A point $P$ is given inside an acute-angled triangle $ABC$. Let $A',B',C'$ be the orthogonal projections of $P$ on sides $BC, CA, AB$ respectively. Determine the locus of points $P$ for which $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$

Express $2.3\overline{57}$ as a common fraction. [i]2017 CCA Math Bonanza Lightning Round #3.1[/i]

Determine the set of all real numbers $p$ for which the polynomial $Q(x) = x^3 + px^2 - px - 1$ has three distinct real roots.

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$

Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?

find the smallest integer $n\geq1$ such that the equation : $$a^2+b^2+c^2-nd^2=0 $$ has $(0,0,0,0)$ as unique solution .

Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$.

a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$ We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$ Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$ b) Prove that the product of $2$ original polynomials is a original polynomial.

Consider these two geoboard quadrilaterals. Which of the following statements is true? [asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 5; ++b) { dot((a,b)); } } draw((0,3)--(0,4)--(1,3)--(1,2)--cycle); draw((2,1)--(4,2)--(3,1)--(3,0)--cycle); label("I",(0.4,3),E); label("II",(2.9,1),W); [/asy] $\text{(A)}\ \text{The area of quadrilateral I is more than the area of quadrilateral II.}$ $\text{(B)}\ \text{The area of quadrilateral I is less than the area of quadrilateral II.}$ $\text{(C)}\ \text{The quadrilaterals have the same area and the same perimeter.}$ $\text{(D)}\ \text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$ $\text{(E)}\ \text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$

You have the following pieces: one $4\times 1$ rectangle, two $3\times 1$ rectangles, three $2\times 1$ rectangles, and four $1\times 1$ squares. Ariel and Bernardo play the following game on a board of $n\times n$, where $n$ is a number that Ariel chooses. In each move, Bernardo receives a piece $R$ from Ariel. Next, Bernardo analyzes if he can place $R$ on the board so that it has no points in common with any of the previously placed pieces (not even a common vertex). If there is such a location for $R$, Bernardo must choose one of them and place $R$. The game stops if it is impossible to place $R$ in the way explained, and Bernardo wins. Ariel wins only if all $10$ pieces have been placed on the board. a) Suppose Ariel gives Bernardo the pieces in decreasing order of size. What is the smallest n that guarantees Ariel victory? b) For the $n$ found in a), if Bernardo receives the pieces in increasing order of size, is Ariel guaranteed victory? Note: Each piece must cover exactly a number of unit squares on the board equal to its own size. The sides of the pieces can coincide with parts of the edge of the board.

The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].

Cheryl rolls a fair dice twice. If the dice's six faces are numbered by $1$, $2$, $3$, $4$, $5$, $6$, what is the probability that the number on one of her rolls is a divisor of the number on the other roll? $\textbf{(A) }\dfrac29\qquad\textbf{(B) }\dfrac5{18}\qquad\textbf{(C) }\dfrac49\qquad\textbf{(D) }\dfrac12\qquad\textbf{(E) }\dfrac{11}{18}$

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$? $\textbf{(A) }10\qquad\textbf{(B) }18\qquad\textbf{(C) }25\qquad\textbf{(D) }36\qquad\textbf{(E) }81$

The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .

For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

The inscribed circle $\omega$ of triangle $ABC$ with center $I$ touches the sides $AB, BC, CA$ at points $C_1, A_1, B_1$. The circle circumsrcibed around $\vartriangle AB_1C_1$ intersects the circumscribed circle of $ABC$ for second time at the point $K$. Let $M$ be the midpoint $BC$, $L$ be the midpoint of $B_1C_1$. The circle circumsrcibed around $\vartriangle KA_1M$ cuts intersects $\omega$ for second time at the point $T$. Prove that the circumscribed circles of triangles $KLT$ and $LIM$ are tangent.

A jeweler can get an alloy that is $40\%$ gold for $200$ dollars per ounce, an alloy that is $60\%$ gold for $300$ dollar per ounce, and an alloy that is $90\%$ gold for $400$ dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is $50\%$ gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.

$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.

The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$. I. Voronovich

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]