags: geometry , JBMO

Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$., and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$ , by $X$ the (other than M ) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$ and $\delta$. Prove that the line $XY$ passes through a fixed point.

2025 NEPALTST, 3

Tags: algebra , functional equation

Find all functions $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\] for all positive real numbers $x$ and $y$. [i](Andrew Brahms, USA)[/i]

1953 AMC 12/AHSME, 24

Tags:

If $ a,b,c$ are positive integers less than $ 10$, then $ (10a \plus{} b)(10a \plus{} c) \equal{} 100a(a \plus{} 1) \plus{} bc$ if: $ \textbf{(A)}\ b \plus{} c \equal{} 10 \qquad\textbf{(B)}\ b \equal{} c \qquad\textbf{(C)}\ a \plus{} b \equal{} 10 \qquad\textbf{(D)}\ a \equal{} b \\ \textbf{(E)}\ a \plus{} b \plus{} c \equal{} 10$